American Mathematical Society

Well-posedness of the water-waves equations

By David Lannes

Abstract

We prove that the water-waves equations (i.e., the inviscid Euler equations with free surface) are well-posed locally in time in Sobolev spaces for a fluid layer of finite depth, either in dimension 2 or 3 under a stability condition on the linearized equations. This condition appears naturally as the Lévy condition one has to impose on these nonstricly hyperbolic equations to insure well-posedness; it coincides with the generalized Taylor criterion exhibited in earlier works. Similarly to what happens in infinite depth, we show that this condition always holds for flat bottoms. For uneven bottoms, we prove that it is satisfied provided that a smallness condition on the second fundamental form of the bottom surface evaluated on the initial velocity field is satisfied.

We work here with a formulation of the water-waves equations in terms of the velocity potential at the free surface and of the elevation of the free surface, and in Eulerian variables. This formulation involves a Dirichlet-Neumann operator which we study in detail: sharp tame estimates, symbol, commutators and shape derivatives. This allows us to give a tame estimate on the linearized water-waves equations and to conclude with a Nash-Moser iterative scheme.

1. Introduction

1.1. Presentation of the problem

The water-waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect, incompressible, irrotational fluid under the influence of gravity. In this paper, we restrict our attention to the case when the surface is a graph parameterized by a function zeta left-parenthesis t comma upper X right-parenthesis , where t denotes the time variable and upper X equals left-parenthesis upper X 1 comma ellipsis comma upper X Subscript d Baseline right-parenthesis element-of double-struck upper R Superscript d the horizontal spatial variables. The method developed here works equally well for any integer d greater-than-or-equal-to 1 , but the only physically relevant cases are of course d equals 1 and d equals 2 . The layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time-independent function b left-parenthesis upper X right-parenthesis . We denote by normal upper Omega Subscript t the fluid domain at time t . The incompressibility of the fluids is expressed by

StartLayout 1st Row with Label left-parenthesis 1.1 right-parenthesis EndLabel d i v upper V equals 0 in normal upper Omega Subscript t Baseline comma t greater-than-or-equal-to 0 comma EndLayout

where upper V equals left-parenthesis upper V 1 comma ellipsis comma upper V Subscript d Baseline comma upper V Subscript d plus 1 Baseline right-parenthesis denotes the velocity field ( upper V 1 comma ellipsis comma upper V Subscript d Baseline being the horizontal, and upper V Subscript d plus 1 the vertical components of the velocity). Irrotationality means that

StartLayout 1st Row with Label left-parenthesis 1.2 right-parenthesis EndLabel c u r l upper V equals 0 in normal upper Omega Subscript t Baseline comma t greater-than-or-equal-to 0 period EndLayout

The boundary conditions on the velocity at the surface and at the bottom are given by the usual assumption that they are both bounding surfaces, i.e. surfaces across which no fluid particles are transported. At the bottom, this is given by

StartLayout 1st Row with Label left-parenthesis 1.3 right-parenthesis EndLabel upper V Subscript n Baseline StartAbsoluteValue colon equals bold n Subscript bold minus Baseline dot upper V EndAbsoluteValue Subscript StartSet y equals b left-parenthesis upper X right-parenthesis EndSet Baseline Subscript StartSet y equals b left-parenthesis upper X right-parenthesis EndSet Baseline equals 0 comma for t greater-than-or-equal-to 0 comma upper X element-of double-struck upper R Superscript d Baseline comma EndLayout

where bold n Subscript bold minus Baseline colon equals StartFraction 1 Over StartRoot 1 plus StartAbsoluteValue nabla Subscript upper X Baseline b EndAbsoluteValue squared EndRoot EndFraction left-parenthesis nabla Subscript upper X Baseline b comma negative 1 right-parenthesis Superscript upper T denotes the outward normal vector to the lower boundary of normal upper Omega Subscript t . At the free surface, the boundary condition is kinematic and is given by

StartLayout 1st Row with Label left-parenthesis 1.4 right-parenthesis EndLabel partial-differential Subscript t Baseline zeta minus StartRoot 1 plus StartAbsoluteValue nabla Subscript upper X Baseline zeta EndAbsoluteValue squared EndRoot upper V Subscript n Baseline vertical-bar Subscript StartSet y equals zeta left-parenthesis upper X right-parenthesis EndSet Baseline equals 0 comma for t greater-than-or-equal-to 0 comma upper X element-of double-struck upper R Superscript d Baseline comma EndLayout

where upper V Subscript n Baseline StartAbsoluteValue colon equals bold n Subscript bold plus Baseline dot upper V EndAbsoluteValue Subscript StartSet y equals zeta left-parenthesis upper X right-parenthesis EndSet Baseline Subscript StartSet y equals zeta left-parenthesis upper X right-parenthesis EndSet , with bold n Subscript bold plus Baseline colon equals StartFraction 1 Over StartRoot 1 plus StartAbsoluteValue nabla Subscript upper X Baseline zeta EndAbsoluteValue squared EndRoot EndFraction left-parenthesis minus nabla Subscript upper X Baseline zeta comma 1 right-parenthesis Superscript upper T denoting the outward normal vector to the free surface.

Neglecting the effects of surface tension yields that the pressure upper P is constant at the interface. Up to a renormalization, we can assume that

StartLayout 1st Row with Label left-parenthesis 1.5 right-parenthesis EndLabel upper P vertical-bar Subscript StartSet y equals zeta left-parenthesis upper X right-parenthesis EndSet Baseline equals 0 for t greater-than-or-equal-to 0 comma upper X element-of double-struck upper R Superscript d Baseline period EndLayout

Finally, the set of equations is closed with Euler’s equation within the fluid,

StartLayout 1st Row with Label left-parenthesis 1.6 right-parenthesis EndLabel partial-differential Subscript t Baseline upper V plus upper V dot nabla Subscript upper X comma y Baseline upper V equals minus g e Subscript d plus 1 Baseline minus nabla Subscript upper X comma y Baseline upper P in normal upper Omega Subscript t Baseline comma t greater-than-or-equal-to 0 comma EndLayout

where minus g e Subscript d plus 1 is the acceleration of gravity.

Early works on the well-posedness of Eqs. (Equation1.1)-(Equation1.6) within a Sobolev class go back to Nalimov Reference27, Yosihara Reference38 and Craig Reference10, as far as 1D-surface waves are concerned. All these authors work in a Lagrangian framework, which allows one to consider surface waves which are not graphs, and rely heavily on the fact that the fluid domain is two dimensional. In this case, complex coordinates are canonically associated to the double-struck upper R squared -coordinates, and the incompressibility and irrotationality conditions (Equation1.1) and (Equation1.2) can be seen as the Cauchy-Riemann equations for the complex mapping upper V 1 minus i upper V 2 . There is therefore a singular integral operator on the top surface recovering boundary values of upper V 2 from boundary values of upper V 1 . The water-waves equations (Equation1.1)-(Equation1.6) can then be reduced to a set of two nonlinear evolution equations, which can be “quasi-linearized” using a subtle cancellation property noticed by Nalimov. It seems that this cancellation property was the main reason why the Lagrangian framework was used. A major restriction of these works is that they only address the case of small perturbations of still water. The reasons for this restriction are quite technical, but the most fundamental is that this smallness assumption ensures that a generalized Taylor criterion is satisfied, thus preventing formation of Taylor instabilities (see Reference33Reference4 and the introduction of Reference36). Physically speaking, this criterion assumes that the surface is not accelerating into the fluid region more rapidly than the normal acceleration of gravity. From a mathematical viewpoint, this condition is crucial because the quasilinear system thus obtained is not strictly hyperbolic (zero is a multiple eigenvalue with a Jordan block) and requires a Lévy condition on the subprincipal symbol to be well-posed; one can see Taylor’s criterion precisely as such a Lévy condition (see Section 4.1 below). In Reference3, Beale et al. proved that the linearization of the water-waves equations around a presumed solution is well-posed, provided this exact solution satisfies the generalized Taylor’s sign condition (which is a weaker assumption than the smallness conditions of Reference27Reference38Reference10). Wu’s major breakthrough was to prove in Reference36 that Taylor’s criterion always holds for solutions of the water-waves equations, as soon as the surface is nonself-intersect. Her energy estimates are also better than those of Reference3 and allow her to solve the full (nonlinear) water-waves equations, locally in time, and without restriction (other than smoothness) on the initial data, but in the case of a layer of fluid of infinite depth. The only existing theorems dealing with the case of finite depth require smallness conditions on the initial data when the bottom is flat Reference10, and an additional smallness condition on the variations of the bottom parameterization b when the bottom is uneven Reference38.

Very few papers deal with the well-posedness of the water-waves equations in Sobolev spaces in the three-dimensional setting (i.e. for a 2D surface). In Reference22, the generalization of the results of Reference3 to the three-dimensional setting is proved. More precisely, the authors show, in the case of a fluid layer of infinite depth, that the linearization of the water-waves equations around a presumed solution is well-posed, provided this exact solution satisfies the generalized Taylor’s sign condition. As in Reference3, the energy estimates provided are not good enough to allow the resolution of the nonlinear water-waves equations by an iterative scheme. In Reference37, S. Wu (still in the case of a fluid layer of infinite depth) solved the nonlinear equations. Her proof relies heavily on Clifford analysis in order to extend to the 3D case (some of) the results provided by harmonic analysis in 2D. In the case of finite depth, no results exist.

1.2. Presentation of the results

In this paper, we deliberately chose to work in the Eulerian (rather than Lagrangian) setting, since it is the easiest to handle, especially when asymptotic properties of the solutions are concerned. Inspired by Reference29Reference13 we use an alternate formulation of the water-waves equation (Equation1.1)-(Equation1.6). From the incompressibility and irrotationality assumptions (Equation1.1) and (Equation1.2), there exists a potential flow phi such that upper V equals nabla Subscript upper X comma y Baseline phi and

StartLayout 1st Row with Label left-parenthesis 1.7 right-parenthesis EndLabel normal upper Delta Subscript upper X comma y Baseline phi equals 0 in normal upper Omega Subscript t Baseline comma t greater-than-or-equal-to 0 semicolon EndLayout

the boundary conditions (Equation1.3) and (Equation1.4) can also be expressed in terms of phi :

StartLayout 1st Row with Label left-parenthesis 1.8 right-parenthesis EndLabel partial-differential Subscript bold n Sub Subscript bold minus Subscript Baseline phi vertical-bar Subscript StartSet y equals b left-parenthesis upper X right-parenthesis EndSet Baseline equals 0 comma for t greater-than-or-equal-to 0 comma upper X element-of double-struck upper R Superscript d Baseline comma EndLayout

and

StartLayout 1st Row with Label left-parenthesis 1.9 right-parenthesis EndLabel partial-differential Subscript t Baseline zeta minus StartRoot 1 plus StartAbsoluteValue nabla Subscript upper X Baseline zeta EndAbsoluteValue squared EndRoot partial-differential Subscript bold n Sub Subscript bold plus Subscript Baseline phi vertical-bar Subscript StartSet y equals zeta left-parenthesis t comma upper X right-parenthesis EndSet Baseline equals 0 comma for t greater-than-or-equal-to 0 comma upper X element-of double-struck upper R Superscript d Baseline comma EndLayout

where we used the notation partial-differential Subscript bold n Sub Subscript bold minus Baseline colon equals bold n Subscript bold minus Baseline dot nabla Subscript upper X comma y and partial-differential Subscript bold n Sub Subscript bold plus Baseline equals bold n Subscript bold plus Baseline dot nabla Subscript upper X comma y . Finally, Euler’s equation (Equation1.6) can be put into Bernouilli’s form

StartLayout 1st Row with Label left-parenthesis 1.10 right-parenthesis EndLabel partial-differential Subscript t Baseline phi plus one-half StartAbsoluteValue nabla Subscript upper X comma y Baseline phi EndAbsoluteValue squared plus g y equals negative upper P in normal upper Omega Subscript t Baseline comma t greater-than-or-equal-to 0 period EndLayout

