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On Bernoulli-Type Elliptic Free Boundary Problems

Daniela De Silva

Communicated by Notices Associate Editor Chikako Mese

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1. Free Boundary Problems

The modern theory of free boundary problems embodies the interplay between sophisticated mathematical analysis and the applied sciences. The discipline has been flourishing since the works of several illustrious mathematicians, such as Lewy, Stampacchia, Lax, Lions, in the early 1950s until the early 1970s. Examples of free boundaries arise in flame propagation, image reconstructions, jet flows, optimal stopping problems in financial mathematics, tumor growth, and in many other different contexts.

In a free boundary problem one or more unknown functions must be determined in different domains with a common boundary portion, and each function is governed in its domain by a set of state laws expressed as partial differential equations (PDEs). The domains are a priori unknown, and their boundaries are therefore called free boundaries and must be determined thanks to a number of free boundary conditions dictated by physical laws or other constraints governing the phase transition.

The origin of the modern mathematical discipline owes much to the Stefan problem, named after the physicist Stefan who toward the end of the 19th century studied ice formations in the polar seas. The problem aims to describe the temperature distribution in a homogeneous medium undergoing a phase change, for example ice melting into water: this is accomplished by coupling the heat equation in the water with a transmission condition, the free boundary condition, on the evolving boundary between its two phases. Interestingly, the classical Stefan problem can be reduced to the so-called parabolic (time-dependent) obstacle problem. The time-independent obstacle problem is a classical problem that belongs to a wide class of free boundary problems with variational structure. It consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed, and which is constrained to lie above a given obstacle under the action of gravity.

To better understand what a variational problem is, we pose the classical question of the celebrated theory of minimal surfaces: we seek for the function with given boundary value on , whose graph has the smallest area

among all possible such graphs. A simpler question, arising from the linearization of the area functional, consists in minimizing the so-called Dirichlet integral

Via the classical method of small perturbations, a PDE, known as the Euler–Lagrange equation, can be associated to such minimization problems. In the case of the Dirichlet integral, the corresponding equation is the Laplace equation .

One fundamental problem in the Calculus of Variations consists in studying critical points for an energy functional of the type

that is, solutions to its associated semilinear equation

where represents a given potential with minimum . Certain classes of potentials have been extensively studied in the literature. One such example is the double-well potential and the corresponding Allen–Cahn equation which appears in the theory of phase-transitions and minimal surfaces.

When the potential is not of class near one of its minimum points, then a minimizer can develop constant patches where it can take that value, and this leads to a free boundary problem. Two such potentials were investigated in great detail. The first one is the Lipschitz potential which corresponds to the classical obstacle problem with obstacle , and we refer the reader to the book of Petrosyan, Shahgholian, and Uraltseva for an introduction to this subject. The second one is the discontinuous potential with its associated Alt–Caffarelli energy, which is known as the Bernoulli free boundary problem AC81.

In this note we present an overview of the theory for minimizers of the Bernoulli problem, which has inspired much of the recent research in the area of free boundaries. This is by far not an exhaustive account, but it should offer an insight on some essential techniques and results in elliptic free boundary problems, and also on some of the interesting open questions in the theory. In the last section, we will describe some related problems and current research directions, and the content will be slightly more technical.

2. The Bernoulli Problem

2.1. Introduction

The classical Bernoulli (or Alt–Caffarelli) one-phase free boundary problem arises from the minimization of the energy functional

among all competitors , , with a given boundary data on . Here is a bounded domain in (say with Lipschitz boundary), and denotes the usual Sobolev space of functions with weak derivatives in as well (see for example GT83 for an account on basic properties of such spaces).

This minimization problem was first investigated systematically by Alt and Caffarelli in the pioneer work AC81, and originates in two-dimensional fluid dynamics. Indeed, in 2D the Euler equations for the motion of an incompressible, irrotational, inviscid fluid of constant density, and in absence of gravity, can be reinterpreted in the steady case in terms of a stream function . Under the assumption that the pressure on the surface of the liquid is constant, the system is reduced to a Bernoulli free boundary problem for .

