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Divergence-Measure Fields: Gauss-Green Formulas and Normal Traces

Gui-Qiang G. Chen
Monica Torres

Communicated by Notices Associate Editor Stephan Ramon Garcia

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It is hard to imagine how most fields of science could provide the stunning mathematical descriptions of their theories and results if the integration by parts formula did not exist. Indeed, integration by parts is an indispensable fundamental operation, which has been used across scientific theories to pass from global (integral) to local (differential) formulations of physical laws. Even though the integration by parts formula is commonly known as the Gauss-Green formula (or the divergence theorem, or Ostrogradsky’s theorem), its discovery and rigorous mathematical proof are the result of the combined efforts of many great mathematicians, starting back in the period when the calculus was invented by Newton and Leibniz in the 17th century.

The one-dimensional integration by parts formula for smooth functions was first discovered by Taylor (1715). The formula is a consequence of the Leibniz product rule and the Newton-Leibniz formula for the fundamental theorem of calculus.

The classical Gauss-Green formula for the multidimensional case is generally stated for vector fields and domains with boundaries. However, motivated by the physical solutions with discontinuity/singularity for partial differential equations (PDEs) and calculus of variations, such as nonlinear hyperbolic conservation laws and Euler-Lagrange equations, the following fundamental issue arises:

Does the Gauss-Green formula still hold for vector fields with discontinuity/singularity (such as divergence-measure fields) and domains with rough boundaries?

The objective of this paper is to provide an answer to this issue and to present a short historical review of the contributions by many mathematicians spanning more than two centuries, which have made the discovery of the Gauss-Green formula possible.

The Classical Gauss-Green Formula

The Gauss-Green formula was originally motivated in the analysis of fluids, electric and magnetic fields, and other problems in the sciences in order to establish the equivalence of integral and differential formulations of various physical laws. In particular, the derivations of the Euler equations and the Navier-Stokes equations in fluid dynamics and Gauss’s laws for the electronic and magnetic fields are based on the validity of the Gauss-Green formula and associated Stokes theorem. As an example, see Figure 1 for the derivation of the Euler equation for the conservation of mass in the smooth case.

Figure 1.

Conservation of mass: the rate of change of the total mass in an open set , , is equal to the total flux of mass across the boundary , , where is the density, is the velocity field, and the boundary value of is regarded as the normal trace of the vector field on . The Gauss-Green formula yields the Euler equation for the conservation of mass: in the smooth case.

The formula that is also later known as the divergence theorem was first discovered by Lagrange⁠Footnote1 J.-L. Lagrange, “Nouvelles recherches sur la nature et la propagation du son,” in Miscellanea Taurinensia (also known as: Mélanges de Turin), vol. 2, 1762, pp. 11–172. He treated a special case of the divergence theorem and transformed triple integrals into double integrals via integration by parts. in 1762 (see Figure 2), however, he did not provide a proof of the result. The theorem was later rediscovered by Gauss⁠Footnote2 C. F. Gauss, “Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata,” Commentationes Societatis Regiae Scientiarium Gottingensis Recentiores 2 (1813), 355–378. In this paper, a special case of the theorem was considered. in 1813 (see Figure 3) and Ostrogradsky⁠Footnote3 M. Ostrogradsky (presented on November 5, 1828; published in 1831), “Première note sur la théorie de la chaleur (First note on the theory of heat),” Mémoires de l’Académie Impériale des Sciences de St. Pétersbourg, Series 6, 1 (1831), 129–133. He stated and proved the divergence theorem in its Cartesian coordinate form. in 1828 (see Figure 4). Ostrogradsky’s method of proof was similar to the approach Gauss used. Independently, Green⁠Footnote4 G. Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, T. Wheelhouse, Nottingham, England, 1828. (see Figure 5) also rediscovered the divergence theorem in the two-dimensional case and published his result in 1828.

Figure 2.

Joseph-Louis Lagrange (January 25, 1736–April 10, 1813).

Figure 3.

Carl Friedrich Gauss (April 30, 1777–February 23, 1855).

Figure 4.

Mikhail Ostrogradsky (September 24, 1801–January 1, 1862).