As in Reference13, we reduce the system (Equation1.7)-(Equation1.10) to a system where all the functions are evaluated at the free surface only. For this purpose, we introduce the trace of the velocity potential phi at the surface

psi left-parenthesis t comma upper X right-parenthesis colon equals phi left-parenthesis t comma upper X comma zeta left-parenthesis t comma upper X right-parenthesis right-parenthesis comma

and the (rescaled) Dirichlet-Neumann operator upper G left-parenthesis zeta comma b right-parenthesis (or simply upper G left-parenthesis zeta right-parenthesis when no confusion can be made on the dependence on the bottom parameterization b ), which is a linear operator defined as

upper G left-parenthesis zeta right-parenthesis psi colon equals StartRoot 1 plus StartAbsoluteValue nabla Subscript upper X Baseline zeta EndAbsoluteValue squared EndRoot partial-differential Subscript bold n Sub Subscript bold plus Subscript Baseline phi vertical-bar Subscript StartSet y equals zeta left-parenthesis t comma upper X right-parenthesis EndSet Baseline period

Taking the trace of (Equation1.10) on the free surface and using the chain rule shows that (Equation1.7)-(Equation1.10) are equivalent to the system

StartLayout 1st Row with Label left-parenthesis 1.11 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row partial-differential Subscript t Baseline zeta minus upper G left-parenthesis zeta right-parenthesis psi equals 0 comma 2nd Row partial-differential Subscript t Baseline psi plus g zeta plus one-half StartAbsoluteValue nabla Subscript upper X Baseline psi EndAbsoluteValue squared minus StartFraction 1 Over 2 left-parenthesis 1 plus StartAbsoluteValue nabla Subscript upper X Baseline zeta EndAbsoluteValue squared right-parenthesis EndFraction left-parenthesis upper G left-parenthesis zeta right-parenthesis psi plus nabla Subscript upper X Baseline zeta dot nabla Subscript upper X Baseline psi right-parenthesis squared equals 0 comma EndLayout EndLayout

which is an evolution equation for the elevation of the free surface zeta left-parenthesis t comma upper X right-parenthesis and the trace of the velocity potential on the free surface psi left-parenthesis t comma upper X right-parenthesis . Our results in this paper are given for this system.

The first part of this work consists in developing simple tools in order to make the proof of the well-posedness of the water-waves equations as simple as possible. It is quite obvious from the equations (Equation1.11) that the Dirichlet-Neumann operator will play a central role in the proof; we give here a self-contained and quite elementary proof of the properties of the Dirichlet-Neumann operator that we shall need. A major difficulty lies in the dependence on zeta of the operator upper G left-parenthesis zeta right-parenthesis dot . It is known that such operators depend analytically on the parameterization of the surface. Coifman and Meyer Reference9 considered small Lipschitz perturbations of a line or plane, and Craig et al. Reference12Reference13 upper C Superscript 1 perturbations of hyperplanes in any dimension. Seen as an operator acting on Sobolev spaces, upper G left-parenthesis zeta right-parenthesis dot is of order one. In Reference13, an estimate of its operator norm is given in the form:

StartLayout 1st Row with Label left-parenthesis 1.12 right-parenthesis EndLabel StartAbsoluteValue upper G left-parenthesis zeta right-parenthesis psi EndAbsoluteValue Subscript upper H Sub Superscript k Subscript Baseline less-than-or-equal-to upper C left-parenthesis k comma StartAbsoluteValue zeta EndAbsoluteValue Subscript upper C Sub Superscript 1 Subscript Baseline right-parenthesis left-parenthesis StartAbsoluteValue zeta EndAbsoluteValue Subscript upper C Sub Superscript k plus 1 Subscript Baseline StartAbsoluteValue psi EndAbsoluteValue Subscript upper H Sub Superscript 1 Subscript Baseline plus StartAbsoluteValue psi EndAbsoluteValue Subscript upper H Sub Superscript k plus 1 Subscript Baseline right-parenthesis comma EndLayout

for all integer k greater-than-or-equal-to 0 (estimates in upper L Superscript q -based Sobolev spaces are also provided). In order to obtain this estimate, the authors give an expression of upper G left-parenthesis zeta right-parenthesis dot as a singular integral operator (inspired by the early works of Garabedian and Schiffer Reference17 and Coifman and Meyer Reference9 on Cauchy integrals) and use a multiple commutator estimate of Christ and Journé Reference6. Estimate (Equation1.12) has the interest of being “tame” (in the sense of Hamilton Reference21; i.e., the control in the norms depending on the regularity index k is linear), but is only proved for flat bottoms and requires too much smoothness on zeta : a control of StartAbsoluteValue zeta EndAbsoluteValue Subscript upper C Sub Superscript k plus 1 is needed in (Equation1.12), and hence of StartAbsoluteValue zeta EndAbsoluteValue Subscript upper H Sub Superscript s , with s greater-than d slash 2 plus k plus 1 , if one works in a Sobolev framework. A rapid look at equations (Equation1.11) shows that one would like to allow only a control of zeta in upper H Superscript k plus 1 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis (i.e., zeta and psi should have the same regularity). Using an expression of upper G left-parenthesis zeta right-parenthesis dot involving tools of Clifford Algebras Reference18 and deep results of Coifman, McIntosh and Meyer Reference8 and Coifman, David and Meyer Reference7, S. Wu obtained in Reference37 another estimate with a sharp dependence on the smoothness of zeta :

StartLayout 1st Row with Label left-parenthesis 1.13 right-parenthesis EndLabel StartAbsoluteValue upper G left-parenthesis zeta right-parenthesis psi EndAbsoluteValue Subscript upper H Sub Superscript s Subscript Baseline less-than-or-equal-to upper C left-parenthesis s comma StartAbsoluteValue zeta EndAbsoluteValue Subscript upper H Sub Superscript s plus 1 Subscript Baseline right-parenthesis StartAbsoluteValue psi EndAbsoluteValue Subscript upper H Sub Superscript s plus 1 Subscript Baseline comma EndLayout

for all real numbers s large enough. If estimate (Equation1.13) is obviously better than (Equation1.12), it has two drawbacks. First, it is not tame, and hence not compatible for later use in a Nash-Moser convergence scheme. Second, its proof requires very deep results, which make its generalization to the present case of finite and uneven bottom highly nontrivial. In this paper, we prove in Theorem 3.6 the following estimate:

StartLayout 1st Row with Label left-parenthesis 1.14 right-parenthesis EndLabel StartAbsoluteValue upper G left-parenthesis zeta right-parenthesis psi EndAbsoluteValue Subscript upper H Sub Superscript k plus 1 slash 2 Subscript Baseline less-than-or-equal-to upper C left-parenthesis k comma StartAbsoluteValue zeta EndAbsoluteValue Subscript upper H Sub Superscript s 0 Subscript Baseline right-parenthesis left-parenthesis StartAbsoluteValue zeta EndAbsoluteValue Subscript upper H Sub Superscript k plus 3 slash 2 Subscript Baseline StartAbsoluteValue nabla Subscript upper X Baseline psi EndAbsoluteValue Subscript upper H Sub Superscript s 0 minus 1 Subscript Baseline plus StartAbsoluteValue nabla Subscript upper X Baseline psi EndAbsoluteValue Subscript upper H Sub Superscript k plus 1 slash 2 Subscript Baseline right-parenthesis comma EndLayout

for all k element-of double-struck upper N , and where s 0 is a fixed positive real number. This estimate has the sharp dependence on zeta of (Equation1.13) and is tame as (Equation1.12). Moreover, it is sharper than the above estimates in the sense that only the gradient of psi is involved; this will prove very useful here. Estimate (Equation1.14) also holds for uneven bottoms and its proof uses only elementary tools of PDE: since the fluid layer is diffeomorphic to the flat strip script upper S colon equals double-struck upper R Superscript d times left-parenthesis negative 1 comma 0 right-parenthesis , we first transform the Laplace equation (Equation1.7) with Dirichlet condition phi equals psi at the surface and homogeneous Neumann condition partial-differential Subscript bold n Sub Subscript bold minus Baseline phi equals 0 at the bottom into an elliptic boundary value problem (BVP) with variable coefficients defined in the flat strip script upper S . The Dirichlet-Neumann operator upper G left-parenthesis zeta right-parenthesis dot can be expressed in terms of the solution to this new BVP (see Prop. 3.4). We give sharp tame estimates for a wide class of such elliptic problems in Theorem 2.9. Choosing the most simple diffeomorphism between the fluid domain and script upper S as in Reference12Reference2 and applying Theorem 2.9 to the elliptic problem thus obtained, we can obtain, via Prop. 3.4, a tame estimate on upper G left-parenthesis zeta right-parenthesis dot . However, this estimate is not sharp since instead of StartAbsoluteValue zeta EndAbsoluteValue Subscript upper H Sub Superscript k plus 3 slash 2 as in (Equation1.14), one would need a control of StartAbsoluteValue zeta EndAbsoluteValue Subscript upper H Sub Superscript k plus 2 . We must therefore gain half a derivative more to obtain (Equation1.14). The trick consists in proving (see Prop. 2.13) that there exists a “regularizing” diffeomorphism between the fluid domain and the flat strip script upper S .

We also need further information on the Dirichlet-Neumann operator. In Theorem 3.10, we give the principal symbol of upper G left-parenthesis zeta right-parenthesis dot : for all f element-of upper H Superscript 1 slash 2 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis ,

StartAbsoluteValue left-parenthesis upper G left-parenthesis zeta right-parenthesis minus g Subscript zeta Baseline left-parenthesis upper X comma upper D right-parenthesis right-parenthesis f EndAbsoluteValue Subscript upper H Sub Superscript j slash 2 Subscript Baseline less-than-or-equal-to upper C s t StartAbsoluteValue f EndAbsoluteValue Subscript upper H Sub Superscript j slash 2 Subscript Baseline comma j equals negative 1 comma 0 comma 1 comma

where g Subscript zeta Baseline left-parenthesis upper X comma xi right-parenthesis colon equals StartRoot StartAbsoluteValue xi EndAbsoluteValue squared plus StartAbsoluteValue nabla Subscript upper X Baseline zeta EndAbsoluteValue squared StartAbsoluteValue xi EndAbsoluteValue squared minus left-parenthesis nabla Subscript upper X Baseline zeta dot xi right-parenthesis squared EndRoot , and where the constant involves the upper L Superscript normal infinity -norm of a finite number of derivatives of zeta . Note in particular that for 1D surfaces, g Subscript zeta Baseline left-parenthesis upper X comma upper D right-parenthesis equals StartAbsoluteValue upper D EndAbsoluteValue , while for 2D surfaces it is a pseudo-differential operator (and not a simple Fourier multiplier). We then give tame estimates of the commutator of upper G left-parenthesis zeta right-parenthesis dot with spatial (in Prop. 3.15) and time (in Prop. 3.19) derivatives. Finally, we give in Theorem 3.20 an explicit expression of the shape derivative of upper G left-parenthesis zeta right-parenthesis dot , i.e. the derivative of the mapping zeta right-arrow from bar upper G left-parenthesis zeta right-parenthesis dot , and tame estimates of this and higher derivatives are provided in Prop. 3.25.

Note that all the above results are proved for a general constant coefficient elliptic operator minus nabla Subscript upper X comma y Baseline dot upper P nabla Subscript upper X comma y Baseline phi equals 0 instead of minus normal upper Delta Subscript upper X comma y in (Equation1.7). This is useful if one wants to work with nondimensionalized equations. This first set of results consists therefore in preliminary tools for the study of the water-waves problem; we would like to stress the fact that they are sharp and only use the classical tools of PDE.

We then turn to investigate the water-waves equations (Equation1.11). The first step consists of course in solving the linearization of (Equation1.11) around some reference state upper U underbar equals left-parenthesis zeta underbar comma psi underbar right-parenthesis , and in giving energy estimates on the solution. Using the explicit expression of the shape derivative of the Dirichlet-Neumann operator given in Theorem 3.20, we can give an explicit expression of the linearized operator script upper L underbar . Having the previous works on the water-waves equations in mind, it is not surprising to find that script upper L underbar is hyperbolic, but that its principal symbol has an eigenvalue of multiplicity two (i.e., it is not strictly hyperbolic). In the works quoted in the previous section, this double eigenvalue is zero. Due to the fact that we work here in Eulerian, as opposed to Lagrangian, variables, this double eigenvalue is not zero anymore, but i bold v underbar dot xi , xi being the dual variable of upper X , and bold v underbar being the horizontal component of the velocity at the surface of the reference state upper U underbar . It is natural to seek a linear change of unknowns which transforms the principal part of script upper L underbar into its canonical expression consisting of an upper triangular 2 times 2 matrix with double eigenvalue i bold v underbar dot xi and a Jordan block. Prop. 4.2 gives a striking result: this a priori pseudo-differential change of unknown is not even differential, and the commutator terms involving the Dirichlet-Neumann operator that should appear in the lower-order terms all vanish! This simplifies greatly the sequel.

