TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 12, December 2007, Pages 5915–5929 S 0002-9947(07)04178-5 Article electronically published on July 23, 2007
THE DIVERGENCE THEOREM FOR UNBOUNDED VECTOR FIELDS
THIERRY DE PAUW AND WASHEK F. PFEFFER
Abstract. In the context of Lebesgue integration, we derive the divergence theorem for unbounded vector fields that can have singularities at every point of a compact set whose Minkowski content of codimension greater than two is finite. The resulting integration by parts theorem is applied to removable sets of holomorphic and harmonic functions.
In the context of Lebesgue integration, powerful divergence theorems (alterna- tively called Gauss-Green theorems) have been proved in [5, 6, 11] for bounded vector fields. Using generalized Riemann integrals, divergence theorems for certain kinds of unbounded vector fields are established in [12] and [10]. These vector fields are allowed to exhibit a controlled growth to infinity in neighborhoods of points ly- ing on small sets — planes of codimension greater than two in [12], and compact sets of finite Hausdorff measure of codimension greater than two in [10]. However, a closer look at these results shows that, due to the growth conditions imposed, they apply only to vector fields with a finite number of singular points (see Remark 2.4 below for details). In contrast, we obtain a divergence theorem for vector fields having singularities at all points of certain compact sets of finite upper Minkowski content of codimen- sion greater than two (see the Main Theorem below). This is achieved by imposing growth conditions about exceptional sets, rather than about individual points of ex- ceptional sets. We employ only the Lebesgue integral, and integrate over bounded BV sets. Our result generalizes substantially the divergence theorem of [11]. As in [11], we show here that mere integrability of divergence implies the Gauss-Green formula; the separate integrability of the relevant partial derivatives is not required. In view of this, we can employ vector fields whose coordinates are not BV functions (see the paragraph following Example 2.6 below). Whether Minkowski contents can be replaced by Hausdorff measures is unclear. The techniques used in [7, Section 3] for characterizing removable sets of linear par- tial differential equations in terms of Hausdorff measures yield results different from ours (cf. Theorems 4.2 and 4.3 below), and they do not appear directly applicable in our approach.
Received by the editors August 11, 2005. 2000 Mathematics Subject Classification. Primary 26B20; Secondary 26B05, 28A75. Key words and phrases. BV sets, Hausdorff measures, Minkowski contents. The first author was a chercheur qualifi´e of the Fonds National de la Recherche Scientifique in Belgium. The second author was supported in part by the Universit´e Catholique de Louvain in Belgium.
c 2007 American Mathematical Society Reverts to public domain 28 years from publication 5915
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The paper is organized as follows. Section 1 contains necessary preliminaries, such as the definitions of BV sets, the upper Minkowski content, and the resulting dimension. It also includes the definition of transversality between a compact set and the essential boundary of a BV set. In Section 2, we introduce derivatives of vector fields defined on measurable sets and formulate the divergence theorem, which is proved in Section 3. The integration by parts formula is derived in Sec- tion 4, and it is applied to removable sets of the Cauchy-Riemann and Laplace equations.
1. Preliminaries
The set of all real numbers is denoted by R,andR+ := {t ∈ R : t>0}.Unless specified otherwise, by a function we always mean a real-valued function. Countable sets are finite (including the empty set) or countably infinite. TheambientspaceofthisnoteisRm where m ≥ 1 is a fixed integer. The usual inner product x · y induces a norm |·| in Rm, and the boldface zero (0) denotes the zero vector of Rm. A cube is a compact nondegenerate cube in Rm. The closure, interior, and diameter of a set E ⊂ Rm are denoted by cl E, int E, and d(E), respectively. The distance from a point x ∈ Rm to a set E ⊂ Rm is denoted by d(x, E). For a set E ⊂ Rm and r>0, we let B(E,r):= x ∈ Rm : d(x, E)
m The sets int∗E := E(1), cl∗E := R − E(0), and ∂∗E := cl∗E − int∗E are called, respectively, the essential interior, essential closure,andessential boundary of E. The relevant closure of E is the set clrE := cl (cl∗E). The extended real number m−1 E := H (∂∗E) is called the perimeter of E. A set of finite perimeter,oraBV set, is a measurable set A ⊂ Rm with |A| < ∞ and A < ∞. Although some of our partial results hold for any BV set, in what follows we deal exclusively with the family BV of all bounded BV sets. If A ∈ BV, m−1 then H almost everywhere in ∂∗A is defined a unit vector field νA such that m m−1 (1.1) div vdL = v · νA dH A ∂∗A 1 m m for each v ∈ C (R ; R ); the vector field νA, called the unit exterior normal of A, m−1 is unique up to an H negligible subset of ∂∗A. Our goal is to relax the boundedness condition in the following theorem, whose elementary proof can be found in [11].
