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

Gui-Qiang G. Chen and Monica Torres

It is hard to imagine how most fields of science could not exist. Indeed, integration by parts is an indispens- provide the stunning mathematical descriptions of their able fundamental operation, which has been used across theories and results if the integration by parts formula did scientific theories to pass from global (integral) to local (differential) formulations of physical laws. Even though Gui-Qiang G. Chen is Statutory Professor in the Analysis of Partial Differential the integration by parts formula is commonly known as Equations and Director of the Oxford Centre for Nonlinear Partial Differential the Gauss-Green formula (or the divergence theorem, or Os- Equations (OxPDE) at the Mathematical Institute of the University of Oxford, trogradsky’s theorem), its discovery and rigorous mathemat- where he is also a Professorial Fellow of Keble College. His email address is ical proof are the result of the combined efforts of many [email protected]. great mathematicians, starting back in the period when the Monica Torres is a professor of mathematics at Purdue University. Her email calculus was invented by Newton and Leibniz in the 17th address is [email protected]. century. Communicated by Notices Associate Editor Stephan Ramon Garcia. The one-dimensional integration by parts formula for For permission to reprint this article, please contact: smooth functions was first discovered by Taylor (1715). [email protected]. The formula is a consequence of the Leibniz product rule DOI: https://doi.org/10.1090/noti2336

1282 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 8 and the Newton-Leibniz formula for the fundamental the- The formula that is also later known as the divergence orem of calculus. theorem was first discovered by Lagrange1 in 1762 (see Fig- The classical Gauss-Green formula for the multidimen- ure 2), however, he did not provide a proof of the result. sional case is generally stated for 퐶1 vector fields and do- The theorem was later rediscovered by Gauss2 in 1813 (see mains with 퐶1 boundaries. However, motivated by the Figure 3) and Ostrogradsky3 in 1828 (see Figure 4). Os- physical solutions with discontinuity/singularity for par- trogradsky’s method of proof was similar to the approach tial differential equations (PDEs) and calculus of varia- Gauss used. Independently, Green4 (see Figure 5) also re- tions, such as nonlinear hyperbolic conservation laws and discovered the divergence theorem in the two-dimensional Euler-Lagrange equations, the following fundamental is- case and published his result in 1828. sue 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 con- tributions by many mathematicians spanning more than two centuries, which have made the discovery of the Gauss- Green formula possible. The Classical Gauss-Green Formula Figure 2. Joseph-Louis Lagrange (January 25, 1736–April 10, 1813). 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 equiva- lence 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 3. Carl Friedrich Gauss (April 30, 1777–February 23, 1855).

1J.-L. Lagrange, “Nouvelles recherches sur la nature et la propagation du son,” in Miscellanea Taurinensia (also known as: Mélangesde 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. 2C. F. Gauss, “Theoria attractionis corporum sphaeroidicorum ellipticorum ho- mogeneorum methodo nova tractata,” Commentationes Societatis Regiae Scientiarium Gottingensis Recentiores 2 (1813), 355–378. In this paper, a special case of the theorem was considered. Figure 1. Conservation of mass: the rate of change of the total 3 d M. Ostrogradsky (presented on November 5, 1828; published in 1831), “Pre- mass in an open set 퐸, ∫ 휌(푡, 푥) d푥, is equal to the total flux d푡 퐸 mière note sur la théorie de la chaleur (First note on the theory of heat),” 푛−1 of mass across the boundary 휕퐸, ∫휕퐸 (휌퐯) ⋅ 휈 d ℋ , where 휌 is Mémoiresde l’AcadémieImpérialedes Sciences de St. Pétersbourg, Se- the density, 퐯 is the velocity field, and the boundary value of ries 6, 1 (1831), 129–133. He stated and proved the divergence theorem in its (휌퐯) ⋅ 휈 is regarded as the normal trace of the vector field 휌퐯 on Cartesian coordinate form. 휕퐸. The Gauss-Green formula yields the Euler equation for 4G. Green, An Essay on the Application of Mathematical Analysis to the the conservation of mass: 휌푡 + div(휌퐯) = 0 in the smooth case. Theories of Electricity and Magnetism, T. Wheelhouse, Nottingham, Eng- land, 1828.