Having transformed the linearized operator script upper L underbar into an operator script upper M underbar whose principal part exhibits the Jordan block structure inherent to the water-waves equations, we turn to study this operator script upper M underbar . The Lévy condition needed on the subprincipal symbol of script upper M underbar in order for the associated Cauchy problem to be well-posed is quite natural, due to the peculiar structure of upper M underbar : a certain function German a underbar depending only on the reference state upper U underbar must satisfy German a underbar greater-than-or-equal-to c 0 greater-than 0 for some positive constant c 0 (this is almost a necessary condition, since the linearized water-waves equations would be ill-posed if one had German a underbar less-than 0 ). It appears in Prop. 4.4 that this sign condition is exactly the generalized Taylor’s sign condition of Reference3Reference22Reference36Reference37. Assuming for the moment that this condition holds, we use the tools developed in the first sections to show, in Prop. 4.5, that the Cauchy problem associated to script upper M underbar is well-posed in Sobolev spaces, and to give energy estimates on the solution. There is a classical loss of information of half a derivative on this solution due to the Jordan block structure, but also a more dramatic loss of information with respect to the reference state upper U underbar , which makes a Picard iterative scheme inefficient for solving the nonlinear equation. Fortunately, the energy estimates given in Prop. 4.5 are tame, and Nash-Moser theory will provide a good iterative scheme. Inverting the change of unknown of Prop. 4.2, tame estimates are deduced in Prop. 4.14 for the solution of the Cauchy problem associated to the linearized operator script upper L underbar . The last step of the proof consists in solving the nonlinear equations (Equation1.11) via a Nash-Moser iterative scheme. This requires proving that Taylor’s sign condition German a underbar greater-than-or-equal-to c 0 greater-than 0 holds at each step of the scheme (and of course that the surface elevation zeta minus b remains positive!). It is quite easy to see that it is sufficient for this condition to be satisfied that the first iterate satisfies it. Wu proved that this is always the case in infinite depth. We prove in Prop. 4.15 that this result remains true in the case of flat bottoms. For uneven bottoms, however, we must assume that the generalized Taylor’s sign condition holds for the initial data. This can be ensured by smallness conditions on the initial data, but we also give a sufficient condition stating that Taylor’s sign condition can be satisfied for initial data of arbitrary size provided that the bottom is “slowly variable” in the sense that

upper I upper I Subscript b Baseline left-parenthesis upper V Subscript 0 tau Baseline comma upper V Subscript 0 tau Baseline right-parenthesis less-than-or-equal-to StartFraction g Over StartRoot 1 plus StartAbsoluteValue nabla Subscript upper X Baseline b EndAbsoluteValue squared EndRoot EndFraction comma

where b is the bottom parameterization, upper I upper I Subscript b the second fundamental form associated to the surface StartSet left-parenthesis upper X comma y right-parenthesis element-of double-struck upper R Superscript d plus 1 Baseline comma y equals b left-parenthesis upper X right-parenthesis EndSet , and upper V Subscript 0 tau the tangential component of the initial velocity field upper V 0 evaluated at the bottom.

Our final result is then given in Theorem 5.3. For flat bottoms (i.e. b left-parenthesis upper X right-parenthesis equals b equals upper C s t less-than 0 ), it can be stated as:

Theorem 1.1

Let zeta 0 element-of upper H Superscript s plus 1 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis and psi 0 be such that nabla Subscript upper X Baseline psi 0 element-of upper H Superscript s Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis Superscript d , with s greater-than upper M ( upper M depending only on d ). Assume moreover that

zeta 0 minus b greater-than-or-equal-to 2 h 0 on double-struck upper R Superscript d Baseline for some h 0 greater-than 0 period

Then there exists upper T greater-than 0 and a unique solution left-parenthesis zeta comma psi right-parenthesis to the water-waves equations Equation1.11 with initial conditions left-parenthesis zeta 0 comma psi 0 right-parenthesis and such that left-parenthesis zeta comma psi minus psi 0 right-parenthesis element-of upper C Superscript 1 Baseline left-parenthesis left-bracket 0 comma upper T right-bracket comma upper H Superscript s Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis times upper H Superscript s Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis right-parenthesis .

Organization of the paper. Section 2 is devoted to the study of the Laplace equation (Equation1.7) in the fluid domain, or more precisely to the equation minus nabla Subscript upper X comma y Baseline dot upper P nabla Subscript upper X comma y Baseline phi equals 0 , where upper P is a constant coefficient, symmetric and coercive left-parenthesis d plus 1 right-parenthesis times left-parenthesis d plus 1 right-parenthesis matrix. In Section 2.1, we show that this equation can be reduced to an elliptic boundary problem with variable coefficients on a flat strip, and sharp tame elliptic estimates for such problems are given in Section 2.2. We then show in Section 2.3 that among the various diffeomorphisms between the fluid domain and the flat strip, there are some that are particularly interesting, which we call “regularizing diffeomorphisms” and which allow the gain of half a derivative with respect to the regularity of the surface parameterization.

Section 3 is entirely devoted to the properties of the Dirichlet-Neumann operator. Basic properties (including the sharp estimate (Equation1.14) mentioned above) are gathered in Section 3.1. In Section 3.2, we are concerned with the derivation of the principal part of the Dirichlet-Neumann operator, and in Section 3.3 with its commutator properties with space or time derivatives. Finally its shape derivatives are studied in Section 3.4.

The linearized water-waves equations are the object of Section 4. We first show in Section 4.1 that the linearized equations can be made trigonal and prove in Section 4.2 that the Cauchy problem associated to the trigonal operator is well-posed in Sobolev spaces, assuming that a Lévy condition on the subprincipal symbol holds. We also provide in this section tame estimates on the solution. The link with the solution of the original linearized water-waves equations is made in Section 4.3, and the Lévy condition is discussed in Section 4.4.

The fully nonlinear water-waves equations are solved in Section 5. A simple Nash-Moser implicit function theorem is first recalled in Section 5.1 and then used in Section 5.2 to obtain our final well-posedness result.

Finally, a technical proof needed in Section 2.1 has been postponed to Appendix A.

1.3. Notation

Here is a set of notation we shall use throughout this paper:

upper C s t always denotes a numerical constant which may change from one line to another. If the constant depends on some parameters lamda 1 comma lamda 2 comma period period period , we denote it by upper C left-parenthesis lamda 1 comma lamda 2 comma period period period right-parenthesis .

For any alpha equals left-parenthesis alpha 1 comma ellipsis comma alpha Subscript d plus 1 Baseline right-parenthesis element-of double-struck upper N Superscript d plus 1 , we write StartAbsoluteValue alpha EndAbsoluteValue equals alpha 1 plus dot dot dot plus alpha Subscript d plus 1 .

For all i equals 1 comma ellipsis comma d , we write partial-differential Subscript i Baseline equals partial-differential Subscript upper X Sub Subscript i ; similarly, we write partial-differential Subscript d plus 1 Baseline equals partial-differential Subscript y , and, for all alpha element-of double-struck upper N Superscript d plus 1 , partial-differential Superscript alpha Baseline equals partial-differential Subscript 1 Superscript alpha 1 Baseline period period period partial-differential Subscript d plus 1 Superscript alpha Super Subscript d plus 1 .

We denote by upper C Subscript b Superscript k Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis the set of functions continuous and bounded on double-struck upper R Superscript d together with their derivatives of order less than or equal to k , endowed with its canonical norm StartAbsoluteValue dot EndAbsoluteValue Subscript k comma normal infinity Baseline equals sigma-summation Underscript StartAbsoluteValue alpha EndAbsoluteValue less-than-or-equal-to k Endscripts StartAbsoluteValue partial-differential Superscript alpha Baseline dot EndAbsoluteValue Subscript upper L Sub Superscript normal infinity . We denote also upper C Subscript b Superscript normal infinity Baseline equals intersection Underscript k Endscripts upper C Subscript b Superscript k .

We denote by left-parenthesis dot comma dot right-parenthesis the usual scalar product on upper L squared left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis .

We denote by normal upper Lamda equals normal upper Lamda left-parenthesis upper D right-parenthesis , or mathematical left-angle upper D mathematical right-angle , the Fourier multiplier with symbol normal upper Lamda left-parenthesis xi right-parenthesis equals mathematical left-angle xi mathematical right-angle equals left-parenthesis 1 plus StartAbsoluteValue xi EndAbsoluteValue squared right-parenthesis Superscript 1 slash 2 .

For all s element-of double-struck upper R , we denote by upper H Superscript s Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis the space of distributions f such that StartAbsoluteValue f EndAbsoluteValue Subscript upper H Sub Superscript s Baseline colon equals left-parenthesis integral normal upper Lamda left-parenthesis xi right-parenthesis Superscript s Baseline StartAbsoluteValue ModifyingAbove f With caret left-parenthesis xi right-parenthesis EndAbsoluteValue squared right-parenthesis Superscript 1 slash 2 Baseline less-than normal infinity , where ModifyingAbove f With caret denotes the Fourier transform of f . We also denote upper H Superscript normal infinity Baseline equals intersection Underscript s Endscripts upper H Superscript s .

If f element-of upper C left-parenthesis left-bracket 0 comma upper T right-bracket comma upper H Superscript s Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis right-parenthesis , we write StartAbsoluteValue f EndAbsoluteValue Subscript upper H Sub Subscript upper T Sub Superscript s Baseline equals sup Underscript t element-of left-bracket 0 comma upper T right-bracket Endscripts StartAbsoluteValue f left-parenthesis t right-parenthesis EndAbsoluteValue Subscript upper H Sub Superscript s .

If upper B is a Banach space and if f comma g element-of upper B , then we write StartAbsoluteValue f comma g EndAbsoluteValue Subscript upper B Baseline equals StartAbsoluteValue f EndAbsoluteValue Subscript upper B Baseline plus StartAbsoluteValue g EndAbsoluteValue Subscript upper B . If upper F equals left-parenthesis f 1 comma ellipsis comma f Subscript n Baseline right-parenthesis Superscript upper T Baseline element-of upper B Superscript n , then StartAbsoluteValue upper F EndAbsoluteValue Subscript upper B Baseline colon equals StartAbsoluteValue f 1 EndAbsoluteValue Subscript upper B plus dot dot dot plus StartAbsoluteValue f Subscript n Baseline EndAbsoluteValue Subscript upper B .

For all s element-of double-struck upper R , left ceiling s right ceiling denotes the first integer strictly larger than s (so that left ceiling 1 right ceiling equals 2 ).

2. Elliptic boundary value problems on a strip

Throughout this section, we work on a domain normal upper Omega defined as

normal upper Omega equals StartSet left-parenthesis upper X comma y right-parenthesis element-of double-struck upper R Superscript d plus 1 Baseline comma b left-parenthesis upper X right-parenthesis less-than y less-than a left-parenthesis upper X right-parenthesis EndSet comma

where a and b satisfy the following condition:

StartLayout 1st Row with Label left-parenthesis 2.1 right-parenthesis EndLabel there-exists h 0 greater-than 0 comma min left-brace negative b comma a minus b right-brace greater-than-or-equal-to h 0 greater-than 0 on double-struck upper R Superscript d EndLayout

(this assumption means that we exclude beaches or islands for the fluid domain, either perturbed or at rest).