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m Theorem 1.1 (Gauss-Green). Let A ∈ BV,andletv :cl∗A → R be bounded. Assume that Ec and Ed are subsets of cl∗A such that m−1 m−1 (1) Ec is H negligible, and Ed is H σ-finite, − ∈ − (2) v(y) v(x) = o(1) as y approaches x cl∗A Ec, (3) v(y) − v(x) = O(1)|y − x| as y approaches x ∈ cl∗A − Ed. 1 m m−1 Then v ∈ L (∂∗A, R ; H ), and there is a measurable function div v defined almost everywhere in A that is uniquely determined by v up to a negligible set. Moreover, if div v ∈ L1(A; Lm),then m m−1 div vdL = v · νA dH . A ∂∗A The function div v in Theorem 1.1 is defined either by means of Whitney’s and Stepanoff’s theorems as in [11, Proof of Theorem 2.9], or using relative differentia- tion introduced in Section 2 below (Definition 2.1 and Proposition 2.3). It is easy to verify that both definitions of div v coincide almost everywhere in A.Wenotethat under the assumptions of Theorem 1.1, the classical divergence of v := (v1,...,vm) m ∈ equal to i=1 ∂vi/∂xi may not be defined at any x A. However, if x is an interior point of A and v is differentiable at x,thendivv(x) of Theorem 1.1 is defined and equals the classical divergence of v at x. For 0 ≤ t ≤ m,thet-dimensional upper Minkowski content of a set E ⊂ Rm is the extended real number B(E,r) M∗t (E) := lim sup m−t . r→0+ r Clearly, M∗0(E) = 0 if and only if E is empty, and M∗0(E) < ∞ if and only if E is finite. There is a κ>0, depending only on m and t, such that Ht(E) ≤ κM∗t(E) for each E ⊂ Rm [9, Section 5.5]. As M∗t(E)=M∗t(cl E)andM∗t(E)=∞ whenever E is unbounded, we apply upper Minkowski contents only to compact sets. Let K ⊂ Rm be a compact set, and let 0 ≤ t ≤ m. The following facts are self-evident: (i) if M∗t(K) > 0, then M∗s(K)=∞ whenever 0 ≤ s
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compact segments of such that the length of each is 3−k and the distance between any two is greater than or equal to 3−k.Chooseapositiver<1/6, and find a positive integer k so that 3−k−1 ≤ 2r<3−k. It is easy to see that k −mk B(Ck,r) <κ2 3 where κ is a constant depending only on the dimension m.Moreover2/3t =1and rm−t ≥ 3−(k+1)(m−t)/2m−t, which implies B(C, r) B(Ck,r) ≤ <κ6m. rm−t rm−t Thus M∗t(C) ≤ κ6m < ∞, and we conclude dim C = t. Definition 1.4. Let A ∈ BV,letK ⊂ Rm be a nonempty compact set, and let t := dim K.Wesaythat K is transverse to ∂∗A if there is a decreasing function 1 β : R+ → R+ such that β(r) dr < ∞ and 0 m−1 H B(K, r) ∩ ∂∗A ∞ (1.2) lim sup m−t < . r→0+ β(r)r m Let A ∈ BV,andletK ⊂ R be a nonempty