Existence of minimizers with a given boundary data is easily obtained via the classical Direct Method of the Calculus of Variations, relying on the lower semicontinuity of the energy and the compactness properties of Sobolev Spaces.

In order to understand the free boundary problem that such a minimizer solves, let us perform a simple one-dimensional computation of the Euler–Lagrange equation associated to . Let (a “nice” function) minimize

among competitors with the same boundary data, and let us consider first a small perturbation of , , with a smooth (i.e., ) function compactly supported in , and small. Then,

and the minimization property gives (we drop the dependence of on the domain),

By integration by parts, and the fact that is arbitrary, we conclude that

Thus, is linear in its positive phase and we need to determine its slope. Say and for some . In order to determine we compare the energy of with the energy of the competitor , and obtain

hence

Similarly, by comparing the energy of with the energy of , we get that , and conclude that has slope 1 at a free boundary point. Heuristically, denoted by

the positivity set of , and by

the free boundary of , we expect that in the -dimensional case minimizers of will solve the following free boundary problem:

which is indeed the case. Clearly, this has to be understood in an appropriate weak sense, as a minimizer , and we have no information about the regularity of the set .

The notion of weak solutions to elliptic equations is well established, and it is known that a weak solution to the interior equation

is a classical solution to the equation (and in fact it is analytic in ), see for example GT83. The novelty here consists in the interpretation of the free boundary condition in a suitable weak sense. The simplest interpretation, which connects to our one-dimensional computation above, is provided by the notion of viscosity solution introduced by Caffarelli, see for example CS05. Precisely, let be a continuous function in (minimizers will turn out to be continuous), and let . We say that is a regular point for if there exists a tangent ball to at , fully contained either in the positive or zero set of . It turns out that if is harmonic in its positive set, then if is regular

in any nontangential region, for some . Here is the normal to at pointing toward the positivity set, and a nontangential region is contained in a cone with vertex at strictly included in the half-space . The free boundary condition in 2.2 can then be interpreted as the Neumann-type boundary condition

The main question then becomes: is it possible to prove that and are sufficiently smooth, so that the free boundary condition can be understood in the classical (pointwise) sense? We answer this question in the following section; we will focus on minimizers and we will not provide details of proofs but rather we will exhibit a successful roadmap to attack such questions, displaying tools which are useful in a variety of free boundary problems.

2.2. Regularity

The first question that we want to address is the regularity of a minimizer to . For simplicity, we focus on interior regularity and work with local minimizers. Precisely, is a (local) minimizer for in , if and for any domain and every function which coincides with in a neighborhood of we have

From the Euler–Lagrange equation 2.2, given the jump discontinuity in the gradient across the free boundary, we expect that the optimal regularity for is Lipschitz continuity. Indeed, the following theorem is proved in AC81. Here and in what follows, denotes a ball of radius centered at . When , we drop the dependence on . Also, constants depending only on dimension are called universal.

Proposition 2.1.

Let minimize in . Then . If and , then

for a universal constant .

The proof of Lipschitz continuity in this one-phase context is reasonably straightforward and relies on comparison with an explicit subsolution.

Moreover, minimizers satisfy the following so-called nondegeneracy property, which is an essential tool in analyzing the regularity of the free boundary AC81.

Proposition 2.2.

Let minimize in . Then, for ,

if , for some universal.

Lipschitz regularity and nondegeneracy yield positive density property for and at free boundary points, i.e, for ,

for small universal. In particular, the free boundary cannot exhibit a cusp-like behavior. Also, these properties are crucial in obtaining the following important compactness result for minimizers AC81.

Proposition 2.3.

Assume are minimizers to in and locally uniformly. Then is a minimizer to , locally in , and locally in the Hausdorff distance.

Analyzing the regularity of the free boundary is a much harder task (which exploits the propositions above) and in fact, it is the heart of the matter in most free boundary problems.

One first regularity result in the Geometric Measure Theory (GMT) sense follows from the observation that has finite perimeter. This can be seen formally from integration by parts as if solves 2.2

with the unit inner normal to . Notice that, the first term on the right-hand side is bounded in view of the Lipschitz continuity, while, by the free boundary condition, the second term represents .