SEPTEMBER 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1283 The divergence theorem in its vector form for the 푛-dimensional case (푛 ≥ 2) can be stated as

∫ div 푭 d 푦 = − ∫ 푭 ⋅ 휈 d ℋ푛−1, (1) 퐸 휕퐸 where 푭 is a 퐶1 vector field, 퐸 is a bounded open set with piecewise smooth boundary, 휈 is the inner unit normal vector to 퐸, the boundary value of 푭 ⋅ 휈 is regarded as the normal trace of the vector field 푭 on 휕퐸, and ℋ푛−1 is the (푛 − 1)-dimensional Hausdorff measure (that is an extension of the surface area measure for 2-dimensional surfaces to general (푛−1)-dimensional boundaries 휕퐸). The formu- lation of (1), where 푭 represents a physical vector quantity, is also the result of the efforts of many mathematicians5 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 Figure 4. Mikhail Ostrogradsky (September 24, 1801–January Vector Fields on Sets of Finite Perimeter 1, 1862). 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 lin- 1 푛 ear functional acting on vector fields 푭 ∈ 퐶푐 (ℝ ). If 퐸 is such that the functional: 푭 → ∫퐸 div 푭 d 푦 is bounded on 푛 퐶푐(ℝ ), then the Riesz representation theorem implies that there exists a Radon measure 휇퐸 such that

1 푛 ∫ div 푭 d 푦 = ∫ 푭 ⋅ d 휇퐸 for all 푭 ∈ 퐶푐 (ℝ ), (2) 퐸 ℝ푛 and the set 퐸 is called a set of finite perimeter in ℝ푛. In this case, the Radon measure 휇퐸 is actually the gradient of −휒퐸 (strictly speaking, the gradient is in the distributional sense), where 휒퐸 is the characteristic function of 퐸. A set of density 훼 ∈ [0, 1] of 퐸 in ℝ푛 is defined by |퐵 (푦) ∩ 퐸| 퐸훼 ∶= {푦 ∈ ℝ푛 ∶ lim 푟 = 훼}, (3) 푟→0 |퐵푟(푦)| where |퐵| represents the Lebesgue measure of any Lebesgue measurable set 퐵. Then 퐸0 is the measure-theoretic exterior of 퐸, while 퐸1 is the measure-theoretic interior of 퐸. Sets of finite perimeter can be quite subtle. For example, the countable union of open balls with centers on the 푛 rational points 푦푘, 푘 = 1, 2, … , of the unit ball in ℝ and with radius 2−푘 is a set of finite perimeter. A set of finite Figure 5. The title page of the original essay of George Green perimeter may have a large set of cusps in the topological (July 14, 1793–May 31, 1841). boundary (e.g., ℋ푛−1(퐸0 ∩ 휕퐸) > 0 or ℋ푛−1(퐸1 ∩ 휕퐸) > 0).

5See C. H. Stolze, “A history of the divergence theorem,” in Historia Mathe- matica, 1978, pp. 437–442, and the references therein.