compact set. If K ∩ cl (∂∗A)=∅, t−m then clearly K is transverse to ∂∗A. Letting β(r):=r ,weseethatK is transverse to ∂∗A whenever dim K>m− 1. On the other hand, [3, Section 2.3, m−1 Theorem 2] and Observation 1.6 below show that for H almost all x ∈ ∂∗A, the singleton {x} is not transverse to ∂∗A. Observation 1.5. Let A ∈ BV,andletK ⊂ Rm be a nonempty compact set trans- M∗s ∞ R → R verse to ∂∗ A.If (K) < , then there is a decreasing function β : + + such that 1 β(r) dr < ∞ and 0 m−1 H B(K, r) ∩ ∂∗A ∞ lim sup m−s < . r→0+ β(r)r Proof. Since t := dim K ≤ s,takingβ as in Definition 1.4 yields
m−1 m−1 H B(K, r) ∩ ∂∗A H B(K, r) ∩ ∂∗A · s−t ∞ lim sup m−s = lim sup m−t r < . r→0+ β(r)r r→0+ β(r)r
R → R 1 Observation 1.6. If β : + + is a decreasing function and 0 β(r) dr is finite, then limr→0+ rβ(r)=0.
Proof. Proceeding toward a contradiction, suppose rkβ(rk) ≥ c>0forastrictly decreasing sequence {rk} in (0, 1) with lim rk =0,andletsk := rk+1/rk.Since rk β(r) dr ≥ β(rk)(rk − rk+1) ≥ c(1 − sk), r k+1 ∞ − the series k=1(1 sk) converges. By [14, Theorem 15.5], k−1 ∞ rk lim = lim si = sk > 0, k→∞ r1 k→∞ i=1 k=1 contrary to our assumption.
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∞ Remark 1.7. Assume ∂∗A and K are C manifolds and dim K ≤ m − 1. If K is transverse to ∂∗A in the sense of differential topology [8, Chapter 3], then K is also transverse to ∂∗A according to Definition 1.4. However, in the measure theoretic setting, the geometric content of transversality is not obvious. Distinct definitions may be introduced, which agree on C∞ manifolds. For instance, in view of Observation 1.6, inequality (1.2) may be relaxed to
m−1 H B(K, r) ∩ ∂∗A lim − − =0. r→0+ rm 1 t The definition we have chosen leads to an integration by parts theorem (Theorem 4.1 below) suitable for our applications. The next definition introduces a concept related to transversality, albeit only formally. The reader should compare it with Example 1.9, Remark 2.7, and Exam- ple 2.8 below. Definition 1.8. Let A ∈ BV,letK ⊂ Rm be a nonempty compact set, and let t := dim K.Wesaythat∂∗A accumulates about K if
m−1 H B(K, r) ∩ ∂∗A ∞ lim sup m−1−t = . r→0+ r m Let A ∈ BV.Thesetofallx ∈ R such that ∂∗A accumulates about {x} is called the singular boundary of A, denoted by ∂sA. It follows from [3, Section 2.3] m−1 that H (∂sA)=0. Example 1.9. Let m =2,andlet 2 −2 −2 Ak := x ∈ R :(k + εk) ≤|x|≤k ∞ R2 − where 0 <εk < 1. The set A := k=1 Ak is BV, and 0 belongs to int∗A or cl∗A according to whether lim εk equals 1 or 0, respectively. In either case, ∂sA = {0}.