We also consider a constant coefficients elliptic operator bold upper P equals minus nabla Subscript upper X comma y Baseline dot upper P nabla Subscript upper X comma y , where upper P is a symmetric matrix satisfying the following condition:

StartLayout 1st Row with Label left-parenthesis 2.2 right-parenthesis EndLabel there-exists p greater-than 0 such that upper P normal upper Theta dot normal upper Theta greater-than-or-equal-to p StartAbsoluteValue normal upper Theta EndAbsoluteValue squared comma for-all normal upper Theta element-of double-struck upper R Superscript d plus 1 Baseline period EndLayout

Finally, we consider boundary value problems of the form

StartLayout 1st Row with Label left-parenthesis 2.3 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row bold upper P u equals h on normal upper Omega comma 2nd Row u StartAbsoluteValue equals f comma partial-differential Subscript n Superscript upper P Baseline u EndAbsoluteValue Subscript StartSet y equals a left-parenthesis upper X right-parenthesis EndSet Baseline Subscript StartSet y equals b left-parenthesis upper X right-parenthesis EndSet Baseline equals g comma EndLayout EndLayout

where h is a function defined on normal upper Omega and f comma g are functions defined on double-struck upper R Superscript d . Moreover, partial-differential Subscript n Superscript upper P Baseline u vertical-bar Subscript StartSet y equals b left-parenthesis upper X right-parenthesis EndSet Baseline denotes the conormal derivative associated to bold upper P of u at the boundary StartSet y equals b left-parenthesis upper X right-parenthesis EndSet ,

StartLayout 1st Row with Label left-parenthesis 2.4 right-parenthesis EndLabel partial-differential Subscript n Superscript upper P Baseline u StartAbsoluteValue equals minus bold n Subscript bold minus Baseline dot upper P nabla Subscript upper X comma y Baseline u EndAbsoluteValue Subscript StartSet y equals b left-parenthesis upper X right-parenthesis EndSet Baseline Subscript StartSet y equals b left-parenthesis upper X right-parenthesis EndSet Baseline comma EndLayout

where bold n Subscript bold minus denotes the outwards normal derivative at the bottom.

Notation 2.1

For all open sets upper U subset-of double-struck upper R Superscript d plus 1 , we denote by double-vertical-bar dot colon upper U double-vertical-bar Subscript p , double-vertical-bar dot colon upper U double-vertical-bar Subscript k comma normal infinity and double-vertical-bar dot colon upper U double-vertical-bar Subscript k comma 2 the canonical norms of upper L Superscript p Baseline left-parenthesis upper U right-parenthesis , upper W Superscript k comma normal infinity Baseline left-parenthesis upper U right-parenthesis and upper H Superscript k Baseline left-parenthesis upper U right-parenthesis respectively. When no confusion is possible on the domain upper U , we write simply double-vertical-bar dot double-vertical-bar Subscript p , double-vertical-bar dot double-vertical-bar Subscript k comma normal infinity and double-vertical-bar dot double-vertical-bar Subscript k comma 2 .

2.1. Reduction to an elliptic equation on a flat strip

Throughout this section, we denote by upper R any diffeomorphism between normal upper Omega and the flat strip script upper S equals double-struck upper R Superscript d Baseline times left-parenthesis 0 comma 1 right-parenthesis , which we assume to be of the form

StartLayout 1st Row with Label left-parenthesis 2.5 right-parenthesis EndLabel upper R colon StartLayout 1st Row 1st Column normal upper Omega 2nd Column right-arrow 3rd Column script upper S 2nd Row 1st Column left-parenthesis upper X comma y right-parenthesis 2nd Column right-arrow from bar 3rd Column left-parenthesis upper X comma r left-parenthesis upper X comma y right-parenthesis right-parenthesis comma EndLayout EndLayout

and we denote its inverse upper R Superscript negative 1 by upper S ,

StartLayout 1st Row with Label left-parenthesis 2.6 right-parenthesis EndLabel upper S colon StartLayout 1st Row 1st Column script upper S 2nd Column right-arrow 3rd Column normal upper Omega 2nd Row 1st Column left-parenthesis upper X overTilde comma y overTilde right-parenthesis 2nd Column right-arrow from bar 3rd Column left-parenthesis upper X overTilde comma s left-parenthesis upper X overTilde comma y overTilde right-parenthesis right-parenthesis period EndLayout EndLayout

We always assume the following on s :

Assumption 2.2

One has s element-of upper W Superscript 1 comma normal infinity Baseline left-parenthesis script upper S right-parenthesis with s vertical-bar Subscript y overTilde equals 0 Baseline equals a and s vertical-bar Subscript y overTilde equals negative 1 Baseline equals b . Moreover, there exists c 0 greater-than 0 such that partial-differential Subscript y overTilde Baseline s greater-than-or-equal-to c 0 on script upper S overbar .

Finally, we need the following definition:

Definition 2.3

Let k element-of double-struck upper N . The mapping s , given by (Equation2.6), is called k -regular if it satisfies Assumption 2.2 and can moreover be decomposed into s equals s 1 plus s 2 with s 1 element-of upper C Subscript b Superscript k Baseline left-parenthesis script upper S overbar right-parenthesis and s 2 element-of upper H Superscript k Baseline left-parenthesis script upper S right-parenthesis , and if partial-differential Subscript y overTilde Baseline s 1 greater-than-or-equal-to c 0 on script upper S overbar .

Remark 2.4

The most simple diffeomorphism upper R between normal upper Omega and script upper S is given by

r left-parenthesis upper X comma y right-parenthesis equals StartFraction y minus a left-parenthesis upper X right-parenthesis Over a left-parenthesis upper X right-parenthesis minus b left-parenthesis upper X right-parenthesis EndFraction comma

and hence s left-parenthesis upper X overTilde comma y overTilde right-parenthesis equals left-parenthesis a left-parenthesis upper X overTilde right-parenthesis minus b left-parenthesis upper X overTilde right-parenthesis right-parenthesis y overTilde plus a left-parenthesis upper X overTilde right-parenthesis . If a element-of upper H Superscript k intersection upper W Superscript 1 comma normal infinity Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis and b element-of upper C Subscript b Superscript k Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis , it is clear that s is k -regular, with s 1 left-parenthesis upper X overTilde comma y overTilde right-parenthesis colon equals minus b left-parenthesis upper X overTilde right-parenthesis y overTilde , s 2 left-parenthesis upper X overTilde comma y overTilde right-parenthesis colon equals left-parenthesis 1 plus y overTilde right-parenthesis a left-parenthesis upper X overTilde right-parenthesis , and c 0 equals h 0 .

To any distribution u defined on normal upper Omega one can associate, using the diffeomorphism upper R and its inverse upper S given by (Equation2.5)-(Equation2.6), a distribution u overTilde defined on script upper S as

StartLayout 1st Row with Label left-parenthesis 2.7 right-parenthesis EndLabel u overTilde equals u ring upper S comma EndLayout

and vice-versa,

StartLayout 1st Row with Label left-parenthesis 2.8 right-parenthesis EndLabel u equals u overTilde ring upper R period EndLayout

The following lemma shows that the constant coefficients elliptic equation bold upper P u equals 0 on normal upper Omega can equivalently be formulated as a variable coefficients elliptic equation bold upper P overTilde u overTilde equals 0 on script upper S .

Lemma 2.5

Suppose that the mapping s , given by Equation2.6 satisfies Assumption 2.2. Let bold upper P equals minus nabla Subscript upper X comma y Baseline dot upper P nabla Subscript upper X comma y with upper P satisfying Equation2.2. Then the equation bold upper P u equals h holds in script upper D prime left-parenthesis normal upper Omega right-parenthesis if and only if the equation bold upper P overTilde u overTilde equals left-parenthesis partial-differential Subscript y overTilde Baseline s right-parenthesis h overTilde holds in script upper D prime left-parenthesis script upper S right-parenthesis , where u overTilde and h overTilde are deduced from u and h via formula Equation2.7, and bold upper P overTilde colon equals minus nabla Subscript upper X overTilde comma y overTilde dot upper P overTilde nabla Subscript upper X overTilde comma y overTilde , with

upper P overTilde equals StartFraction 1 Over partial-differential Subscript y overTilde Baseline s EndFraction Start 2 By 2 Matrix 1st Row 1st Column partial-differential Subscript y overTilde Baseline s upper I d Subscript d times d Baseline 2nd Column 0 2nd Row 1st Column minus nabla Subscript upper X overTilde Baseline s Superscript upper T Baseline 2nd Column 1 EndMatrix upper P Start 2 By 2 Matrix 1st Row 1st Column partial-differential Subscript y overTilde Baseline s upper I d Subscript d times d Baseline 2nd Column minus nabla Subscript upper X overTilde Baseline s 2nd Row 1st Column 0 2nd Column 1 EndMatrix period

Moreover, one has, for all normal upper Theta element-of double-struck upper R Superscript d plus 1 ,

upper P overTilde normal upper Theta dot normal upper Theta greater-than-or-equal-to p overTilde StartAbsoluteValue normal upper Theta EndAbsoluteValue squared comma with p overTilde equals upper C s t p StartFraction c 0 squared Over double-vertical-bar partial-differential Subscript y overTilde Baseline s double-vertical-bar Subscript normal infinity Baseline left-parenthesis 1 plus double-vertical-bar nabla Subscript upper X overTilde comma y overTilde Baseline s double-vertical-bar Subscript normal infinity Superscript 2 Baseline right-parenthesis EndFraction period

Proof.

By definition, bold upper P u equals h in script upper D prime left-parenthesis normal upper Omega right-parenthesis if and only if

StartLayout 1st Row with Label left-parenthesis 2.9 right-parenthesis EndLabel integral Underscript normal upper Omega Endscripts bold upper P u phi equals integral Underscript normal upper Omega Endscripts h phi comma for-all phi element-of script upper D left-parenthesis normal upper Omega right-parenthesis period EndLayout

By definition of bold upper P , one also has

StartLayout 1st Row 1st Column integral Underscript normal upper Omega Endscripts bold upper P u phi equals integral Underscript normal upper Omega Endscripts upper P nabla Subscript upper X comma y Baseline u dot nabla Subscript upper X comma y Baseline phi 2nd Row 1st Column Blank 2nd Column equals 3rd Column integral Underscript normal upper Omega Endscripts upper P StartBinomialOrMatrix left-parenthesis nabla Subscript upper X overTilde Baseline u overTilde right-parenthesis ring upper R plus nabla Subscript upper X Baseline r left-parenthesis partial-differential Subscript y overTilde Baseline u overTilde right-parenthesis ring upper R Choose partial-differential Subscript y Baseline r left-parenthesis partial-differential Subscript y overTilde Baseline u overTilde right-parenthesis ring upper R EndBinomialOrMatrix dot StartBinomialOrMatrix left-parenthesis nabla Subscript upper X overTilde Baseline phi overTilde right-parenthesis ring upper R plus nabla Subscript upper X Baseline r left-parenthesis partial-differential Subscript y overTilde Baseline phi overTilde right-parenthesis ring upper R Choose partial-differential Subscript y Baseline r left-parenthesis partial-differential Subscript y overTilde Baseline phi overTilde right-parenthesis ring upper R EndBinomialOrMatrix 3rd Row 1st Column Blank 2nd Column equals 3rd Column integral Underscript script upper S Endscripts StartAbsoluteValue partial-differential Subscript y overTilde Baseline s EndAbsoluteValue upper P StartBinomialOrMatrix nabla Subscript upper X overTilde Baseline plus left-parenthesis nabla Subscript upper X Baseline r right-parenthesis ring upper S partial-differential Subscript y overTilde Baseline Choose left-parenthesis partial-differential Subscript y Baseline r right-parenthesis ring upper S partial-differential Subscript y overTilde Baseline EndBinomialOrMatrix u overTilde dot StartBinomialOrMatrix nabla Subscript upper X overTilde Baseline plus left-parenthesis nabla Subscript upper X Baseline r right-parenthesis ring upper S partial-differential Subscript y overTilde Baseline Choose left-parenthesis partial-differential Subscript y Baseline r right-parenthesis ring upper S partial-differential Subscript y overTilde Baseline EndBinomialOrMatrix phi overTilde period EndLayout