1284 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 8 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 퐸 of finite perimeter, 휒퐸, 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 (퐵푉), is necessary, which consists of all functions in 퐿1 whose derivatives are Radon measures.6 This space has compactness properties that allow, for instance, to show the existence of minimal surfaces and the well-posedness of 퐵푉 solutions for hyperbolic conservation laws. Moreover, the Gauss-Green formula (4) is Figure 6. 퐸 is an open set of ﬁnite perimeter where cusp 푃 is a also valid for 퐵푉 vector fields over Lipschitz domains. See point of density 0. [13, 14, 20] and the references therein. The set 퐸 can also have points in the boundary which belong to 퐸훼 for 훼 ∉ {0, 1}. For example, the four corners of Divergence-Measure Fields and Hyperbolic 1 Conservation Laws a square are points of density . 4 A vector field 푭 ∈ 퐿푝(Ω), 1 ≤ 푝 ≤ ∞, is called a divergence- Even though the boundary of a set 퐸 of finite perime- measure field if div 푭 is a signed Radon measure with finite ter can be very rough, De Giorgi’s structure theorem indi- total variation in Ω. Such vector fields form Banach spaces, cates that it has nice tangential properties so that there is 푝 denoted as 풟ℳ (Ω) for 1 ≤ 푝 ≤ ∞. a notion of measure-theoretic tangent plane. More rigor- These spaces arise naturally in the field of hyperbolic ously, the topological boundary 휕퐸 of 퐸 contains an (푛−1)- conservation laws. Consider hyperbolic systems of conser- rectifiable set, known as the reduced boundary of 퐸, de- vation laws of the form: noted as 휕∗퐸, which can be covered by a countable union 1 푛−1 ⊤ 푛 푚 of 퐶 surfaces, up to a set of ℋ -measure zero. More gen- 퐮푡 + ∇푥 ⋅ 퐟(퐮) = 0 for 퐮 = (푢1, ..., 푢푚) ∶ ℝ → ℝ , (5) 푛−1 푠 ∗ erally, Federer’s theorem states that ℋ (휕 퐸 ⧵ 휕 퐸) = 0, 푛 푑 푠 푛 0 1 where (푡, 푥) ∈ ℝ+ ∶= [0, ∞) × ℝ , 푛 ∶= 푑 + 1, 퐟 = where 휕 퐸 = ℝ ⧵ (퐸 ∪ 퐸 ) is called the essential boundary. 푚 푚 1 (퐟1, 퐟2, ..., 퐟푑), and 퐟푗 ∶ ℝ → ℝ , 푗 = 1, … , 푑. A prototype ∗ ∗ 푠 Note that 휕 퐸 ⊂ 퐸 2 and 휕 퐸 ⊂ 휕 퐸 ⊂ 휕퐸. of such systems is the system of Euler equations for com- ∗ It can be shown that every 푦 ∈ 휕 퐸 has an inner unit pressible fluids, which consists of the conservation equa- normal vector 휈퐸(푦) and a tangent plane in the measure- tions of mass, momentum, and energy in continuum me- theoretic sense, and (2) reduces to chanics such as gas dynamics and elasticity. One of the main features of the hyperbolic system is that the speeds of 푛−1 ∫ div 푭 d 푦 = − ∫ 푭(푦) ⋅ 휈퐸(푦) d ℋ (푦). (4) propagation of solutions are finite; another feature is that, ∗ 퐸 휕 퐸 no matter how smooth a solution starts with initially, it This Gauss-Green formula for Lipschitz vector fields 푭 over generically develops singularity and becomes a discontin- sets of finite perimeter was proved by De Giorgi (1954–55) uous/singular solution. To single out physically relevant and Federer (1945, 1958) in a series of papers. See [13,15] solutions requires the solutions of system (5) to fall within and the references therein. the following class of entropy solutions: Gauss-Green Formulas and Traces for Sobolev An entropy solution 퐮(푡, 푥) of system (5) is character- and 퐵푉 Functions on Lipschitz Domains ized by the Lax entropy inequality: for any entropy pair It happens in many areas of analysis, such as PDEs and (휂, 퐪) with 휂 being a convex function of 퐮, calculus of variations, that it is necessary to work with the 휂(퐮(푡, 푥)) + ∇ ⋅ 퐪(퐮(푡, 푥)) ≤ 0 in the sense of distributions. 푝 푡 푥 functions that are not Lipschitz, but only in 퐿 , 1 ≤ 푝 ≤ ∞. (6) In many of these cases, the functions have weak derivatives Here, a function 휂 ∈ 퐶1(ℝ푚, ℝ) is called an entropy of that also belong to 퐿푝. That is, the corresponding 푭 in (4) is a Sobolev vector field. The necessary and sufficient con- 6 ditions for the existence of traces (i.e., boundary values) of The definition of the 퐵푉 space is a generalization of the classical notion of the class of one-dimensional functions 푓(푥) with finite total variation (TV) over an Sobolev functions defined on the boundary of the domain 푏 interval [푎, 푏] ⊂ ℝ: 푇푉푎 (푓) = ∑풫 |푓(푥푗+1) − 푓(푥푗)|, where the supremum have been obtained so that (4) is a valid formula over open runs over the set of all partitions 풫 = {(푥0, … , 푥퐽 ) ∶ 푎 ≤ 푥0 ≤ ⋯ ≤ 푥퐽 ≤ sets with Lipschitz boundary. 푏, 1 ≤ 퐽 < ∞}.