In particular, one can define the reduced part of the free boundary, , and in view of the density estimates . For the notion of perimeter of a set and of reduced boundary we refer to Giusti Giu84.

The main purpose of this section will be to explain the strategy behind the following “state of the art” strong regularity result for the free boundary of a minimizer . It contains contributions of several authors AC81CJK04JS15KN77 and we will make that precise in what follows.

Theorem 2.4.

Let be a minimizer of . Then, is locally an analytic surface, except on a closed singular set of Hausdorff dimension , i.e.,

The strategy to prove Theorem 2.4 is based on a blow-up analysis. Let us assume for simplicity that and let us analyze the behavior of near 0. A crucial feature of this problem is its invariance under Lipschitz rescaling

that is is a minimizer if and only if is a minimizer. We thus consider a sequence of rescalings , with as , and in view of the Lipschitz continuity (Proposition 2.1) we obtain by Ascoli–Arzela (up to extracting a subsequence):

By nondegeneracy and is called a blow-up limit. Moreover, by the compactness, Proposition 2.3,

i.e., it minimizes over any compact set in . Finally, again by Proposition 2.3, in the appropriate sense. Therefore, properties of can be transferred to near . Our objective is now to better understand blow-up global minimizers. A fundamental property follows from the next result, a monotonicity formula due to Weiss Wei98:

Theorem 2.5.

If is a minimizer to in , then

is increasing in . Moreover is constant if and only if is homogeneous of degree .

Indeed the rescaling defined in 2.3 satisfies

Thus, if we obtain that:

and as

hence is homogeneous of degree 1. Thus, we need to classify global minimizers that are homogeneous of degree one. In view of 2.2, the obvious candidate is (up to rotation) . In fact, the following classification theorem holds.

Theorem 2.6 (Classification of Global Homogenous minimizers in low dimensions.).

In dimension the only global 1-homogeneous minimizer is (up to rotations) .

In dimension this was already obtained in AC81. The case was settled by Caffarelli, Jerison, and Kenig in CJK04, while Jerison and Savin established the result in dimension JS15. Furthermore, in collaboration with Jerison DSJ09, we provided an explicit example of a global homogeneous minimizer with a singularity at the origin, and we will discuss it in more detail in the next section. The question of whether Theorem 2.6 holds in dimension remains open.

Now, to settle the original question of the regularity of the free boundary, we need to investigate the following perturbative issue:

Assume is “close” to the optimal configuration , what can we deduce about the regularity ?

Since the optimal configuration has a free boundary which is a hyperplane, we refer to this question as a

“flatness implies regularity”

question. The answer is contained in the next theorem, first established in AC81 and then extended by Caffarelli to the context of free boundary viscosity solutions in his breakthrough trilogy in the late 80’s (for an account of such works see for example CS05).

Theorem 2.7.

Let be a minimizer to in . There exists a universal constant , such that if

and

then is in .

With this theorem combined with the blow up analysis, we can conclude that minimizers in dimension have free boundaries. This is the desired regularity as by classical elliptic theory we conclude also that is up to , hence the free boundary condition, on , can be understood in a pointwise sense. Moreover, once regularity of the free boundary is proven, then the celebrated work of Kindherleher and Nirenberg KN77 yields analyticity of the free boundary. This is achieved via a hodograph transform that changes the problem into a fixed boundary nonlinear elliptic problem. Once full regularity in dimension is established, a standard dimension reduction argument due to Federer (see for example Giu84) leads to the conclusive claim in Theorem 2.4. More precisely, the singular set has Hausdorff dimension , where is the first dimension in which a singular homogeneous minimizer occurs. In view of the result in DSJ09, we have that is either 5, 6, or 7.

In conclusion we have identified the key steps in proving (partial or full) regularity of the free boundary, and this strategy is applicable in a variety of free boundary problems:

Optimal regularity and compactness of minimizers;

Weiss-type monotonicity formula;

Classification of global homogeneous solutions;

Flatness implies regularity;

Dimension reduction & GMT techniques;

Higher regularity.