Integrating by parts yields therefore that integral Underscript normal upper Omega Endscripts bold upper P u phi is equal to

minus integral Underscript script upper S Endscripts phi overTilde StartBinomialOrMatrix nabla Subscript upper X overTilde Baseline dot plus partial-differential Subscript y overTilde Baseline left-parenthesis left-parenthesis nabla Subscript upper X Baseline r right-parenthesis ring upper S dot right-parenthesis Choose partial-differential Subscript y overTilde Baseline left-parenthesis left-parenthesis partial-differential Subscript y Baseline r right-parenthesis ring upper S dot right-parenthesis EndBinomialOrMatrix dot StartAbsoluteValue partial-differential Subscript y overTilde Baseline s EndAbsoluteValue upper P StartBinomialOrMatrix nabla Subscript upper X overTilde Baseline plus left-parenthesis nabla Subscript upper X Baseline r right-parenthesis ring upper S partial-differential Subscript y overTilde Choose left-parenthesis partial-differential Subscript y Baseline r right-parenthesis ring upper S partial-differential Subscript y overTilde EndBinomialOrMatrix u overTilde

and thus to

minus integral Underscript script upper S Endscripts phi overTilde nabla Subscript upper X overTilde comma y overTilde Baseline dot Start 2 By 2 Matrix 1st Row 1st Column upper I d 2nd Column 0 2nd Row 1st Column left-parenthesis left-parenthesis nabla Subscript upper X Baseline r right-parenthesis ring upper S right-parenthesis Superscript upper T Baseline 2nd Column left-parenthesis partial-differential Subscript y Baseline r right-parenthesis ring upper S EndMatrix StartAbsoluteValue partial-differential Subscript y overTilde Baseline s EndAbsoluteValue upper P Start 2 By 2 Matrix 1st Row 1st Column upper I d 2nd Column left-parenthesis nabla Subscript upper X Baseline r right-parenthesis ring upper S 2nd Row 1st Column 0 2nd Column left-parenthesis partial-differential Subscript y Baseline r right-parenthesis ring upper S EndMatrix nabla Subscript upper X overTilde comma y overTilde Baseline u overTilde period

By definition of r and s , one has r left-parenthesis upper X overTilde comma s left-parenthesis upper X overTilde comma y overTilde right-parenthesis right-parenthesis equals y overTilde for all left-parenthesis upper X overTilde comma y overTilde right-parenthesis element-of script upper S . Differentiating this identity with respect to upper X overTilde and y overTilde respectively yields

left-parenthesis nabla Subscript upper X Baseline r right-parenthesis ring upper S plus left-parenthesis partial-differential Subscript y Baseline r right-parenthesis ring upper S nabla Subscript upper X overTilde Baseline s equals 0 comma partial-differential Subscript y overTilde Baseline s left-parenthesis partial-differential Subscript y Baseline r right-parenthesis ring upper S equals 1 period

Using these expressions in the above expressions gives the equality

StartLayout 1st Row with Label left-parenthesis 2.10 right-parenthesis EndLabel integral Underscript normal upper Omega Endscripts bold upper P u phi equals integral Underscript script upper S Endscripts bold upper P overTilde u overTilde phi overTilde comma EndLayout

where bold upper P overTilde is as given in the statement of the lemma. Since one clearly has

integral Underscript normal upper Omega Endscripts h phi equals integral Underscript script upper S Endscripts partial-differential Subscript y Baseline s h overTilde phi overTilde comma

the first claim of the lemma follows from (Equation2.9) and (Equation2.10).

We now prove the coercivity of bold upper P overTilde . One has, for all normal upper Theta element-of double-struck upper R Superscript d plus 1 ,

upper P overTilde normal upper Theta dot normal upper Theta equals StartFraction 1 Over partial-differential Subscript y overTilde Baseline s EndFraction upper P upper A normal upper Theta dot upper A normal upper Theta comma with upper A colon equals Start 2 By 2 Matrix 1st Row 1st Column partial-differential Subscript y overTilde Baseline s upper I d Subscript d times d Baseline 2nd Column minus nabla Subscript upper X overTilde Baseline s 2nd Row 1st Column 0 2nd Column 1 EndMatrix comma

and owing to (Equation2.2) we have therefore

StartLayout 1st Row with Label left-parenthesis 2.11 right-parenthesis EndLabel upper P overTilde normal upper Theta dot normal upper Theta greater-than-or-equal-to StartFraction p Over partial-differential Subscript y overTilde Baseline s EndFraction StartAbsoluteValue upper A normal upper Theta EndAbsoluteValue squared period EndLayout

The matrix upper A is invertible, and its inverse is given by

upper A Superscript negative 1 Baseline equals StartFraction 1 Over partial-differential Subscript y overTilde Baseline s EndFraction Start 2 By 2 Matrix 1st Row 1st Column upper I d Subscript d times d Baseline 2nd Column nabla Subscript upper X overTilde Baseline s 2nd Row 1st Column 0 2nd Column partial-differential Subscript y overTilde Baseline s EndMatrix comma

so that normal upper Theta equals upper A Superscript negative 1 Baseline upper A normal upper Theta can be bounded as

StartAbsoluteValue normal upper Theta EndAbsoluteValue less-than-or-equal-to upper C s t StartFraction 1 Over c 0 EndFraction left-parenthesis 1 plus double-vertical-bar nabla Subscript upper X overTilde comma y overTilde Baseline s double-vertical-bar Subscript normal infinity Baseline right-parenthesis StartAbsoluteValue upper A normal upper Theta EndAbsoluteValue period

Together with (Equation2.11), this estimate yields the result of the lemma.

The next lemma shows how the boundary conditions are transformed by the diffeomorphism upper R .

Lemma 2.6

Suppose that the mapping s , given by Equation2.6, satisfies Assumption 2.2. For all u element-of upper C Superscript 1 Baseline left-parenthesis normal upper Omega overbar right-parenthesis , one has

u StartAbsoluteValue equals u overTilde EndAbsoluteValue Subscript StartSet y equals a EndSet Baseline Subscript StartSet y overTilde equals 0 EndSet Baseline and partial-differential Subscript n Superscript upper P Baseline u StartAbsoluteValue equals StartFraction 1 Over StartRoot 1 plus StartAbsoluteValue nabla Subscript upper X Baseline b EndAbsoluteValue squared EndRoot EndFraction partial-differential Subscript n Superscript upper P overTilde Baseline u overTilde EndAbsoluteValue Subscript StartSet y equals b EndSet Baseline Subscript StartSet y overTilde equals negative 1 EndSet Baseline period

Proof.

The first assertion of the lemma is straightforward. We now prove the second. By definition,

StartLayout 1st Row 1st Column partial-differential Subscript n Superscript upper P overTilde Baseline u overTilde vertical-bar Subscript y overTilde equals negative 1 Baseline 2nd Column equals 3rd Column minus left-parenthesis minus e Subscript d plus 1 Baseline right-parenthesis dot upper P overTilde nabla Subscript upper X overTilde comma y overTilde Baseline u overTilde vertical-bar Subscript y overTilde equals negative 1 Baseline 2nd Row 1st Column Blank 2nd Column equals 3rd Column minus left-parenthesis minus e Subscript d plus 1 Baseline right-parenthesis dot upper P overTilde StartBinomialOrMatrix nabla Subscript upper X Baseline u StartAbsoluteValue plus nabla Subscript upper X overTilde Baseline s EndAbsoluteValue Subscript y equals b Baseline Subscript y overTilde equals negative 1 Baseline partial-differential Subscript y Baseline u vertical-bar Subscript y equals b Baseline Choose partial-differential Subscript y overTilde Baseline s StartAbsoluteValue partial-differential Subscript y Baseline u EndAbsoluteValue Subscript y overTilde equals negative 1 Baseline Subscript y equals b Baseline EndBinomialOrMatrix period EndLayout

Replacing upper P overTilde by its expression given in Lemma 2.5, one obtains easily that

StartLayout 1st Row 1st Column partial-differential Subscript n Superscript upper P overTilde Baseline u overTilde vertical-bar Subscript y overTilde equals negative 1 Baseline 2nd Column equals 3rd Column minus StartBinomialOrMatrix nabla Subscript upper X Baseline s vertical-bar Subscript y overTilde equals negative 1 Baseline Choose negative 1 EndBinomialOrMatrix dot upper P nabla Subscript upper X comma y Baseline u vertical-bar Subscript y equals b Baseline 2nd Row 1st Column Blank 2nd Column equals 3rd Column StartRoot 1 plus StartAbsoluteValue nabla Subscript upper X Baseline b EndAbsoluteValue squared EndRoot partial-differential Subscript n Superscript upper P Baseline u vertical-bar Subscript y equals b Baseline comma EndLayout

which ends the proof of the lemma.

Lemmas 2.5 and 2.6 show that the study of the boundary problems (Equation2.3) can be deduced from the study of elliptic boundary value problems on a flat strip:

Proposition 2.7

Suppose that the mapping s , given by Equation2.6, satisfies Assumption 2.2. Then u is a (variational, classical) solution of Equation2.3 if and only if u overTilde given by Equation2.7 is a (variational, classical) solution of

StartLayout 1st Row with Label left-parenthesis 2.12 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row bold upper P overTilde u overTilde equals left-parenthesis partial-differential Subscript y overTilde Baseline s right-parenthesis h overTilde on script upper S comma 2nd Row u overTilde StartAbsoluteValue equals f comma partial-differential Subscript n Superscript upper P overTilde Baseline u overTilde EndAbsoluteValue Subscript y overTilde equals 0 Baseline Subscript y overTilde equals negative 1 Baseline equals StartRoot 1 plus StartAbsoluteValue nabla Subscript upper X Baseline b EndAbsoluteValue squared EndRoot g comma EndLayout EndLayout

where bold upper P overTilde is as given in Lemma 2.5.

The next section is therefore devoted to the study of the well-posedness of such variable coefficients elliptic boundary value problems on a flat strip. Before this, let us state a lemma dealing with the smoothness of the coefficients of upper P overTilde . Its proof is given in Appendix A.

Lemma 2.8

Let k element-of double-struck upper N and assume that the mapping s , given by Equation2.6, is left-parenthesis 1 plus k right-parenthesis -regular. Then one can write upper P overTilde equals upper P overTilde Subscript 1 Baseline plus upper P overTilde Subscript 2 with upper P overTilde Subscript 1 Baseline element-of upper C Subscript b Superscript k Baseline left-parenthesis script upper S overbar right-parenthesis Superscript left-parenthesis d plus 1 right-parenthesis squared , upper P overTilde Subscript 2 Baseline element-of upper H Superscript k Baseline left-parenthesis script upper S right-parenthesis Superscript left-parenthesis d plus 1 right-parenthesis squared and

StartLayout 1st Row 1st Column double-vertical-bar upper P overTilde Subscript 1 Baseline double-vertical-bar Subscript k comma normal infinity 2nd Column less-than-or-equal-to 3rd Column upper C left-parenthesis StartFraction 1 Over c 0 EndFraction comma double-vertical-bar s 1 double-vertical-bar Subscript k plus 1 comma normal infinity Baseline right-parenthesis comma 2nd Row 1st Column double-vertical-bar upper P overTilde Subscript 2 Baseline double-vertical-bar Subscript k comma 2 2nd Column less-than-or-equal-to 3rd Column upper C left-parenthesis StartFraction 1 Over c 0 EndFraction comma double-vertical-bar s 1 double-vertical-bar Subscript 1 plus k comma normal infinity Baseline comma double-vertical-bar s 2 double-vertical-bar Subscript 1 comma normal infinity Baseline right-parenthesis double-vertical-bar s 2 double-vertical-bar Subscript 1 plus k comma 2 Baseline period EndLayout

2.2. Variable coefficients elliptic equations on a flat strip

We have seen in the previous section that the theory of elliptic equations on a general strip of type (Equation2.3) can be deduced from the study of elliptic equations on a flat strip, but with variable coefficients. In this section, we study the following generic problem:

StartLayout 1st Row with Label left-parenthesis 2.13 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row bold upper Q u colon equals minus nabla Subscript upper X comma y Baseline dot upper Q nabla Subscript upper X comma y Baseline u equals h on script upper S comma 2nd Row u StartAbsoluteValue equals f comma partial-differential Subscript n Superscript upper Q Baseline u EndAbsoluteValue Subscript y equals 0 Baseline Subscript y equals negative 1 Baseline equals g comma EndLayout EndLayout

where we recall that partial-differential Subscript n Superscript upper Q denotes the conormal derivative associated to bold upper Q ,

StartLayout 1st Row with Label left-parenthesis 2.14 right-parenthesis EndLabel partial-differential Subscript n Superscript upper Q Baseline u StartAbsoluteValue equals minus e Subscript d plus 1 Baseline dot upper Q nabla Subscript upper X comma y Baseline u EndAbsoluteValue Subscript y equals 0 Baseline Subscript y equals 0 Baseline comma partial-differential Subscript n Superscript upper Q Baseline u StartAbsoluteValue equals minus left-parenthesis minus e Subscript d plus 1 Baseline right-parenthesis dot upper Q nabla Subscript upper X comma y Baseline u EndAbsoluteValue Subscript y equals negative 1 Baseline Subscript y equals negative 1 Baseline period EndLayout

We also assume that upper Q satisfies the following coercivity assumption:

StartLayout 1st Row with Label left-parenthesis 2.15 right-parenthesis EndLabel there-exists q greater-than 0 such that upper Q left-parenthesis upper X comma y right-parenthesis normal upper Theta dot normal upper Theta greater-than-or-equal-to q StartAbsoluteValue normal upper Theta EndAbsoluteValue squared comma for-all normal upper Theta element-of double-struck upper R Superscript d plus 1 Baseline comma for-all left-parenthesis upper X comma y right-parenthesis element-of script upper S period EndLayout

The main result of this section is the following theorem.