SEPTEMBER 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1285 system (5) if there exists an entropy flux 퐪 = (푞1, … , 푞푑) ∈ discussed in Degiovanni-Marzocchi-Musesti [12], Silhav´yˇ 퐶1(ℝ푚,ℝ푑) such that [16, 17], Chen-Torres-Ziemer [9], and Chen-Comi-Torres [4]. ∇푞푗(퐮) = ∇휂(퐮)∇퐟푗(퐮) for 푗 = 1, 2, ..., 푑. (7) Then (휂, 퐪)(퐮) is called an entropy pair of system (5). Gauss-Green Formulas and Normal Traces Friedrichs-Lax (1971) observed that most systems of for 풟ℳ∞ Fields conservation laws that result from continuum mechanics We start with the following simple example. are endowed with a globally defined, strictly convex en- 2 tropy. In particular, for the compressible Euler equations Example 1. Consider the vector field 푭 ∶ Ω = ℝ ∩ {푦1 > 2 in Lagrangian coordinates, 휂 = −푆 is such an entropy, 0} → ℝ : 푆 1 where is the physical thermodynamic entropy of the 푭(푦1, 푦2) = (0, sin( )). (9) fluid. Then the Lax entropy inequality for the physical en- 푦1 ∞ tropy 휂 = −푆 is an exact statement of the second law of Then 푭 ∈ 풟ℳ (Ω) with div 푭 = 0 in Ω. thermodynamics (cf. [3, 11]). Similar notions of entropy However, in an open half-disk 퐸 = {(푦1, 푦2) ∶ 푦1 > 2 2 have also been used in many fields such as kinetic theory, 0, 푦1 + 푦2 < 1} ⊂ Ω, the previous Gauss-Green formulas statistical physics, ergodic theory, information theory, and do not apply. Indeed, since 푭(푦1, 푦2) is highly oscillatory stochastic analysis. when 푦1 → 0 which is not well-defined at 푦1 = 0, it is not Indeed, the available existence theories show that the clear how the normal trace 푭 ⋅ 휈 on {푦1 = 0} can be under- solutions of (5) are entropy solutions obeying the Lax en- stood in the classical sense so that the equality between 1 1 tropy inequality (6). This implies that, for any entropy ∫퐸 div 푭 dℋ = 0 and ∫휕퐸 푭 ⋅ 휈 dℋ holds. This example pair (휂, 퐪) with 휂 being a convex function, there exists a shows that a suitable notion of normal traces is required 푛 nonnegative measure 휇휂 ∈ ℳ(ℝ+) such that to be developed. ∞ div(푡,푥)(휂(퐮(푡, 푥)), 퐪(퐮(푡, 푥))) = −휇휂. (8) A generalization of (1) to 풟ℳ (Ω) fields and bounded Moreover, for any 퐿∞ entropy solution 퐮, if the system is sets with Lipschitz boundary was derived in Anzellotti [1] 2 and Chen-Frid [5] by different approaches. A further gener- endowed with a strictly convex entropy, then, for any 퐶 ∞ entropy pair (휂, 퐪) (not necessarily convex for 휂), there ex- alization of (1) to 풟ℳ (Ω) fields and arbitrary bounded ists a signed Radon measure 휇 ∈ ℳ(ℝ푛 ) such that (8) sets of finite perimeter, 퐸 ⋐ Ω, was first obtained in Chen- 휂 + ˇ still holds (see [3]). For these cases, (휂(퐮), 퐪(퐮))(푡, 푥) is Torres [8] and Silhav´y[16] independently; see also [9, 10]. 푝 푛 a 풟ℳ (ℝ+) vector field, as long as (휂(퐮), 퐪(퐮))(푡, 푥) ∈ Theorem 1. Let 퐸 be a set of finite perimeter. Then 푝 푛 푛 퐿 (ℝ+,ℝ ) for some 푝 ∈ [1, ∞]. Equation (8) is one of the main motivations to develop ∫ 휙 d div 푭 + ∫ 푭 ⋅ ∇휙 d 푦 a 풟ℳ theory in Chen-Frid [5, 6]. In particular, one of 퐸1 퐸1 the major issues is whether integration by parts can be per- 푛−1 1 푛 formed in (6) to explore to the fullest extent possible all = − ∫ 휙 픉i ⋅ 휈퐸 d ℋ for every 휙 ∈ 퐶푐 (ℝ ), (10) ∗ the information about the entropy solution 퐮. Thus, a con- 휕 퐸 cept of normal traces for 풟ℳ fields 푭 is necessary to be where the interior normal trace 픉i ⋅ 휈퐸 is a bounded func- developed. The existence of normal traces is also funda- tion defined on the reduced boundary of 퐸 (i.e., 픉i ⋅ 휈퐸 ∈ ∞ ∗ 푛−1 1 mental for initial-boundary value problems for hyperbolic 퐿 (휕 퐸; ℋ )), and 퐸 is the measure-theoretic interior of systems (5) and for the analysis of structure and regularity 퐸 as defined in (3). of entropy solutions 퐮 (see, e.g., [3, 5, 6, 11,19]). One approach for the proof of (10) is based on a prod- Motivated by hyperbolic conservation laws, the interior uct rule for 풟ℳ∞ fields (see [8]). Another approach in and exterior normal traces need to be constructed as the [9], following [5], is based on a new approximation theo- limit of classical normal traces on one-sided smooth ap- rem for sets of finite perimeter, which shows that the level proximations of the domain. Then the surface of a shock sets of convolutions 푤푘 ∶= 휒퐸 ∗ 휚푘 by the standard posi- wave or vortex sheet can be approximated with smooth sur- tive and symmetric mollifiers provide smooth approxima- faces to obtain the interior and exterior fluxes on the shock −1 tions essentially from the interior (by choosing 푤푘 (푡) for wave or vortex sheet. 1 1 < 푡 < 1) and the exterior (for 0 < 푡 < ). Thus, the inte- Other important connections for the 풟ℳ theory are the 2 2 characterization of phase transitions (coexistent phases rior normal trace 픉i ⋅ 휈퐸 is constructed as the limit of classical normal traces over the smooth approximations of 퐸. with discontinuities across the boundaries), the concept of 1 Since the level set 푤−1(푡) (with a suitable fixed 0 < 푡 < ) stress, the notion of Cauchy flux, and the principle of bal- 푘 2 ance law to accommodate the discontinuities and singular- can intersect the measure-theoretic exterior 퐸0 of 퐸, a criti- 푛−1 −1 0 ities in the continuum media in continuum mechanics, as cal step for this approach is to show that ℋ (푤푘 (푡)∩퐸 )

