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A Sharp Divergence Theorem with Nontangential Traces

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A Sharp Divergence Theorem with Nontangential Traces A Sharp Divergence Theorem with Nontangential Traces Dorina Mitrea, Irina Mitrea, and Marius Mitrea 1. A Brief Historical Perspective functions on a compact interval [푎, 푏], the Fundamental The Fundamental Theorem of Calculus, one of the most Theorem of Calculus reads: spectacular scientific achievements, stands as beautiful, 푏 powerful, and relevant today as it did more than three ∫ 퐹′(푥) 푑푥 = 퐹(푏) − 퐹(푎) (1.1) centuries ago when it first emerged onto the mathemati- 푎 cal scene. Typically, Isaac Newton and Gottfried Leibniz for every 퐹 ∈ AC([푎, 푏]). are credited with developing much of the mathematical It is a stark example of how local information, encoded in machinery associated with this result into a coherent the- the instantaneous rate of change (aka derivative) 퐹′, can ory for infinitesimal quantities (the bedrock of modern cal- be pieced together via integration to derive conclusions of culus), a mathematical landscape within which the Fun- a global nature about the variation of 퐹 over [푎, 푏], a fun- damental Theorem of Calculus stands out as the crown- damental paradigm in calculus. ing achievement. In its sharp one-dimensional version, Intriguingly, while (1.1) is essentially optimal, dealing involving the class AC([푎, 푏]) of absolutely continuous with higher-dimensional versions of the Fundamental The- Dorina Mitrea is a professor of mathematics at Baylor University. Her email orem of Calculus remains an active area of research in con- 푛 address is [email protected]. temporary mathematics. In its standard version, with ℒ Irina Mitrea is a professor of mathematics at Temple University. Her email denoting the 푛-dimensional Lebesgue measure in ℝ푛, the address is [email protected]. Divergence Theorem asserts that Marius Mitrea is a professor of mathematics at Baylor University. His email 푛 1 address is [email protected]. if Ω ⊆ ℝ is a bounded domain of class 풞 , Communicated by Notices Associate Editor Daniela De Silva. with outward unit normal 휈 and surface mea- ⃗ 푛 ⃗| (1.2) For permission to reprint this article, please contact: sure 휎, then ∫Ω div 퐹 푑ℒ = ∫휕Ω 휈 ⋅ (퐹|휕Ω) 푑휎 푛 [email protected]. for each vector field 퐹⃗ ∈ [풞1(Ω)] . DOI: https://doi.org/10.1090/noti2149 OCTOBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1295 ′ 푛 Since the divergence of a continuously differentiable vec- where ퟏΩ is the characteristic function of Ω and 풟 (ℝ ) de- tor field 퐹⃗ may be computed pointwise as (div 퐹)(푥)⃗ = notes the space of distributions in ℝ푛. Conversely, since 푦−푥 푛 푛 −1 ⃗ 푛−1 both ∇ퟏΩ and −휈휎 are vector distributions in ℝ of order lim ℒ (퐵(푥, 푟)) ∫휕퐵(푥,푟) ( ) ⋅ 퐹(푦) 푑ℋ (푦), where 푟→0+ 푟 ≤ 1, their actions canonically extend to vector fields from ℋ 푛−1 stands for the (푛 − 1)-dimensional Hausdorff mea- 푛 [풞1(ℝ푛)] , in which scenario we recover (1.2). Hence, ℝ푛 (div 퐹)(푥)⃗ 푐 sure in , it follows that the quantity is em- (1.3) amounts to an equivalent reformulation of the classi- ⃗ blematic of the tendency of a vector field 퐹 to collect (sink cal Divergence Theorem (1.2), which has a purely geomet- effect) or disperse (source effect) at a point 푥. In view of ric measure theoretic nature. In particular, (1.3) brings this feature, the Divergence Formula in (1.2) may be re- into focus the fact that the distributional gradient of the garded as a conservation law, asserting that the solid inte- characteristic function of a bounded 풞1 domain is a locally gral of all such sources and sinks associated with a given finite Borel vector-valued measure in ℝ푛. vector field is equal to the net flow of said vector field As far as the latter property is concerned, R. Caccioppoli, 1 through the solid’s boundary. Hence, in complete anal- E. De Giorgi, and H. Federer registered a decisive leap for- ogy to the Fundamental Theorem of Calculus mentioned ward by considering the largest class of Euclidean subsets earlier, the Divergence Theorem describes how the infini- enjoying said property, i.e., the class of sets of locally fi- tesimal sink/source effects created by a vector field may be nite perimeter. It turns out that this consists of Lebesgue pieced together inside a given domain to produce a global, 푛 measurable subsets Ω of ℝ with the property that ퟏΩ is macroscopic effect along the boundary. 푛 푛 of locally bounded variation in ℝ , i.e., ퟏΩ ∈ BVloc(ℝ ). The classical result recorded in (1.2) is usually associ- 푛 In turn, membership to BVloc(ℝ ) is conceived in such a ated with the names of J.