Theorem 2.9

Let k element-of double-struck upper N , m 0 equals left ceiling StartFraction d plus 1 Over 2 EndFraction right ceiling . Let f element-of upper H Superscript k plus 3 slash 2 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis , g element-of upper H Superscript k plus 1 slash 2 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis and h element-of upper H Superscript k Baseline left-parenthesis script upper S right-parenthesis .

i.

If upper Q element-of upper W Superscript 1 plus k Baseline left-parenthesis script upper S right-parenthesis Superscript left-parenthesis d plus 1 right-parenthesis squared satisfies Equation2.15, then there exists a unique solution u element-of upper H Superscript k plus 2 Baseline left-parenthesis script upper S right-parenthesis to Equation2.13. Moreover, double-vertical-bar u double-vertical-bar Subscript k plus 2 comma 2 Baseline less-than-or-equal-to upper C left-parenthesis StartFraction 1 Over q EndFraction comma double-vertical-bar upper Q double-vertical-bar Subscript 1 plus k comma normal infinity Baseline right-parenthesis left-parenthesis double-vertical-bar h double-vertical-bar Subscript k comma 2 Baseline plus StartAbsoluteValue f EndAbsoluteValue Subscript upper H Sub Superscript k plus 3 slash 2 Subscript Baseline plus StartAbsoluteValue g EndAbsoluteValue Subscript upper H Sub Superscript k plus 1 slash 2 Subscript Baseline right-parenthesis period

ii.

If upper Q 1 element-of upper C Subscript b Superscript k plus 1 Baseline left-parenthesis script upper S overbar right-parenthesis Superscript left-parenthesis d plus 1 right-parenthesis squared and upper Q 2 element-of upper H Superscript 1 plus k intersection upper W Superscript m 0 comma normal infinity Baseline left-parenthesis script upper S right-parenthesis Superscript left-parenthesis d plus 1 right-parenthesis squared are such that upper Q colon equals upper Q 1 plus upper Q 2 satisfies Equation2.15, then there exists a unique solution u element-of upper H Superscript k plus 2 Baseline left-parenthesis script upper S right-parenthesis to Equation2.13. Moreover, when k greater-than-or-equal-to m 0 , StartLayout 1st Row 1st Column double-vertical-bar u double-vertical-bar Subscript k plus 2 comma 2 2nd Column less-than-or-equal-to 3rd Column upper C Subscript k Baseline times left-parenthesis double-vertical-bar h double-vertical-bar Subscript k comma 2 Baseline plus StartAbsoluteValue f EndAbsoluteValue Subscript upper H Sub Superscript k plus 3 slash 2 Subscript Baseline plus StartAbsoluteValue g EndAbsoluteValue Subscript upper H Sub Superscript k plus 1 slash 2 Subscript Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column plus 3rd Column upper C Subscript k Baseline times left-parenthesis double-vertical-bar h double-vertical-bar Subscript m 0 minus 1 comma 2 Baseline plus StartAbsoluteValue f EndAbsoluteValue Subscript upper H Sub Superscript m 0 plus 1 slash 2 Subscript Baseline plus StartAbsoluteValue g EndAbsoluteValue Subscript upper H Sub Superscript m 0 minus 1 slash 2 Subscript Baseline right-parenthesis double-vertical-bar upper Q 2 double-vertical-bar Subscript 1 plus k comma 2 Baseline comma EndLayout

where upper C Subscript k Baseline equals upper C left-parenthesis StartFraction 1 Over q EndFraction comma double-vertical-bar upper Q 1 double-vertical-bar Subscript 1 plus k comma normal infinity Baseline comma double-vertical-bar upper Q 2 double-vertical-bar Subscript m 0 plus 1 comma normal infinity Baseline right-parenthesis .

Remark 2.10

i.

The proof below shows that the quantity double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript k plus 1 comma 2 can be estimated more precisely than double-vertical-bar u double-vertical-bar Subscript k plus 2 comma 2 . Namely, one can replace the quantities StartAbsoluteValue f EndAbsoluteValue Subscript upper H Sub Superscript k plus 3 slash 2 and StartAbsoluteValue f EndAbsoluteValue Subscript upper H Sub Superscript m 0 plus 1 slash 2 in both estimates of the theorem by StartAbsoluteValue nabla Subscript upper X Baseline f EndAbsoluteValue Subscript upper H Sub Superscript k plus 1 slash 2 and StartAbsoluteValue nabla Subscript upper X Baseline f EndAbsoluteValue Subscript upper H Sub Superscript m 0 minus 1 slash 2 respectively. This remark is very useful when giving estimates on the Dirichlet-Neumann operator.

ii.

The second estimate of the theorem remains of course valid when k less-than m 0 , but in that case, the first estimate of the theorem is more precise.

Proof.

Even though script upper S is unbounded, the proof follows the same lines as the usual proofs of existence and regularity estimates of solutions to elliptic equations on regular bounded domains (Reference26Reference19), but special care must be paid to use the specific Sobolev regularity of the coefficients of bold upper Q . We only prove the second point of the theorem since the first one can be obtained by skipping the fourth step of the proof below.

Step 1. Construction of a variational solution to (Equation2.13). We first introduce f Superscript normal ♯ Baseline left-parenthesis y comma dot right-parenthesis colon equals chi left-parenthesis y StartAbsoluteValue upper D EndAbsoluteValue right-parenthesis f , where chi is a smooth compactly supported function such that chi left-parenthesis 0 right-parenthesis equals 1 . Classically, one has f Superscript normal ♯ Baseline vertical-bar Subscript y equals 0 Baseline equals f and

StartLayout 1st Row with Label left-parenthesis 2.16 right-parenthesis EndLabel double-vertical-bar nabla Subscript upper X comma y Baseline f Superscript normal ♯ Baseline double-vertical-bar Subscript 1 comma 2 Baseline less-than-or-equal-to upper C s t StartAbsoluteValue nabla Subscript upper X Baseline f EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 Subscript Baseline comma StartAbsoluteValue partial-differential Subscript n Superscript upper Q Baseline f Subscript vertical-bar Sub Subscript y equals negative 1 Subscript Superscript normal ♯ Baseline EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 Subscript Baseline less-than-or-equal-to upper C s t double-vertical-bar upper Q double-vertical-bar Subscript 1 comma normal infinity Baseline StartAbsoluteValue nabla Subscript upper X Baseline f EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 Subscript Baseline period EndLayout

It follows that u is a variational solution of (Equation2.13) if and only if u Superscript normal ♯ Baseline colon equals u minus f Superscript normal ♯ is a variational solution to

StartLayout 1st Row with Label left-parenthesis 2.17 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row bold upper Q u Superscript normal ♯ Baseline equals h minus bold upper Q left-parenthesis f Superscript normal ♯ Baseline right-parenthesis colon equals h Superscript normal ♯ Baseline comma 2nd Row u Superscript normal ♯ Baseline StartAbsoluteValue equals 0 comma partial-differential Subscript n Superscript upper Q Baseline u Superscript normal ♯ Baseline EndAbsoluteValue Subscript y equals 0 Baseline Subscript y equals negative 1 Baseline equals g overTilde comma EndLayout EndLayout

where g overTilde colon equals g minus partial-differential Subscript n Superscript upper Q Baseline f Superscript normal ♯ Baseline vertical-bar Subscript y equals negative 1 Baseline .

Define the space upper V as upper V colon equals ModifyingAbove script upper D left-parenthesis double-struck upper R Superscript d Baseline times left-bracket negative 1 comma 0 right-parenthesis right-parenthesis With bar , where the closure is taken relative to the upper H Superscript 1 Baseline left-parenthesis script upper S right-parenthesis -norm. It is a classical consequence of Lax-Milgram’s theorem that there exists a unique u Superscript normal ♯ Baseline element-of upper V such that

StartLayout 1st Row with Label left-parenthesis 2.18 right-parenthesis EndLabel integral Underscript script upper S Endscripts upper Q nabla Subscript upper X comma y Baseline u Superscript normal ♯ Baseline dot nabla Subscript upper X comma y Baseline v equals integral Underscript script upper S Endscripts h Superscript normal ♯ Baseline v minus integral Underscript y equals negative 1 Endscripts g overTilde v comma for-all v element-of upper V period EndLayout

Step 2. Regularity of the variational solution. We show that u Superscript normal ♯ Baseline element-of upper H squared left-parenthesis script upper S right-parenthesis using the classical method of Nirenberg’s tangential differential quotients. For all v element-of upper V and i equals 1 comma ellipsis comma d , one has rho Subscript i comma h Baseline v element-of upper V , where rho Subscript i comma h Baseline v is defined as

rho Subscript i comma h Baseline v colon equals StartFraction tau Subscript i comma h Baseline v minus v Over h EndFraction comma with tau Subscript i comma h Baseline phi colon equals phi left-parenthesis dot plus h e 1 right-parenthesis comma for-all phi element-of script upper D left-parenthesis double-struck upper R Superscript d Baseline times left-bracket negative 1 comma 0 right-parenthesis right-parenthesis semicolon

we also recall that the adjoint operator of rho Subscript i comma h is minus rho Subscript i comma negative h and that one has the product rule v 1 rho Subscript i comma h Baseline v 2 equals rho Subscript i comma h Baseline left-parenthesis v 1 v 2 right-parenthesis minus rho Subscript i comma h Baseline v 1 tau Subscript i comma h Baseline v 2 . Using (Equation2.18) with rho Subscript i comma h Baseline v instead of v , one gets therefore

integral Underscript script upper S Endscripts left-parenthesis tau Subscript i comma negative h Baseline upper Q right-parenthesis nabla Subscript upper X comma y Baseline rho Subscript i comma negative h Baseline u Superscript normal ♯ Baseline dot nabla Subscript upper X comma y Baseline v equals integral Underscript negative 1 Endscripts g overTilde rho Subscript i comma h Baseline v minus integral Underscript script upper S Endscripts left-parenthesis h Superscript normal ♯ Baseline rho Subscript i comma h Baseline v minus rho Subscript i comma negative h Baseline upper Q nabla Subscript upper X comma y Baseline u Superscript normal ♯ Baseline dot nabla Subscript upper X comma y Baseline v right-parenthesis period

By the trace theorem and Poincaré’s inequality, we get StartAbsoluteValue rho Subscript i comma h Baseline v EndAbsoluteValue Subscript upper H Sub Superscript negative 1 slash 2 Baseline less-than-or-equal-to upper C s t double-vertical-bar nabla Subscript upper X comma y Baseline v double-vertical-bar Subscript 2 , so that the r.h.s of the above inequality can be bounded from above by

StartLayout 1st Row with Label left-parenthesis 2.19 right-parenthesis EndLabel upper C s t left-parenthesis StartAbsoluteValue g overTilde EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 Subscript Baseline plus double-vertical-bar h Superscript normal ♯ Baseline double-vertical-bar Subscript 2 Baseline plus double-vertical-bar upper Q double-vertical-bar Subscript 1 comma normal infinity Baseline double-vertical-bar nabla Subscript upper X comma y Baseline u Superscript normal ♯ Baseline double-vertical-bar Subscript 2 Baseline right-parenthesis double-vertical-bar nabla Subscript upper X comma y Baseline v double-vertical-bar Subscript 2 Baseline period EndLayout

Taking v equals rho Subscript i comma negative h Baseline u Superscript normal ♯ as a test function in (Equation2.19), using condition (Equation2.15), and letting h right-arrow 0 , one gets therefore