2. The result As BV sets of positive measure may have no interior points, we need to extend the definition of derivative to points of essential interior.
m m Definition 2.1. Let E ⊂ R ,x∈ int∗E,andv :cl∗E → R .Wesaythatv is relatively differentiable at x if there is a linear map Dv(x):Rm → Rm such that v(y) − v(x) − Dv(x)(y − x) = o(1)|y − x|
as y ∈ cl∗E approaches x. Relatively differentiable maps are approximately differentiable in the sense of [5, Section 3.1.2]. Although the converse is false, relative and approximate differentia- bility exhibit many similarities. In particular, an argument analogous to proving the uniqueness of approximate derivatives [3, Section 6.1.3, Theorem 3] shows that the linear map Dv(x) of Definition 2.1 is unique; see [13, Observation 2.5.6] for details. The number div v(x):=traceofDv(x) is called the relative divergence of v at x. Since our concept of relative differen- tiability coincides with the usual definition of differentiability whenever x is the interior point of E, using the standard notation will cause no confusion.
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m m Let B ⊂ E ⊂ R and x ∈ int∗B.Ifv :cl∗E → R is relatively differentiable at x, then so is the restriction w := v cl∗B and Dw(x)=Dv(x). In other words, relative differentiation commutes with restrictions:
D(v cl∗B)=(Dv) int∗B. Applying arguments similar to those used for approximate differentiation in [5, Section 3.3], we obtain a version of Stepanoff’s theorem for relative differentiability (Proposition 2.3 below).
m m Lemma 2.2. Let E ⊂ R and C ⊂ int∗E.Supposev :cl∗E → R satisfies the following conditions: (i) there are positive numbers c and δ such that v(x) − v(y) ≤ c|x − y|
for each y ∈ C and each x ∈ B(y, δ) ∩ cl∗E; (ii) the restriction v C is Lipschitz. m m If a Lipschitz extension w : R → R of v C is differentiable at z ∈ C ∩ int∗C, then v is relatively differentiable at z and Dv(z)=Dw(z).
m m Proof. Select a Lipschitz extension w : R → R of v C and a z ∈ C ∩ int∗C at which w is differentiable. Choose a positive ε ≤ 1, and making c larger and δ smaller, assume Lip(w) ≤ c, δ < ε,and w(x) − w(z) − Dw(z)(x − z) ≤ ε|x − z| for each x ∈ B(z,δ). Let r := |x − z|,andobserve x B(x, εrx) ∩ C lim =1 → x z B(x, εrx)
[13, Lemma 1.5.6]. Thus making δ still smaller, we may assume B(x, εrx)∩C = ∅ for each x ∈ B(z,δ). Given x ∈ B(z,δ) ∩ cl∗E with x = z,chooseay ∈ B(x, εrx) ∩ C. Since w(z)=v(z),w(y)=v(y), and |y − x| <ε|x − z| <δ<ε,weobtain v(x) − v(z) − Dw(z)(x − z) ≤ v(x) − v(z) − w(x) − w(z) + w(x) − w(z) − Dw(z)(x − z) ≤ v(x) − v(y) + w(y) − w(x) + ε|x − z|≤ε(2c +1)|x − z|.
m m Proposition 2.3. Let E ⊂ R ,C⊂ int∗E,andv :cl∗E → R .Ifforeach x ∈ C, v(y) − v(x) = O(1)|y − x| as y ∈ cl∗E approaches x,thenv is relatively differentiable at almost all x ∈ C. Proof. For i =1, 2,...,denotebyC the set of all x ∈ C such that i v(x) − v(y) ≤ i|x − y| ∈ ∩ ∞ for each y B(x, 1/i) cl∗E,andobserveC = i=1 Ci.EachsetCi is the union of sets Ci,j with d(Ci,j ) < 1/i for j =1, 2,.... It follows that each v Ci,j is Lipschitz. Indeed given x, y ∈ C ,wehave i,j v(x) − v(y) ≤ i|x − y|,
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since x ∈ Ci and y ∈ B(x, 1/i) ∩ cl∗E. By Rademacher’s theorem [3, Section 3.1.2, m Theorem 2], a Lipschitz extension wi,j of v Ci,j to R is differentiable almost everywhere. Lemma 2.2 implies that v is relatively differentiable at almost all x ∈ Ci,j , and hence at almost all x ∈ C. m Main Theorem (Gauss-Green). Let A ∈ BV and v :cl∗A → R . Assume that Ec and Ed are subsets of cl∗A such that m−1 m−1 (1) E c is H negligible, and Ed is H σ-finite, − ∈ − (2) v(y) v(x) = o(1) as y approaches x cl∗A Ec, (3) v(y) − v(x) = O(1)|y − x| as y approaches x ∈ cl∗A − Ed.