Figure 5.

The title page of the original essay of George Green (July 14, 1793–May 31, 1841).

The divergence theorem in its vector form for the -dimensional case can be stated as

where is a vector field, is a bounded open set with piecewise smooth boundary, is the inner unit normal vector to , the boundary value of is regarded as the normal trace of the vector field on , and is the -dimensional Hausdorff measure (that is an extension of the surface area measure for -dimensional surfaces to general -dimensional boundaries ). The formulation of 1, where represents a physical vector quantity, is also the result of the efforts of many mathematicians⁠Footnote5 See C. H. Stolze, “A history of the divergence theorem,” in Historia Mathematica, 1978, pp. 437–442, and the references therein. including Gibbs, Heaviside, Poisson, Sarrus, Stokes, and Volterra. In conclusion, formula 1 is the result of more than two centuries of efforts by great mathematicians!

Gauss-Green Formulas and Traces for Lipschitz Vector Fields on Sets of Finite Perimeter

We first go back to the issue arisen earlier for extending the Gauss-Green formula to very rough sets. The development of geometric measure theory in the middle of the 20th century opened the door to the extension of the classical Gauss-Green formula over sets of finite perimeter (whose boundaries can be very rough and contain cusps, corners, among others; cf. Figure 6) for Lipschitz vector fields.

Indeed, we may consider the left side of 1 as a linear functional acting on vector fields . If is such that the functional: is bounded on , then the Riesz representation theorem implies that there exists a Radon measure such that

and the set is called a set of finite perimeter in . In this case, the Radon measure is actually the gradient of (strictly speaking, the gradient is in the distributional sense), where is the characteristic function of . A set of density of in is defined by

where represents the Lebesgue measure of any Lebesgue measurable set . Then is the measure-theoretic exterior of , while is the measure-theoretic interior of .

Figure 6.

is an open set of finite perimeter where cusp is a point of density .

Sets of finite perimeter can be quite subtle. For example, the countable union of open balls with centers on the rational points , , of the unit ball in and with radius is a set of finite perimeter. A set of finite perimeter may have a large set of cusps in the topological boundary (e.g., or ). The set can also have points in the boundary which belong to for . For example, the four corners of a square are points of density .

Even though the boundary of a set of finite perimeter can be very rough, De Giorgi’s structure theorem indicates that it has nice tangential properties so that there is a notion of measure-theoretic tangent plane. More rigorously, the topological boundary of contains an -rectifiable set, known as the reduced boundary of , denoted as , which can be covered by a countable union of surfaces, up to a set of -measure zero. More generally, Federer’s theorem states that , where is called the essential boundary. Note that and .

It can be shown that every has an inner unit normal vector and a tangent plane in the measure-theoretic sense, and 2 reduces to

This Gauss-Green formula for Lipschitz vector fields over sets of finite perimeter was proved by De Giorgi (1954–55) and Federer (1945, 1958) in a series of papers. See 1315 and the references therein.

Gauss-Green Formulas and Traces for Sobolev and Functions on Lipschitz Domains

It happens in many areas of analysis, such as PDEs and calculus of variations, that it is necessary to work with the functions that are not Lipschitz, but only in , . In many of these cases, the functions have weak derivatives that also belong to . That is, the corresponding in 4 is a Sobolev vector field. The necessary and sufficient conditions for the existence of traces (i.e., boundary values) of Sobolev functions defined on the boundary of the domain have been obtained so that 4 is a valid formula over open sets with Lipschitz boundary.

The development of the theory of Sobolev spaces has been fundamental in analysis. However, for many further applications, this theory is still not sufficient. For example, the characteristic function of a set of finite perimeter, , is not a Sobolev function. Physical solutions in gas dynamics involve shock waves and vortex sheets that are discontinuities with jumps. Thus, a larger space of functions, called the space of functions of bounded variation (), is necessary, which consists of all functions in whose derivatives are Radon measures.⁠Footnote6 The definition of the space is a generalization of the classical notion of the class of one-dimensional functions with finite total variation (TV) over an interval : where the supremum runs over the set of all partitions .