1286 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 8 converges to zero as 푘 → ∞. A key point for this proof is covering of 휕퐸 ∩ 퐸0. Moreover, (12) is a formula up to the ∞ the fact that, if 푭 is a 풟ℳ field, then the Radon measure boundary, since we do not assume that the domain of inte- | div 푭| is absolutely continuous with respect to ℋ푛−1, as gration is compactly contained in the domain of 푭. More first observed by Chen-Frid [5]. general product rules for div(휙푭) can be proved to weaken 2 If 퐸 = {푦 ∈ ℝ ∶ |푦| < 1} ⧵ {푦1 > 0, 푦2 = 0}, the the regularity of 휙; see [4, 7] and the references therein. above formulas apply, but the integration is not over the Gauss-Green Formulas and Normal Traces original representative consisting of the disk with radius 푝 1 2 풟ℳ {푦1 > 0, 푦2 = 0} removed, since 퐸 = {푦 ∈ ℝ ∶ |푦| < 1} for Fields 푝 is the open disk. In many applications including those in For 풟ℳ fields with 1 ≤ 푝 < ∞, the situation becomes materials science, we may want to integrate on a domain more delicate. with fractures or cracks; see also Figure 7. Since the cracks Example 2 (Whitney [Example 1, p. 100]7). Consider the are part of the topological boundary and belong to the vector field 푭 ∶ ℝ2 ⧵ {(0, 0)} → ℝ2 (see Figure 8): measure-theoretic interior 퐸1, the formulas in (10) do not (푦 , 푦 ) provide such information. In order to establish a Gauss- 1 2 푭(푦1, 푦2) = 2 2 . (13) Green formula that includes such cases, we restrict to open 푦1 + 푦2 퐸 ℋ푛−1(휕퐸⧵퐸0) < ∞ 푝 2 2 sets of finite perimeter with . Therefore, Then 푭 ∈ 풟ℳloc(ℝ ) for 1 ≤ 푝 < 2. If 퐸 = (0, 1) , it is 휕퐸 can still have a large set of cusps or points of density 0 observed that (i.e., points belonging to 퐸0). 1 휋 0 = div 푭(퐸) ≠ − ∫ 푭 ⋅ 휈퐸 d ℋ = , 휕퐸 2

where 휈퐸 is the inner unit normal to the square. However, 휀 푛 if 퐸휀 ∶= {푦 ∈ 퐸 ∶ dist(푦, 휕퐸) > 휀} and 퐸 ∶= {푦 ∈ ℝ ∶ dist(푦, 퐸) < 휀} for any 휀 > 0, then

1 0 = div 푭(퐸) = − lim ∫ 푭 ⋅ 휈퐸 d ℋ , 휀→0 휀 휕퐸휀

1 2휋 = div 푭(퐸) = − lim ∫ 푭 ⋅ 휈퐸휀 d ℋ . 휀→0 휕퐸휀 In this sense, the equality is achieved on both sides of the formula. Indeed, for a 풟ℳ푝 field with 푝 ≠ ∞, | div 푭| is abso- ′ lutely continuous with respect to ℋ푛−푝 for 푝′ > 1 with 1 1 + = 1, but not with respect to ℋ푛−1 in general. This 푝′ 푝 Figure 7. 퐸 is an open set with several fractures (dotted lines). implies that the approach in [9] does not apply directly to obtain normal traces for 풟ℳ푝 fields for 푝 ≠ ∞. It has been shown in [7] that, if 퐸 is a bounded open Then the following questions arise: set satisfying • Can the previous formulas be proved in general for any 푝 ℋ푛−1(휕퐸 ⧵ 퐸0) < ∞ (11) 푭 ∈ 풟ℳ (Ω) and for any open set 퐸 ⊂ Ω? ∞ • Even though almost all the level sets of the distance and 푭 ∈ 풟ℳ (퐸), then there exists a family of sets 퐸푘 ⋐ 퐸 1 푛−1 ∗ function are sets of finite perimeter, can the formulas such that 퐸푘 → 퐸 in 퐿 and sup푘 ℋ (휕 퐸푘) < ∞ such that, for any 휙 ∈ 퐶1(ℝ푛), with smooth approximations of 퐸 be obtained, in place 푐 휀 of 퐸휀 and 퐸 ? 푛−1 ∫ 휙 d div 푭 + ∫ 푭 ⋅ ∇휙 d 푦 =− lim ∫ 휙푭 ⋅ 휈퐸 d ℋ • If 퐸 has a Lipschitz boundary, do regular Lipschitz de- 푘→∞ 푘 퐸 퐸 휕퐸푘 formations of 퐸, as defined in Chen-Frid [5,6], exist?