In addition, assume that there is a countable collection {Ki} of disjoint compact subsets of Ec ∪ (clrA − cl∗A) satisfying the following conditions:
(4) each Ki is nonempty and transverse to ∂∗A; ∗ − M ti ∞ (5) for each Ki, there is a nonnegative ti
(∗)Foreachx ∈ Ki and for each neighborhood U of x,themapv is unbounded in (U − Ki) ∩ cl∗A.
We claim that if (∗) is satisfied for each Ki, then the collection {Ki} is finite. We prove the claim by contradiction. Select xi ∈ Ki and assuming {xi} is an infinite sequence, find a subsequence, still denoted by {xi},thatconvergesto x ∈ clrA. By assumption (7) of the Main Theorem, each Ki has a neighborhood Ui − ∩ ∗ ∩ ∅ such that v is locally bounded in (Ui Ki) cl∗A. It follows from ( )thatUi Kj = ∈ for every j = i;inparticularx i Ki. Inductively, we can find disjoint open sets ⊂ ⊂ ∈ − ∩ Vi so that Ki Vi Uifor i =1, 2,....Thereisa yi (Vi Ki) cl∗A so that | − | −{ } xi yi < 1/i and v(yi) >i.NowV := i Vi x, yi is an open set containing − i Ki and v is not bounded in cl∗A V . This contradicts assumption (6) of the Main Theorem.
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A similar argument shows that [12, Theorem 5.4] and [10, Gauss-Green Theorem] apply only to vector fields with finitely many singular points — an observation which simplifies dramatically the statements and proofs of these results.
m Example 2.5. Let A := B(0, 1), and define v :cl∗A → R by the formula x|x|−m if x = 0, v(x):= 0ifx = 0.
A direct calculation shows that div v(x)=0foreachx ∈ A −{0},andthat v · ν dHm−1 = A. Clearly, (7) is the only assumption of the Main Theorem ∂∗A A which is not satisfied. The next example shows that identity (2.1) may hold for vector fields that do not satisfy assumptions of the Main Theorem. Example 2.6. Let m =2andp>0. For x =(ξ,η)inR2,let |x|−p(η, −ξ)ifx = 0, v(x)= 0ifx = 0. A direct calculation shows that div v(x)=0foreachx ∈ R2 −{0},and 2 v · νB(0,r) dH =0 ∂∗B(0,r) for each r>0. It follows from Theorem 1.1 that identity (2.1) holds for each A ∈ BV whose topological boundary does not contain 0.Yet,v does not satisfy condition (7) of the Main Theorem when p ≥ 2. Note that for A := B(0, 1) and 1 ≤ p<2, the Main Theorem applies to the vector field v of Example 2.6, although the coordinates of v are not BV functions. Remark 2.7. A simple modification of the proof presented in Section 3 below shows that the Main Theorem remains valid when conditions (4) and (7) are replaced, respectively, by conditions: (4*) each Ki is nonempty and ∂∗A does not accumulate about Ki; R → R 1 ∞ (7*) there are decreasing functions βi : + + such that 0 βi(r) dr < and if y approaches K ,i=1, 2,...,then i t+2−m v(y) = o(1)βi d(y, Ki) d(y, Ki) or t+2−m v(y) = O(1)βi d(y, Ki) d(y, Ki) ∗t ∗t according to whether M (Ki) > 0orM (Ki) = 0, respectively. Only the obvious reformulations of Lemmas 3.1, 3.2, and 3.4 below must be proved. We leave the details to the reader. Example 2.8. Let A be the bounded BV set of Example 1.9, and let x|x|−3/2 if x ∈ R2 −{0}, v(x):= 0ifx = 0. A direct calculation shows that div v belongs to L1(A; L2) while v does not belong 1 2 1 −3/4 to L (∂∗A, R ; H ). Since condition (7*) holds with β(r):=r ,itiseasyto
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see that condition (4*) is the only unsatisfied assumption of the Main Theorem modified according to Remark 2.7.