This space has compactness properties that allow, for instance, to show the existence of minimal surfaces and the well-posedness of solutions for hyperbolic conservation laws. Moreover, the Gauss-Green formula 4 is also valid for vector fields over Lipschitz domains. See 131420 and the references therein.

Divergence-Measure Fields and Hyperbolic Conservation Laws

A vector field , , is called a divergence-measure field if is a signed Radon measure with finite total variation in . Such vector fields form Banach spaces, denoted as for .

These spaces arise naturally in the field of hyperbolic conservation laws. Consider hyperbolic systems of conservation laws of the form:

where , , and . A prototype of such systems is the system of Euler equations for compressible fluids, which consists of the conservation equations of mass, momentum, and energy in continuum mechanics such as gas dynamics and elasticity. One of the main features of the hyperbolic system is that the speeds of propagation of solutions are finite; another feature is that, no matter how smooth a solution starts with initially, it generically develops singularity and becomes a discontinuous/singular solution. To single out physically relevant solutions requires the solutions of system 5 to fall within the following class of entropy solutions:

An entropy solution of system 5 is characterized by the Lax entropy inequality: for any entropy pair with being a convex function of ,

Here, a function is called an entropy of system 5 if there exists an entropy flux such that

Then is called an entropy pair of system 5.

Friedrichs-Lax (1971) observed that most systems of conservation laws that result from continuum mechanics are endowed with a globally defined, strictly convex entropy. In particular, for the compressible Euler equations in Lagrangian coordinates, is such an entropy, where is the physical thermodynamic entropy of the fluid. Then the Lax entropy inequality for the physical entropy is an exact statement of the second law of thermodynamics (cf. 311). Similar notions of entropy have also been used in many fields such as kinetic theory, statistical physics, ergodic theory, information theory, and stochastic analysis.

Indeed, the available existence theories show that the solutions of 5 are entropy solutions obeying the Lax entropy inequality 6. This implies that, for any entropy pair with being a convex function, there exists a nonnegative measure such that

Moreover, for any entropy solution , if the system is endowed with a strictly convex entropy, then, for any entropy pair (not necessarily convex for ), there exists a signed Radon measure such that 8 still holds (see 3). For these cases, is a vector field, as long as for some .

Equation 8 is one of the main motivations to develop a theory in Chen-Frid 56. In particular, one of the major issues is whether integration by parts can be performed in 6 to explore to the fullest extent possible all the information about the entropy solution . Thus, a concept of normal traces for fields is necessary to be developed. The existence of normal traces is also fundamental for initial-boundary value problems for hyperbolic systems 5 and for the analysis of structure and regularity of entropy solutions (see, e.g., 3561119).

Motivated by hyperbolic conservation laws, the interior and exterior normal traces need to be constructed as the limit of classical normal traces on one-sided smooth approximations of the domain. Then the surface of a shock wave or vortex sheet can be approximated with smooth surfaces to obtain the interior and exterior fluxes on the shock wave or vortex sheet.

Other important connections for the theory are the characterization of phase transitions (coexistent phases with discontinuities across the boundaries), the concept of stress, the notion of Cauchy flux, and the principle of balance law to accommodate the discontinuities and singularities in the continuum media in continuum mechanics, as discussed in Degiovanni-Marzocchi-Musesti 12, Šilhavý 1617, Chen-Torres-Ziemer 9, and Chen-Comi-Torres 4.

Gauss-Green Formulas and Normal Traces for Fields

We start with the following simple example.

Example 1.

Consider the vector field :

Then with in .

However, in an open half-disk , the previous Gauss-Green formulas do not apply. Indeed, since is highly oscillatory when which is not well-defined at , it is not clear how the normal trace on can be understood in the classical sense so that the equality between and holds. This example shows that a suitable notion of normal traces is required to be developed.

A generalization of 1 to fields and bounded sets with Lipschitz boundary was derived in Anzellotti 1 and Chen-Frid 5 by different approaches. A further generalization of 1 to fields and arbitrary bounded sets of finite perimeter, , was first obtained in Chen-Torres 8 and Šilhavý 16 independently; see also 910.

Theorem 1.