2.3. Singular minimizers

In this section we address the question of better understanding global nontrivial solutions to 2.2, which are homogeneous of degree 1. We focus on those which are generated by the so-called Lawson cones. For example, consider the (unique up to scalar multiple) positive harmonic function in the axis-symmetric cone

which is 0 on . is connected and its complement consists of two circular cones with central axis in the direction . For , there exists a unique such that is homogeneous of degree 1. The inner normal derivative of , , is homogeneous of degree zero and by rotational symmetry it is constant. Thus we can choose so that on . Then, after extending it to 0 outside , is clearly a global critical point to our Bernoulli problem 2.2. In dimension , it was ruled out in AC81 that is a minimizer. Later, Caffarelli, Jerison, and Kenig showed that the same is true in dimension CJK04. In collaboration with Jerison, we proved in DSJ09 that is a minimizer in dimension , exhibiting an explicit minimizing solution with a singular free boundary. The proof is based on comparison with explicit families of sub- and supersolutions, expressed in terms of certain hypergeometric series.

In Hon15, Hong studied other Lawson-type cones, showing that they are all unstable in dimension . Roughly speaking, stability in a region where are smooth, consists in requiring minimality with respect to the appropriate class of small perturbations supported in . It amounts to establish whether the following inequality holds true for all test functions :

Here denotes the normal to pointing toward the positive phase. Notice that, at a free boundary point,

where is the mean curvature of oriented toward the positive phase, thus the free boundary condition yields . While this stability analysis is supporting evidence that is the first dimension in which singularities occur, the question remains open.

In a recent work with Jerison and Shahgholian DSJS22, we observed that in fact the singularity of can be perturbed away. Precisely, we proved the following result. Here we use the notation: for

Theorem 2.8.

Let be a global minimizer to homogeneous of degree 1, with singular set . There exist global minimizers to 2.1, such that

(i)

, and

(ii)

are analytic hypersurfaces;

(iii)

for any , the ray , , intersects in a single point and the intersection is transverse; similarly, for any , the ray , , intersects in a single point and the intersection is transverse;

(iv)

the graphs foliate the half-space , i.e.,

(v)

if is a global minimizer to and (resp. ), then

for some , unless .

In fact, we can provide precise asymptotic developments for in terms of the homogeneous solutions of the so-called linearized equation associated to our problem. However, this is rather technical and beyond the scope of this note.

Finally, the question of whether or not blowups at singular points are unique remains open. A priori, it is possible that the free boundary around a singular point asymptotically approaches one singular cone at a certain set of small scales, but approaches a different singular cone at another set of scales. In ESV20, Engelstein, Spolaor, and Velichkov proved uniqueness of blowup and -regularity at points where one blowup has an isolated singularity.

2.4. Connection with minimal surfaces

The regularity theory described above parallels very well the theory for area minimizing sets (in the sense of De Giorgi), both in results and techniques. For a comprehensive treatment of such theory we refer the reader to Giusti’s monograph Giu84. In the context of the perimeter functional , Theorem 2.4 can be stated as follows.

Theorem 2.9.

Let be a Caccioppoli set that minimizes . Then, is locally an analytic surface, except on a singular set of Hausdorff dimension , i.e.,

Moreover, this result is sharp as, in dimension , the Simons cone is a global minimizing cone with a singularity at the origin. Also in this context the singularity can be perturbed away as proved by Hardt and Simon.

In dimension an actual connection between the two problems was unveiled by Traizet in Tra14. Precisely, he established a one-to-one correspondence between the following two classes of objects:

critical points of with a smooth free boundary and such that in ,

complete, embedded minimal surfaces in which are symmetric with respect to the horizontal plane and such that is a graph over the unbounded domain in the plane bounded by (minimal bigraph).

Further results for critical points which resemble those available in the theory of minimal surfaces are also known (due to Hauswirth, Helein and Pacard, Khavinson, Lundberg and Teodorescu, Jerison and Kamburov). However, for the sake of brevity, in this note we only focus on minimizers.

2.5. Two-phase problems

In several applications it is important to consider minimizers of , or of slightly more general classes of energy functionals, which change sign, for instance when modeling the flow of two liquids in jet and cavities. In the case of the basic functional , the corresponding Euler–Lagrange equation reads,

where