푛−1 The answer to all three questions is affirmative. =∫ 휙 픉 ⋅ 휈퐸 d ℋ , 휕퐸⧵퐸0 Theorem 2 (Chen-Comi-Torres [4]). Let 퐸 ⊂ Ω be a (12) bounded open set, and let 푭 ∈ 풟ℳ푝(Ω) for 1 ≤ 푝 ≤ ∞. ∞ 0 푛−1 0 ∞ 푝′ 푛 where 픉 ⋅ 휈퐸 is well-defined in 퐿 (휕퐸 ⧵ 퐸 ;ℋ ) as the Then, for any 휙 ∈ (퐶 ∩ 퐿 )(Ω) with ∇휙 ∈ 퐿 (Ω; ℝ ) for 1 1 interior normal trace on 휕퐸 ⧵ 퐸0. 푝′ ≥ 1 + = 1 so that ′ , there exists a sequence of bounded This approximation result for the bounded open set 퐸 푝 푝 can be accomplished by performing delicate covering argu- 7H. Whitney, Geometric Integration Theory, Princeton University Press, ments, especially by applying the Besicovitch theorem to a Princeton, 1957.

SEPTEMBER 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1287 퐿1(휕퐸) as 휀 approaches zero (see [4, Theorem 8.19]).8 This shows that any bounded open set with Lipschitz boundary admits a regular Lipschitz deformation in the sense of Chen- Frid [5, 6]. Therefore, we can write more explicit Gauss- Green formulas for Lipschitz domains. Theorem 3 (Chen-Comi-Torres [4]). If 퐸 ⋐ Ω is an open set with Lipschitz boundary and 푭 ∈ 풟ℳ푝(Ω) for 1 ≤ 푝 ≤ ∞, ′ then, for every 휙 ∈ 퐶0(Ω) with ∇휙 ∈ 퐿푝 (Ω; ℝ푛), there exists a set 풩 ⊂ ℝ of Lebesgue measure zero such that, for every nonnegative sequence {휀푘} ⊄ 풩 satisfying 휀푘 → 0,

∫ 휙 d div 푭 + ∫ 푭 ⋅ ∇휙 d 푦 퐸 퐸

∇휌 휕퐸 푛−1 = − lim ∫ (휙푭 ⋅ )(Ψ휀 (푦))퐽 (Ψ휀 )(푦) d ℋ (푦), 푘→∞ 푘 푘 휕퐸 |∇휌| where 휌(푦) ∈ 퐶2(ℝ푛 ⧵휕퐸)∩Lip(ℝn) is a regularized distance9 (푦1,푦2) 휌(푦) 푑(푦) Figure 8. The vector ﬁeld 푭(푦1, 푦2) = in Example 2. 푦2+푦2 for 퐸, so that the ratio functions and are positive and 1 2 푑(푦) 휌(푦) uniformly bounded for all 푦 ∈ ℝ푛 ⧵휕푈 for 푑(푦) to be dist(푦, 휕퐸) for 푦 ∈ 퐸 and −dist(푦, 휕퐸) for 푦 ∉ 퐸. The question now arises as to whether the limit can be realized on the right-hand side of the previous formulas as an integral on 휕퐸. In general, this is not possible. However, in some cases, it is possible to represent the normal trace with a measure supported on 휕퐸. In order to see this, for 푭 ∈ 풟ℳ푝(Ω), 1 ≤ 푝 ≤ ∞, and a bounded Borel set 퐸 ⊂ Ω, we follow [1,6,16] to define the normal trace distribution of 푭 on 휕퐸 as

⟨푭 ⋅ 휈, 휙⟩휕퐸

푛 ∶= ∫ 휙 d div 푭 + ∫ 푭 ⋅ ∇휙 d 푦 for any 휙 ∈ Lip푐(ℝ ). 퐸 퐸 (15)