3. Proof of the Main Theorem Lemma 3.1. Let E ⊂ Rm be measurable, and let K ⊂ Rm be a compact set with M∗t(K) < ∞ for 0 ≤ t | | Lm Then lim B(K,r) v d =0as r>0 approaches zero. Proof. Choose an r>0sothatB(K, s)/sm−t Lemma 3.2. Let A ∈ BV,andletK ⊂ Rm be a compact set with M∗t(K) < ∞ m m−1 for 0 ≤ t If K is transverse to ∂∗A,then lim |v| dHm−1 =0. r→0+ B(K,r)∩∂∗A Proof. By the definition of transversality and our assumptions, there are c>0 R → R 1 ∞ and a decreasing function β : + + with 0 β(r) dr < such that given a sufficiently small positive number r, m−1 m−t t+1−m H B(K, s) ∩ ∂∗A ≤ cβ(s)s and v(y) ≤ cd(y, K) for each positive s ≤ r and each y ∈ B(K, r). Select such an r>0, and let B := B(K, r2−i) − B(K, r2−i−1) i ∪ ∞ Hm−1 for i =0, 1,....AsB(K, r)=K i=0 Bi and (K) = 0, we obtain ∞ |v| dHm−1 ≤ c d(y, K)t+1−m dHm−1(y) ∩ ∩ B(K,r) ∂∗A i=0 Bi ∂∗A ∞ ≤ c2 (r2−i−1)t+1−mβ(r2−i)(r2−i)m−t i=0 − ∞ r2 i r ≤ 2mc2 β(s) ds =2mc2 β(s) ds. −i−1 i=0 r2 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5924 T. DE PAUW AND W. F. PFEFFER Lemma 3.3. Let 0 ≤ t ≤ m − 1,andletK be a compact set. Then B(K, r) ∈ BV for L1 almost all r>0,and B(K, r) ≤ − M∗t lim inf − − (m t) (K). r→0+ rm 1 t Proof. Since x → d(x, K) is a Lipschitz function whose Lipschitz constant is one, the coarea theorem [3, Section 5.5] implies that B(K, r) ∈ BV for L1 almost all r>0, and that r P (r):= B(K, s) ds ≤ B(K, r) 0 for all r>0. Integrating by parts yields r B(K, s)st+1−m ds δ r = P (r)rt+1−m − P (δ)δt+1−m − (t +1− m) P (s)st−m ds δ B(K, r) r B(K, s) ≤ − − r m−t +(m 1 t) m−t ds r δ s for all r>δ>0. As there is nothing to prove otherwise, assume M∗t(K) < ∞. Given ε>0, find an R>0sothat B(K, s) < M∗t(K)+ε sm−t for each positive s Lemma 3.4. Let A ∈ BV,andletK ⊂ Rm be a compact set with M∗t(K) < ∞ m m−1 for 0 ≤ t If K is transverse to ∂∗A, then there is a sequence {ri} in R+ such that m−1 lim A ∩ B(K, ri) =0 and lim |v| dH =0. ∂∗ A∩B(K,ri) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE DIVERGENCE THEOREM 5925 Proof. By Lemma 3.3, there is a decreasing sequence {r } in R such that i + B(K, ri) ∗t lim ri = 0 and lim ≤ mM (K); rm−1−t i in particular, lim A ∩ B(K, ri) = 0 by [13, Proposition 1.9.2]. Choose ε>0and let b := 1 + mM∗t(K). If condition (i) holds find an integer k ≥ 1sothat ε v(y) ≤ d(y, K)t+1−m and B(K, r ) ≤ brm−1−t b i i whenever d(y, K) ∂∗ A ∩ B(K, ri) ⊂ cl∗A ∩ ∂∗B(K, ri) ∪ B(K, ri) ∩ ∂∗A , established in [13, Proposition 1.8.5]. ∞ ∞ ∞ Lemma 3.5. If A1,A 2,... are BV sets and i=1 Ai < ,thenA = i=1 Ai is ≤ ∞ − ∞ Hm−1 aBVsetwith A i=1 Ai ,andtheset∂∗A i=1 ∂∗Ai is negligible. Proof. As all is clear if m = 1, we assume m ≥ 2. By the lower semicontinuity of perimeter [3, Section 5.2.1], k k ∞ A≤lim inf Ai ≤ lim inf Ai = Ai < ∞, k k i=1 i=1 i=1 and the isoperimetric inequality [3, Section 5.6.2] implies |A| < ∞.ThusA is a BV set, and a direct verification shows k ∞ ∞ ∞ ∂∗A ⊂ ∂∗Ai ∪ ∂∗ Ai ⊂ ∂∗Ai ∪ ∂∗ Ai. i=1 i=k+1 i=1 i=k+1 − ∞ ⊂ ∞ Hence ∂∗A i=1 ∂∗Ai ∂∗ i=k+1 Ai for k =1, 2,.... The lemma follows: ∞ ∞ ∞ m−1 lim sup H ∂∗ Ai = lim sup Ai ≤ lim Ai =0. k k k i=k+1 i=k+1 i=k+1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5926 T. DE PAUW AND W. F. PFEFFER m−1 Proof of the Main Theorem. As v is continuous H almost everywhere in cl∗A, it is both Lm and Hm−1 measurable. Conditions (6) and (7) of the Main Theorem and Lemma 3.1 imply v ∈ L1(A, Rm; Lm). Choose positive numbers ε and δ,and using Lemma 3.4, find r > 0sothat i −i m−1 −i A ∩ B(Ki,ri) <δ2 and |v| dH <ε2 . ∂∗ A∩B(Ki,ri) ∩ ≤ By Lemma 3.5, the set Aδ := A i B(Ki,ri)isaBVsetwith Aδ δ,and |v| dHm−1 ≤ |v| dHm−1 <ε. ∂∗Aδ ∂∗ A∩B(Ki,ri) i | | Hm−1 ∞ As − v d < follows from condition (6) of the Main Theorem, ∂∗(A Aδ) |v| dHm−1 ≤ |v| dHm−1 + |v| dHm−1 < ∞. ∂∗A ∂∗Aδ ∂∗(A−Aδ) Now assume that div v belongs to L1(A; Lm). For sufficiently small δ, the isoperi- metric inequality and absolute continuity of the Lebesgue integral yield |div v| dLm <ε. Aδ Applying Theorem 1.1 to v cl∗(A − Aδ), we obtain · Hm−1 − Lm · Hm−1 − Lm v νA d div vd = v νAδ d div vd ∂∗A A ∂∗Aδ Aδ ≤ |v| dHm−1 + |div v| dLm < 2ε, ∂∗Aδ Aδ and the Main Theorem follows from the arbitrariness of ε. 4. Applications Theorem 4.1 (Integration by parts). Let U ⊂ Rm be an open set, let v : U → Rm, and let Ec and Ed be subsets of U such that m−1 m−1 (1) E c is H negligible, and Ed is H σ-finite, − ∈ − (2) v(y) v(x) = o(1) as y approaches x U Ec, (3) v(y) − v(x) = O(1)|y − x| as y approaches x ∈ U − Ed. Further let {Ci} be a countable collection of disjoint subsets of Ec satisfying the following conditions: (4) each Ci is nonempty and relatively closed in U, (5) given Ci and a compact set K ⊂ Ci,thereisatK with 0 ≤ tK License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE DIVERGENCE THEOREM 5927 then ϕdiv vdLm = − ∇ϕ · vdLm U U ∈ 1 Lm ∇ ∈ 1 Rm Lm whenever div v Lloc(U; ), ϕ Lloc(U, ; ),andϕv has compact support. Proof. The local integrability of v follows easily from assumptions (1)–(7) and Lemma 3.1. Find an A ∈ BV so that S := supp (ϕv) is contained in int A and cl A ⊂ U.Sinceϕ is bounded on cl A and S ∩ ∂∗A = ∅,itiseasytoverifythat w := (ϕv) cl∗A,thesetsS ∩ Ec and S ∩ Ed, and the collection {S ∩ Ci} satisfy conditions (1)–(7) of the Main Theorem. As div w = ∇ϕ · v + ϕ div v, we see that div w ∈ L1(A; Lm). Applying the Main Theorem to w and observing that w ∂∗A = 0 completes the argument. Employing the ideas of [2, Section 4], we apply the