-L. Lagrange, who first established way that the Riesz Representation Theorem can naturally a special case of the Divergence Theorem in 1762 working be applied to the functional on the propagation of sound waves (cf. [14]); C. F. Gauss, who independently considered a particular case in 1813 푛 1 푛 푛 Λ(퐹)⃗ ∶= ∫ div 퐹⃗ 푑ℒ for all 퐹⃗ ∈ [풞푐 (ℝ )] , (1.4) (cf. [9]); M. V. Ostrogradsky, who gave the first proof of the Ω general theorem in 1826 (cf. [20]); G. Green, who used a to conclude that there exist a Borel measure 휎 in ℝ푛, related formula in 1828 (cf. [10]); A. Cauchy, who in 1846 ∗ which is actually supported on 휕Ω, and a 휎 -measurable first published, without proof, the nowadays familiar form ∗ vector-valued function 휈 ∶ ℝ푛 → ℝ푛 satisfying |휈| = 1 at of Green’s theorem (cf. [1]); B. Riemann, who provided a 휎 -a.e. point in ℝ푛, such that proof of Green’s formula in his 1851 inaugural disserta- ∗ tion (see [21]); Lord Kelvin, who in 1850 discovered a spe- 1 푛 푛 Λ(퐹)⃗ = ∫ 휈 ⋅ 퐹⃗ 푑휎∗ for all 퐹⃗ ∈ [풞푐 (ℝ )] . (1.5) cial version of Stokes’ theorem (in the three-dimensional ℝ푛 setting, also known as the curl theorem); and E.´ Cartan, 휈 who first published the general form of Stokes’ theorem (in The function is referred to as the geometric measure the- Ω 휎 the language of differential forms on manifolds) in 1945, oretic outward unit normal to . Bearing in mind that ∗ 휕Ω among others. However, a precise attribution is fraught is actually supported on , the following version of the with difficulty since the Divergence Theorem in its mod- Divergence Theorem emerges from (1.4) and (1.5): ern format has undergone successive waves of reformula- ⃗ 푛 ⃗ tions and generalizations, as well as more rigorous proofs, ∫ div 퐹 푑ℒ = ∫ 휈 ⋅ 퐹 푑휎∗ Ω 휕Ω (1.6) with inputs from a multitude of sources (general historical 푛 for each vector field 퐹⃗ ∈ [풞1(ℝ푛)] . accounts may be found in [12] and [22]). 푐 Specializing the Divergence Formula in (1.2) to the The real achievement of De Giorgi and Federer is further case when 퐹⃗ is the restriction to Ω of vector fields from refining (1.6) by establishing that actually 푛 [풞 ∞(ℝ푛)] (where 풞 ∞(ℝ푛) denotes the space of smooth, 푐 푐 휎 = ℋ 푛−1⌊휕 Ω, compactly supported functions in ℝ푛) yields the state- ∗ ∗ (1.7) ment: where 휕∗Ω denotes the measure theoretic boundary of Ω, if Ω ⊆ ℝ푛 is a bounded domain of class 풞1, defined as with outward unit normal 휈 and surface mea- (1.3) ℒ푛(퐵(푥, 푟) ∩ Ω) 푛 푛 ′ 푛 휕∗Ω ∶= {푥 ∈ ℝ ∶ lim sup 푛 > 0 and sure 휎, then ∇ퟏΩ = −휈휎 in [풟 (ℝ )] , 푟→0+ 푟 ℒ푛(퐵(푥, 푟) ⧵ Ω) lim sup 푛 > 0}. 푟→0+ 푟 1For instance, imagining 퐹⃗ as the velocity field of an incompressible fluid flow occupying a fixed region Ω, this informally states that “what goes in must come Hence, near points in 휕∗Ω there is enough mass (relative to out.” the scale) both in Ω and in ℝ푛 ⧵Ω. Let us also note that, in 1296 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 9 principle, 휕∗Ω can be a much smaller set than the topolog- Lipschitz domains, and also for a class of Reifenberg-flat ical boundary 휕Ω. Substituting (1.7) back into (1.6) then domains (cf. [3], [13]), but to go beyond this one needs yields the following result: genuinely new techniques. Progress in this regard has been registered in [11], which treats a much larger class of do- Theorem 1.1 (De Giorgi–Federer’s Divergence Theorem [4, mains than Lipschitz, without any flatness assumptions. 5,7,8]). Let Ω ⊆ ℝ푛 be a set of locally finite perimeter. Let 휈 be However, the version of the Divergence Theorem estab- the geometric measure theoretic outward unit normal to Ω, and 푛 lished in [11] requires that the (nontangential) trace of the let 휎 ∶= ℋ 푛−1⌊휕Ω. Then for each vector field 퐹⃗ ∈ [풞1(ℝ푛)] 푐 vector field 퐹⃗ on the boundary is 푝th power integrable for one has some 푝 > 1. This requirement is an artifact of the proof, ∫ (div 퐹⃗)| 푑ℒ푛 = ∫ 휈 ⋅ (퐹⃗| ) 푑휎. (1.8) which relies on the boundedness of the Hardy–Littlewood Ω 휕∗Ω 푝 Ω 휕∗Ω maximal function on 퐿 with 푝 > 1. The goal of this article is to describe a new brand of In a nutshell, one of the key results of the De Giorgi– Divergence Theorem developed in [17] exhibiting the fol- Federer theory is the identity lowing features (all of which are absent from De Giorgi– 푛−1 ∇ ퟏΩ = −휈 ℋ ⌊휕∗Ω Federer’s version of the Divergence Theorem recorded in as distributions in ℝ푛, whenever Theorem 1.1): • when 푛 = 1 and Ω is a finite interval on the real line, Ω ⊆ ℝ푛 , is a set of locally finite perimeter our theorem reduces precisely to the sharp version of the which may then readily be reinterpreted as the Divergence Fundamental Theorem of Calculus formulated in (1.1); Formula (1.8) simply by untangling jargon.
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