(−푦 ,푦 ) The formula presented above shows that the trace distri- Figure 9. The vector ﬁeld 푭(푦 , 푦 ) = 2 1 in Example 3. 1 2 2 2 휕퐸 {휙 ∈ 푦1+푦 bution on can be extended to a functional on 2 ′ (퐶0 ∩ 퐿∞)(Ω) ∶ ∇휙 ∈ 퐿푝 (Ω, ℝ푛)} so that we can always ∞ represent the normal trace distribution as the limit of clas- open sets 퐸푘 with 퐶 boundary such that 퐸푘 ⋐ 퐸, ⋃푘 퐸푘 = 퐸, and sical normal traces on smooth approximations of 퐸. Then the question is whether there exists a Radon measure 휇 con- 푛−1 ∫ 휙 d div 푭 + ∫ 푭 ⋅ ∇휙 d 푦 = − lim ∫ 휙푭 ⋅ 휈퐸 d ℋ , centrated on 휕퐸 such that ⟨푭 ⋅ 휈, 휙⟩휕퐸 = ∫휕퐸 휙 d 휇. Unfor- 푘→∞ 푘 퐸 퐸 휕퐸푘 tunately, this not the case in general. (14) where 휈퐸푘 is the classical inner unit normal vector to 퐸푘. 8 A simpler construction of smooth approximations 퐸휀 can also be obtained when 퐸 For the open set 퐸 with Lipschitz boundary, it can is a strongly Lipschitz domain—a stronger requirement than a general Lip- schitz domain; see S. Hofmann, M. Mitrea, and M. Taylor, “Geometric and be proved that the deformations of 퐸 obtained with the transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, method of regularized distance are bi-Lipschitz. We can and other classes of finite perimeter domains,” J. Geometric Anal. 17 (2007), also employ an alternative construction by Neˇcas(1962) 593–647. 9Such a regularized distance has been constructed in Ball-Zarnescu [2, Propo- to obtain smooth approximations 퐸휀 of a bounded open 퐸 sition 3.1]; also see G. M. Lieberman, “Regularized distance and its applica- set with Lipschitz boundary in such a way that the defor- tions,” Pacific J. Math. 117(1985), no. 2, 329–352, and L. E. Fraenkel, “On mation Ψ휀(푥) mapping 휕퐸 to 휕퐸휀 is bi-Lipschitz, and the regularized distance and related functions,” Proc. R. Soc. Edinb. Sect. A 휕퐸 Jacobians of the deformations, 퐽 (Ψ휀), converge to 1 in Math. 83 (1979), 115–122.