StartLayout 1st Row with Label left-parenthesis 2.20 right-parenthesis EndLabel double-vertical-bar partial-differential Subscript i j Superscript 2 Baseline u Superscript normal ♯ Baseline double-vertical-bar Subscript 2 Baseline less-than-or-equal-to upper C s t StartFraction 1 Over q EndFraction left-parenthesis StartAbsoluteValue g overTilde EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 Subscript Baseline plus double-vertical-bar h Superscript normal ♯ Baseline double-vertical-bar Subscript 2 Baseline plus double-vertical-bar upper Q double-vertical-bar Subscript 1 comma normal infinity Baseline double-vertical-bar nabla Subscript upper X comma y Baseline u Superscript normal ♯ Baseline double-vertical-bar Subscript 2 Baseline right-parenthesis EndLayout

for all 1 less-than-or-equal-to i comma j less-than-or-equal-to d plus 1 such that i plus j less-than-or-equal-to 2 d plus 1 . The missing term partial-differential Subscript y y Superscript 2 Baseline u Superscript normal ♯ is obtained as usual using the equation,

StartLayout 1st Row with Label left-parenthesis 2.21 right-parenthesis EndLabel minus partial-differential Subscript y y Superscript 2 Baseline u Superscript normal ♯ Baseline equals StartFraction 1 Over q Subscript d plus 1 comma d plus 1 Baseline EndFraction left-parenthesis bold upper Q u Superscript normal ♯ Baseline plus sigma-summation Underscript i plus j less-than-or-equal-to 2 d plus 1 Endscripts partial-differential Subscript i Baseline left-parenthesis q Subscript i comma j Baseline partial-differential Subscript j Baseline u Superscript normal ♯ Baseline right-parenthesis plus left-parenthesis partial-differential Subscript y Baseline q Subscript d plus 1 comma d plus 1 Baseline right-parenthesis partial-differential Subscript y Baseline u Superscript normal ♯ Baseline right-parenthesis comma EndLayout

where upper Q equals left-parenthesis q Subscript i comma j Baseline right-parenthesis Subscript i comma j , from which it follows easily that

StartLayout 1st Row with Label left-parenthesis 2.22 right-parenthesis EndLabel double-vertical-bar partial-differential Subscript y y Superscript 2 Baseline u Superscript normal ♯ Baseline double-vertical-bar Subscript 2 Baseline less-than-or-equal-to StartFraction upper C s t Over q EndFraction left-parenthesis double-vertical-bar h Superscript normal ♯ Baseline double-vertical-bar Subscript 2 Baseline plus double-vertical-bar upper Q double-vertical-bar Subscript 1 comma normal infinity Baseline left-parenthesis sigma-summation Underscript i plus j less-than-or-equal-to 2 d plus 1 Endscripts double-vertical-bar partial-differential Subscript i j Superscript 2 Baseline u Superscript normal ♯ Baseline double-vertical-bar Subscript 2 Baseline plus double-vertical-bar nabla Subscript upper X comma y Baseline u Superscript normal ♯ Baseline double-vertical-bar Subscript 2 Baseline right-parenthesis right-parenthesis period EndLayout

From (Equation2.20) and (Equation2.22), it follows that u Superscript normal ♯ Baseline element-of upper H squared left-parenthesis script upper S right-parenthesis and satisfies

double-vertical-bar nabla Subscript upper X comma y Baseline u Superscript normal ♯ Baseline double-vertical-bar Subscript 1 comma 2 Baseline less-than-or-equal-to upper C left-parenthesis StartFraction 1 Over q EndFraction comma double-vertical-bar upper Q double-vertical-bar Subscript 1 comma normal infinity Baseline right-parenthesis left-parenthesis StartAbsoluteValue g overTilde EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 Subscript Baseline plus double-vertical-bar h Superscript normal ♯ Baseline double-vertical-bar Subscript 2 Baseline plus double-vertical-bar nabla Subscript upper X comma y Baseline u Superscript normal ♯ Baseline double-vertical-bar Subscript 2 Baseline right-parenthesis period

Replacing u Superscript normal ♯ Baseline comma h Superscript normal ♯ and g overTilde by their expressions in the above inequality, and using the estimates (Equation2.16) yields

StartLayout 1st Row with Label left-parenthesis 2.23 right-parenthesis EndLabel double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript 1 comma 2 Baseline less-than-or-equal-to upper C left-parenthesis StartFraction 1 Over q EndFraction comma double-vertical-bar upper Q double-vertical-bar Subscript 1 comma normal infinity Baseline right-parenthesis left-parenthesis double-vertical-bar h double-vertical-bar Subscript 2 Baseline plus double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript 2 Baseline plus StartAbsoluteValue nabla Subscript upper X Baseline f EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 Subscript Baseline plus StartAbsoluteValue g EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 Subscript Baseline right-parenthesis period EndLayout

Step 3. Further regularity. We show by finite induction on k that for all u element-of upper H Superscript 2 plus k Baseline left-parenthesis script upper S right-parenthesis , k equals 0 comma ellipsis comma m 0 minus 1 , one has

StartLayout 1st Row with Label left-parenthesis 2.24 right-parenthesis EndLabel 1st Column double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript 1 plus k comma 2 Baseline less-than-or-equal-to upper C left-parenthesis StartFraction 1 Over q EndFraction comma double-vertical-bar upper Q double-vertical-bar Subscript 1 plus k comma normal infinity Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column times 3rd Column left-parenthesis double-vertical-bar bold upper Q u double-vertical-bar Subscript k comma 2 Baseline plus double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript k comma 2 Baseline plus StartAbsoluteValue nabla Subscript upper X Baseline u Subscript vertical-bar Sub Subscript y equals 0 Subscript Baseline EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 plus k Subscript Baseline plus StartAbsoluteValue partial-differential Subscript n Superscript upper Q Baseline u Subscript vertical-bar Sub Subscript y equals negative 1 Subscript Baseline EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 plus k Subscript Baseline right-parenthesis period EndLayout

By Step 2, this assertion is true when k equals 0 . Let m 0 greater-than k greater-than-or-equal-to 1 and assume it is also true for 0 less-than-or-equal-to l less-than-or-equal-to k minus 1 . For all i equals 1 comma ellipsis comma d , we apply (Equation2.24) Subscript l with l equals k minus 1 to the function rho Subscript i comma h Baseline u :

StartLayout 1st Row 1st Column double-vertical-bar nabla Subscript upper X comma y Baseline rho Subscript i comma h Baseline u double-vertical-bar Subscript k comma 2 2nd Column less-than-or-equal-to 3rd Column upper C left-parenthesis StartFraction 1 Over q EndFraction comma double-vertical-bar upper Q double-vertical-bar Subscript k comma normal infinity Baseline right-parenthesis left-parenthesis double-vertical-bar bold upper Q rho Subscript i comma h Baseline u double-vertical-bar Subscript k minus 1 comma 2 Baseline plus double-vertical-bar nabla Subscript upper X comma y Baseline rho Subscript i comma h Baseline u double-vertical-bar Subscript k minus 1 comma 2 Baseline 2nd Row with Label left-parenthesis 2.25 right-parenthesis EndLabel 1st Column Blank 2nd Column Blank 3rd Column plus StartAbsoluteValue nabla Subscript x Baseline rho Subscript i comma h Baseline u Subscript vertical-bar Sub Subscript y equals 0 Subscript Baseline EndAbsoluteValue Subscript upper H Sub Superscript k minus 1 slash 2 Subscript Baseline plus vertical-bar partial-differential Subscript n Superscript upper Q Baseline left-parenthesis rho Subscript i comma h Baseline u right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript y equals negative 1 Baseline Subscript upper H Sub Superscript negative 1 slash 2 plus k Subscript Baseline right-parenthesis period EndLayout

We now estimate the four terms which appear in the r.h.s. of (Equation2.25). Since bold upper Q rho Subscript i comma h Baseline u equals rho Subscript i comma h Baseline left-parenthesis bold upper Q u right-parenthesis plus nabla Subscript upper X comma y Baseline dot left-parenthesis rho Subscript i comma h Baseline upper Q right-parenthesis nabla Subscript upper X comma y Baseline tau Subscript i comma h Baseline u , one has

StartLayout 1st Row with Label left-parenthesis 2.26 right-parenthesis EndLabel double-vertical-bar bold upper Q rho Subscript i comma h Baseline u double-vertical-bar Subscript k minus 1 comma 2 Baseline less-than-or-equal-to double-vertical-bar bold upper Q u double-vertical-bar Subscript k comma 2 Baseline plus upper C s t double-vertical-bar upper Q double-vertical-bar Subscript k plus 1 comma normal infinity Baseline double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript k comma 2 Baseline period EndLayout

The second and third terms of (Equation2.25) are very easily controlled. For the fourth one, we use the explicit expression of partial-differential Subscript n Superscript upper Q Baseline left-parenthesis rho Subscript i comma h Baseline u right-parenthesis , use the trace theorem, and proceed as for the derivation of (Equation2.26) to obtain

StartLayout 1st Row with Label left-parenthesis 2.27 right-parenthesis EndLabel StartAbsoluteValue partial-differential Subscript n Superscript upper Q Baseline left-parenthesis rho Subscript i comma h Baseline u right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript y equals negative 1 Baseline Subscript upper H Sub Superscript k minus 1 slash 2 Subscript Baseline less-than-or-equal-to EndAbsoluteValue partial-differential Subscript n Superscript upper Q Baseline u Subscript vertical-bar Sub Subscript y equals negative 1 Subscript Baseline vertical-bar Subscript upper H Sub Superscript k plus 1 slash 2 Subscript Baseline plus upper C s t double-vertical-bar upper Q double-vertical-bar Subscript k plus 1 comma normal infinity Baseline double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript k comma 2 Baseline period EndLayout

From (Equation2.25), (Equation2.26) and (Equation2.27) (and letting h right-arrow 0 ), it follows that double-vertical-bar partial-differential Subscript i Baseline u double-vertical-bar Subscript k plus 1 comma 2 is bounded from above by the right-hand side of (Equation2.24). In order to complete the proof, we still need an estimate of partial-differential Subscript y Superscript 2 Baseline u in upper H Superscript k Baseline left-parenthesis script upper S right-parenthesis . As in Step 2, such an estimate is obtained using (Equation2.21).

Step 4. Further regularity. We show by induction on k that (Equation2.24) can be generalized for k greater-than-or-equal-to m 0 as

StartLayout 1st Row 1st Column double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript 1 plus k comma 2 2nd Column less-than-or-equal-to 3rd Column upper C Subscript k Baseline times left-parenthesis double-vertical-bar bold upper Q u double-vertical-bar Subscript k comma 2 Baseline plus double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript k comma 2 Baseline plus double-vertical-bar upper Q 2 double-vertical-bar Subscript 1 plus k comma 2 Baseline double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript m 0 comma 2 Baseline 2nd Row with Label left-parenthesis 2.28 right-parenthesis EndLabel 1st Column Blank 2nd Column Blank 3rd Column plus StartAbsoluteValue nabla Subscript upper X Baseline u Subscript vertical-bar Sub Subscript y equals 0 Subscript Baseline EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 plus k Subscript Baseline plus StartAbsoluteValue partial-differential Subscript n Superscript upper Q Baseline u Subscript vertical-bar Sub Subscript y equals negative 1 Subscript Baseline EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 plus k Subscript Baseline right-parenthesis comma EndLayout

where upper C Subscript k Baseline equals upper C left-parenthesis StartFraction 1 Over q EndFraction comma double-vertical-bar upper Q 1 double-vertical-bar Subscript 1 plus k comma normal infinity Baseline comma double-vertical-bar upper Q 2 double-vertical-bar Subscript m 0 plus 1 comma normal infinity Baseline right-parenthesis .