Let be a set of finite perimeter. Then

where the interior normal trace is a bounded function defined on the reduced boundary of (i.e., ), and is the measure-theoretic interior of as defined in 3.

One approach for the proof of 10 is based on a product rule for fields (see 8). Another approach in 9, following 5, is based on a new approximation theorem for sets of finite perimeter, which shows that the level sets of convolutions by the standard positive and symmetric mollifiers provide smooth approximations essentially from the interior (by choosing for ) and the exterior (for ). Thus, the interior normal trace is constructed as the limit of classical normal traces over the smooth approximations of . Since the level set (with a suitable fixed ) can intersect the measure-theoretic exterior of , a critical step for this approach is to show that converges to zero as . A key point for this proof is the fact that, if is a field, then the Radon measure is absolutely continuous with respect to , as first observed by Chen-Frid 5.

If , the above formulas apply, but the integration is not over the original representative consisting of the disk with radius removed, since is the open disk. In many applications including those in materials science, we may want to integrate on a domain with fractures or cracks; see also Figure 7. Since the cracks are part of the topological boundary and belong to the measure-theoretic interior , the formulas in 10 do not provide such information. In order to establish a Gauss-Green formula that includes such cases, we restrict to open sets of finite perimeter with . Therefore, can still have a large set of cusps or points of density (i.e., points belonging to ).

Figure 7.

is an open set with several fractures (dotted lines).

It has been shown in 7 that, if is a bounded open set satisfying

and , then there exists a family of sets such that in and such that, for any ,

where is well-defined in as the interior normal trace on .

This approximation result for the bounded open set can be accomplished by performing delicate covering arguments, especially by applying the Besicovitch theorem to a covering of . Moreover, 12 is a formula up to the boundary, since we do not assume that the domain of integration is compactly contained in the domain of . More general product rules for can be proved to weaken the regularity of ; see 47 and the references therein.

Gauss-Green Formulas and Normal Traces for Fields

For fields with , the situation becomes more delicate.

Example 2 (Whitney [Example 1, p. 100]⁠Footnote7 H. Whitney, Geometric Integration Theory, Princeton University Press, Princeton, 1957.).

Consider the vector field (see Figure 8):

Then for . If , it is observed that

where is the inner unit normal to the square. However, if and for any , then

In this sense, the equality is achieved on both sides of the formula.

Figure 8.

The vector field in Example 2.

Figure 9.

The vector field in Example 3.

Indeed, for a field with , is absolutely continuous with respect to for with , but not with respect to in general. This implies that the approach in 9 does not apply directly to obtain normal traces for fields for .

Then the following questions arise:

Can the previous formulas be proved in general for any and for any open set ?

Even though almost all the level sets of the distance function are sets of finite perimeter, can the formulas with smooth approximations of be obtained, in place of and ?

If has a Lipschitz boundary, do regular Lipschitz deformations of , as defined in Chen-Frid 56, exist?

The answer to all three questions is affirmative.

Theorem 2 (Chen-Comi-Torres 4).

Let be a bounded open set, and let for . Then, for any with for so that , there exists a sequence of bounded open sets with boundary such that , , and

where is the classical inner unit normal vector to .

For the open set with Lipschitz boundary, it can be proved that the deformations of obtained with the method of regularized distance are bi-Lipschitz. We can also employ an alternative construction by Nečas (1962) to obtain smooth approximations of a bounded open set with Lipschitz boundary in such a way that the deformation mapping to is bi-Lipschitz, and the Jacobians of the deformations, , converge to in as approaches zero (see 4, Theorem 8.19).⁠Footnote8 A simpler construction of smooth approximations can also be obtained when is a strongly Lipschitz domain—a stronger requirement than a general Lipschitz domain; see S. Hofmann, M. Mitrea, and M. Taylor, “Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains,” J. Geometric Anal. 17 (2007), 593–647. This shows that any bounded open set with Lipschitz boundary admits a regular Lipschitz deformation in the sense of Chen-Frid 56. Therefore, we can write more explicit Gauss-Green formulas for Lipschitz domains.

Theorem 3 (Chen-Comi-Torres 4).

If is an open set with Lipschitz boundary and