1288 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 8 Example 3. Consider the vector field 푭 ∶ ℝ2 ⧵ {(0, 0)} → trace distribution is given in [6, Theorem 3.1, (3.2)] and ℝ2: [18, Theorem 2.4, (2.5)] as the limit of averages over the (−푦 , 푦 ) 푭(푦 , 푦 ) = 2 1 for 1 ≤ 훼 < 3. neighborhoods of the boundary. In [4], the normal trace 1 2 2 2 훼/2 (푦1 + 푦2) is presented as the limit of classical normal traces over 푝 2 2 smooth approximations of the domain. Roughly speaking, Then 푭 ∈ 풟ℳloc(ℝ ) with 1 ≤ 푝 < for 1 < 훼 < 3 and 훼−1 the approach in [4] is to differentiate under the integral 푝 = ∞ for 훼 = 1, and div 푭 = 0 on 퐸 = (−1, 1) × (−1, 0). sign in the formulas [6, Theorem 3.1, (3.2)] and [18, The- For 휙 ∈ Lip(휕퐸) with supp 휙 ⊂ {푦 = 0, |푦 | < 1}, as shown 2 1 orem 2.4, (2.5)] in order to represent the normal trace as in Silhav´y[18,ˇ Example 2.5], the limit of boundary integrals (i.e., integrals of the classi- 푭 ⋅ 휈 ⟨푭 ⋅ 휈, 휙⟩휕퐸 cal normal traces over appropriate smooth approxi- 1 1−훼 mations of the domain). ∫−1 휙(푡, 0)sgn(푡)|푡| d 푡 for 1≤훼<2, ∶= { Entropy Solutions, Hyperbolic Conservation lim ∫ 휙(푡, 0)sgn(푡)|푡|1−훼 d 푡 for 2≤훼<3. 휀→0 {|푡|>휀} Laws, and 풟ℳ Fields This indicates that the normal trace ⟨푭 ⋅ 휈, ⋅⟩휕퐸 of the vector One of the main issues in the theory of hyperbolic con- field 푭(푦1, 푦2) is a measure when 1 ≤ 훼 < 2, but is not a servation laws (5) is to study the behavior of entropy so- measure when 2 ≤ 훼 < 3. Also see Figure 9 when 훼 = 2. lutions determined by the Lax entropy inequality (6) to Indeed, it can be shown (see [4, Theorem 4.1]) that explore to the fullest extent possible questions relating to large-time behavior, uniqueness, stability, structure, and ⟨푭 ⋅ 휈, 휙⟩휕퐸 can be represented as a measure if and only 푝 traces of entropy solutions, with neither specific reference if 휒퐸푭 ∈ 풟ℳ (Ω). Moreover, if ⟨푭 ⋅ 휈, ⋅⟩휕퐸 is a measure, then: to any particular method for constructing the solutions nor additional regularity assumptions. (i) for 푝 = ∞, | ⟨푭 ⋅ 휈, ⋅⟩ | ≪ ℋ푛−1 휕퐸 (i.e., ℋ푛−1 휕퐸 It is clear that understanding more properties of 풟ℳ restricted to 휕퐸) (see [5, Proposition 3.1]); 푛 fields can advance our understanding of the behavior ofen- (ii) for ≤ 푝 < ∞, | ⟨푭 ⋅ 휈, ⋅⟩ |(퐵) = 0 for any 푛−1 휕퐸 tropy solutions for hyperbolic conservation laws and other ′ Borel set 퐵 ⊂ 휕퐸 with 휎-finite ℋ푛−푝 measure (see related nonlinear equations by selecting appropriate en- [16, Theorem 3.2(i)]). tropy pairs. Successful examples include the stability of This characterization can be used to find classes of vector Riemann solutions, which may contain rarefaction waves, fields for which the normal trace can be represented bya contact discontinuities, and/or vacuum states, in the class measure. An important observation is that, for a constant of entropy solutions of the Euler equations for gas dynam- vector field 푭 ≡ 풗 ∈ ℝ푛, ics; the decay of periodic entropy solutions; the initial and 푛 boundary layer problems; the initial-boundary value problems; and the structure of entropy solutions of nonlinear ⟨풗 ⋅ 휈, ⋅⟩휕퐸 = − div(휒퐸풗) = − ∑ 푣푗퐷푦푗 휒퐸. 푗=1 hyperbolic conservation laws. See [3,5,6,8,11,19] and the 푛 references therein. Thus, in order that ∑ 푣 퐷 휒 is a measure, it is not 푗=1 푗 푦푗 퐸 Further connections and applications of 풟ℳ fields in- 휒 퐵푉 necessary to assume that 퐸 is a function, since cancel- clude the solvability of the vector field 푭 for the equation lations could be possible so that the previous sum could div 푭 = 휇 for given 휇, image processing via the dual of 퐵푉, still be a measure. Indeed, such an example has been con- 2 and the analysis of minimal surfaces over weakly-regular structed (see [4, Remark 4.14]) for a set 퐸 ⊂ ℝ without 10 푝 2 domains. finite perimeter and a vector field 푭 ∈ 풟ℳ (ℝ ) for any Moreover, the 풟ℳ theory is useful for the develop- 푝 ∈ [1, ∞] such that the normal trace of 푭 is a measure on ments of new techniques and tools for entropy analysis, 휕퐸. measure-theoretic analysis, partial differential equations, In general, even for an open set with smooth boundary, free boundary problems, calculus of variations, geomet- ⟨푭 ⋅ 휈, ⋅⟩ ≠ ⟨푭 ⋅ 휈, ⋅⟩ div 푭 휕퐸 휕퐸, since the Radon measure in ric analysis, and related areas, which involve the solutions (15) is sensitive to small sets and is not absolutely continu- with discontinuities, singularities, among others. ous with respect to the Lebesgue measure in general. Finally, we remark that a Gauss-Green formula for 풟ℳ푝 fields and extended divergence-measure fields (i.e., 푭 is a 10See Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear vector-valued measure whose divergence is a Radon mea- Evolution Equations, AMS, Providence, RI, 2001; N. C. Phuc and M. Torres, sure) was first obtained in Chen-Frid [6] for Lipschitz do- “Characterizations of signed measures in the dual of 퐵푉 and related isometric ˇ isomorphisms,” Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no. mains. In Silhav´y[18], a Gauss-Green formula for ex- 1, 385–417; G. P. Leonardi and G. Saracco, “Rigidity and trace properties of tended divergence-measure fields was shown to be also divergence-measure vector fields,” Adv. Calc. Var. 2021 (to appear), and the held over general open sets. A formula for the normal references therein.

SEPTEMBER 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1289 [11] Constantine M. Dafermos, Hyperbolic Conservation Laws ACKNOWLEDGMENTS. The authors would like to in Continuum Physics, 4th ed., Grundlehren der Mathema- thank Professors John M. Ball and John F. Toland, as tischen Wissenschaften, vol. 325, Springer-Verlag, Berlin, well as the anonymous referees, for constructive sugges- 2016, DOI 10.1007/978-3-662-49451-6. MR3468916 tions to improve the presentation of this article. The [12] Marco Degiovanni, Alfredo Marzocchi, and Alessan- research of Gui-Qiang G. Chen was supported in part dro Musesti, Cauchy fluxes associated with tensor fields hav- by the UK Engineering and Physical Sciences Research ing divergence measure, Arch. Ration. Mech. Anal. 147 Council Award EP/L015811/1, and the Royal Society– (1999), no. 3, 197–223, DOI 10.1007/s002050050149. Wolfson Research Merit Award (UK). 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