The procedure is absolutely similar to Step 3. It is strictly unchanged until Eq. (Equation2.26) where we now use Moser’s tame estimates on products (e.g. Reference1):

Lemma 2.11

Let l element-of double-struck upper N and u comma v element-of upper H Superscript l Baseline left-parenthesis script upper S right-parenthesis intersection upper L Superscript normal infinity Baseline left-parenthesis script upper S right-parenthesis . Then one has

double-vertical-bar u v double-vertical-bar Subscript l comma 2 Baseline less-than-or-equal-to upper C s t left-parenthesis double-vertical-bar u double-vertical-bar Subscript l comma 2 Baseline double-vertical-bar v double-vertical-bar Subscript normal infinity Baseline plus double-vertical-bar v double-vertical-bar Subscript l comma 2 Baseline double-vertical-bar u double-vertical-bar Subscript normal infinity Baseline right-parenthesis period

This yields

StartLayout 1st Row 1st Column double-vertical-bar bold upper Q rho Subscript i comma h Baseline u double-vertical-bar Subscript k minus 1 comma 2 Baseline less-than-or-equal-to double-vertical-bar bold upper Q u double-vertical-bar Subscript k comma 2 Baseline 2nd Row 1st Column Blank 2nd Column plus 3rd Column upper C s t left-parenthesis left-parenthesis double-vertical-bar upper Q 1 double-vertical-bar Subscript k plus 1 comma normal infinity Baseline plus double-vertical-bar upper Q 2 double-vertical-bar Subscript 1 comma normal infinity Baseline right-parenthesis double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript k comma 2 Baseline plus double-vertical-bar upper Q 2 double-vertical-bar Subscript 1 plus k comma 2 Baseline double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript normal infinity Baseline right-parenthesis period EndLayout

Estimate (Equation2.27) is modified along the same lines and it follows from the Sobolev embedding upper H Superscript m 0 Baseline left-parenthesis script upper S right-parenthesis subset-of upper L Superscript normal infinity Baseline left-parenthesis script upper S right-parenthesis that double-vertical-bar partial-differential Subscript i Baseline u double-vertical-bar Subscript k plus 1 comma 2 is bounded from above by the right-hand side of (Equation2.28).

An estimate on partial-differential Subscript y Superscript 2 Baseline u in upper H Superscript k Baseline left-parenthesis script upper S right-parenthesis is then provided as before using (Equation2.21), which concludes the induction.

Step 5. Endgame. From the variational formulation of the problem, one easily gets the following lemma, whose proof we omit.

Lemma 2.12

Let h element-of upper L squared left-parenthesis script upper S right-parenthesis , f element-of upper H Superscript 1 slash 2 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis and g element-of upper H Superscript negative 1 slash 2 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis . If u element-of upper H squared left-parenthesis script upper S right-parenthesis solves the boundary value problem Equation2.13, then

double-vertical-bar nabla Subscript upper X comma y Baseline u double-vertical-bar Subscript 2 Baseline less-than-or-equal-to upper C left-parenthesis StartFraction 1 Over q EndFraction comma double-vertical-bar upper Q double-vertical-bar Subscript normal infinity Baseline right-parenthesis left-parenthesis double-vertical-bar h double-vertical-bar Subscript 2 Baseline plus StartAbsoluteValue nabla Subscript upper X Baseline f EndAbsoluteValue Subscript upper H Sub Superscript negative 1 slash 2 Subscript Baseline plus StartAbsoluteValue g EndAbsoluteValue Subscript upper H Sub Superscript negative 1 slash 2 Subscript Baseline right-parenthesis

and

double-vertical-bar u double-vertical-bar Subscript 1 comma 2 Baseline less-than-or-equal-to upper C left-parenthesis StartFraction 1 Over q EndFraction comma double-vertical-bar upper Q double-vertical-bar Subscript normal infinity Baseline right-parenthesis left-parenthesis double-vertical-bar h double-vertical-bar Subscript 2 Baseline plus StartAbsoluteValue f EndAbsoluteValue Subscript upper H Sub Superscript 1 slash 2 Subscript Baseline plus StartAbsoluteValue g EndAbsoluteValue Subscript upper H Sub Superscript negative 1 slash 2 Subscript Baseline right-parenthesis period

Iterating estimates (Equation2.24) and (Equation2.28) and using the lemma gives the theorem.

2.3. Regularizing diffeomorphisms

If u solves the boundary value problem (Equation2.3), then one can give precise estimates on u overTilde equals u ring upper S , owing to Prop. 2.7 and 2.8, and using Theorem 2.9. However, these estimates depend strongly on the diffeomorphism upper S chosen to straighten the fluid domain. The trivial diffeomorphism given in Remark 2.4 is not the best choice possible: in order to control the upper H Superscript k Baseline left-parenthesis script upper S right-parenthesis -norm of its Sobolev component, one needs to control the upper H Superscript k Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis -norm of the surface parameterization a . The next proposition shows that there exist “regularizing” diffeomorphisms for which a linear control of the upper H Superscript k minus 1 slash 2 -norm suffices.

Proposition 2.13

Let k element-of double-struck upper N , k minus one-half greater-than 1 plus StartFraction d Over 2 EndFraction , and let b element-of upper C Subscript b Superscript k Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis , a element-of upper H Superscript k minus 1 slash 2 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis . If there exists h 0 greater-than 0 such that a minus b greater-than-or-equal-to h 0 on double-struck upper R Superscript d , then there exists a diffeomorphism upper S of the form Equation2.6 such that

s is k -regular (with c 0 equals h 0 slash 2 );

one has partial-differential Subscript y overTilde Baseline s vertical-bar Subscript y overTilde equals 0 Baseline equals a minus b ;

one has s 1 equals minus b left-parenthesis upper X overTilde right-parenthesis y overTilde and double-vertical-bar s 2 double-vertical-bar Subscript k comma 2 Baseline less-than-or-equal-to upper C s t StartAbsoluteValue a EndAbsoluteValue Subscript upper H Sub Superscript negative 1 slash 2 plus k Subscript Baseline period

Remark 2.14

i.

The diffeomorphism upper S provided by this lemma is a perturbation of the trivial diffeomorphism given in Remark 2.4. The upper C Subscript b Superscript k -component s 2 remains unchanged, and the behavior at the surface is exactly the same. However, the Sobolev component s 2 is half a derivative smoother here than for the trivial diffeomorphism (where it has the smoothness of a ). This is why we say that the diffeomorphism is “regularizing”.

ii.

Note that if a element-of upper C left-parenthesis left-bracket 0 comma upper T right-bracket comma upper H Superscript negative 1 slash 2 plus k Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis right-parenthesis for some upper T greater-than 0 , and if the condition a minus b greater-than h 0 is satisfied uniformly in t element-of left-bracket 0 comma upper T right-bracket , then one has partial-differential Subscript t Baseline s equals partial-differential Subscript t Baseline s 2 and double-vertical-bar partial-differential Subscript t Baseline s 2 double-vertical-bar Subscript k comma 2 Baseline less-than-or-equal-to upper C left-parenthesis StartAbsoluteValue partial-differential Subscript t Baseline a left-parenthesis t right-parenthesis EndAbsoluteValue Subscript upper H Sub Subscript upper T Sub Superscript negative 1 slash 2 plus k Subscript Baseline right-parenthesis . This will be used in the proof of Prop. 3.19.

iii.

If a underbar element-of upper H Superscript k plus 3 slash 2 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis with k greater-than d slash 2 is such that a minus b greater-than-or-equal-to h 0 greater-than 0 on double-struck upper R Superscript d , then one can find a neighbourhood script upper U Subscript a underbar of a underbar in upper H Superscript k plus 3 slash 2 such that for all a element-of script upper U Subscript a underbar , a minus b greater-than-or-equal-to three-fourths h 0 . To each of these a element-of script upper U Subscript a underbar , one can associate a regularizing diffeomorphism upper S Subscript a Baseline left-parenthesis upper X comma y right-parenthesis equals left-parenthesis upper X comma s Subscript a Baseline left-parenthesis upper X comma y right-parenthesis right-parenthesis by Prop. 2.13. The proof shows that if script upper U Subscript a underbar is small enough, then the mapping a right-arrow from bar s Subscript a is affine. This mapping is therefore smooth and (using the notation of the proof) one can check that for all h element-of upper H Superscript k plus 3 slash 2 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis , one has d Subscript a underbar Baseline s dot h equals left-parenthesis y overTilde plus 1 right-parenthesis h Subscript lamda underbar for some lamda underbar greater-than 0 . Hence, d Subscript a underbar Baseline s dot h StartAbsoluteValue equals h comma d Subscript a underbar Baseline s dot h EndAbsoluteValue Subscript y overTilde equals 0 Baseline Subscript y overTilde equals negative 1 Baseline equals 0 comma partial-differential Subscript y overTilde Baseline d Subscript a underbar Baseline s dot h vertical-bar Subscript y overTilde equals 0 Baseline equals h period

Proof.

Note that the Jacobian of the mapping left-parenthesis upper X overTilde comma y overTilde right-parenthesis element-of script upper S right-arrow from bar left-parenthesis upper X overTilde comma s left-parenthesis upper X overTilde comma y overTilde right-parenthesis right-parenthesis element-of normal upper Omega is equal to StartAbsoluteValue partial-differential Subscript y overTilde Baseline s EndAbsoluteValue . Therefore, if s satisfies the properties stated in the lemma, upper S is indeed a diffeomorphism between script upper S and normal upper Omega .

Let s 1 element-of upper C Subscript b Superscript k Baseline left-parenthesis script upper S overbar right-parenthesis be given by

s 1 left-parenthesis upper X overTilde comma y overTilde right-parenthesis equals minus b left-parenthesis upper X overTilde right-parenthesis y overTilde comma for-all left-parenthesis upper X overTilde comma y overTilde right-parenthesis element-of script upper S semicolon

we look for s 2 element-of upper H Superscript k Baseline left-parenthesis script upper S right-parenthesis such that s colon equals s 1 plus s 2 satisfies

StartLayout 1st Row with Label left-parenthesis 2.29 right-parenthesis EndLabel partial-differential Subscript y overTilde Baseline s greater-than-or-equal-to StartFraction h 0 Over 2 EndFraction on script upper S overbar comma s 2 StartAbsoluteValue equals a comma partial-differential Subscript y overTilde Baseline s 2 EndAbsoluteValue Subscript y overTilde equals 0 Baseline Subscript y overTilde equals 0 Baseline equals a comma s 2 vertical-bar Subscript y overTilde equals negative 1 Baseline equals 0 period EndLayout

We construct such a mapping s 2 using a Poisson kernel extension of a . Let chi be a smooth, compactly supported, function defined on double-struck upper R and such that chi left-parenthesis 0 right-parenthesis equals 1 and chi prime left-parenthesis 0 right-parenthesis equals 0 . For any lamda greater-than 0 , and a element-of upper H Superscript k minus 1 slash 2 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis , we define a Subscript lamda Baseline element-of upper H Superscript k Baseline left-parenthesis script upper S right-parenthesis as a Subscript lamda Baseline left-parenthesis dot comma y overTilde right-parenthesis equals chi left-parenthesis lamda ModifyingAbove y With tilde mathematical left-angle upper D mathematical right-angle right-parenthesis a . From this definition it follows also that for all left-parenthesis upper X overTilde comma y overTilde right-parenthesis element-of script upper S overbar , one has

StartLayout 1st Row 1st Column StartAbsoluteValue partial-differential Subscript y overTilde Baseline a Subscript lamda Baseline left-parenthesis upper X overTilde comma y overTilde right-parenthesis EndAbsoluteValue 2nd Column equals 3rd Column StartAbsoluteValue lamda chi prime left-parenthesis lamda ModifyingAbove y With tilde mathematical left-angle upper D mathematical right-angle right-parenthesis mathematical left-angle upper D mathematical right-angle a EndAbsoluteValue less-than-or-equal-to lamda StartAbsoluteValue chi prime EndAbsoluteValue Subscript normal infinity Baseline integral Underscript double-struck upper R Superscript d Baseline Endscripts mathematical left-angle xi mathematical right-angle StartAbsoluteValue ModifyingAbove a With caret left-parenthesis xi right-parenthesis EndAbsoluteValue d xi 2nd Row with Label left-parenthesis 2.30 right-parenthesis EndLabel 1st Column Blank 2nd Column less-than-or-equal-to 3rd Column upper C s t lamda StartAbsoluteValue chi Superscript prime Baseline EndAbsoluteValue Subscript normal infinity Baseline StartAbsoluteValue a EndAbsoluteValue Subscript upper H Sub Superscript k minus 1 slash 2 Subscript Baseline comma EndLayout

since k minus 1 slash 2 greater-than 1 plus d slash 2