integration by parts theorem to removable sets for holomorphic and harmonic functions. The following theorem generalizes the classical results of Besicovitch [1]. Theorem 4.2. Let U be an open subset of the complex plane C,letf : U → C, and let Ec and Ed be subsets of U such that 1 1 (1) Ec is H negligible, and Ed is H σ-finite, − ∈ − (2) f(y) f(x) = o(1) as y approaches x U Ec, (3) f(y) − f(x) = O(1)|y − x| as y approaches x ∈ U − Ed. Further let {Ci} be a countable collection of disjoint subsets of Ec satisfying the following conditions: (4) each Ci is nonempty and relatively closed in U, (5) given Ci and a compact set K ⊂ Ci,thereisatK with 0 ≤ tK < 1 and ∗t M K (K) < ∞, − (6) f is locally bounded in U V for each open set V containing i Ci, (7) given Ci and a compact set K ⊂ Ci,asy approaches K, − − f(y) = o(1)d(y, K)tK 1 or f(y) = O(1)d(y, K)tK 1 ∗ ∗ according to whether M tK (K) > 0 or M tK (K)=0, respectively. If the complex derivative f (z) exists for almost all z ∈ U,thenf can be redefined on Ec so that it is holomorphic in U. Proof. Denote by g and h the real and imaginary part of f, respectively. The vector − ∈ ∞ fields v := (g, h),w:= (h, g), together with any function ϕ Cc (U), satisfy 1 R2 L2 the assumptions of Theorem 4.1; in particular, v and w belong to Lloc(U, ; ). Since div v =divw = 0 almost everywhere in U by the Cauchy-Riemann equations, integrating by parts yields v ·∇ϕ = − ϕ div v =0 and w ·∇ϕ = − ϕ div w =0. U U U U In other words, the pair (f,g) is a distributional solution of the Cauchy-Riemann equations, which form an elliptic system. The regularity theorem [15, Example 8.14] implies that almost everywhere f equals a holomorphic function. As f is continuous outside an H1 negligible set, the theorem is established. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5928 T. DE PAUW AND W. F. PFEFFER Theorem 4.3. Let U ⊂ Rm be an open set, let u : U → R be locally bounded, and let Ec and Ed be subsets of U such that m−1 m−1 (1) Ec is H negligible and relatively closed in U,andEd is H σ-finite, 1 (2) u ∈ C (U − Ec), (3) ∇u(y) −∇u(x) = O(1)|y − x| as y approaches x ∈ U − Ed. Further let {Ci} be a countable collection of disjoint subsets of Ec satisfying the following conditions: (4) each Ci is nonempty and relatively closed in U, (5) given Ci and a compact set K ⊂ Ci,thereisatK with 0 ≤ tK If u(x)=0for almost all x ∈ U,thenu can be redefined on Ec so that it is harmonic in U. ∈ ∞ ∇ Proof. Choose a ψ Cc (U), and observe that v := ψ and ϕ := u, together with the sets Ec and Ed := Ec, satisfy the assumptions of Theorem 4.1 where the collection {Ci} is empty. As U − Ec is an open set, ∇u extended to U by zero satisfies conditions (2),(3),(6), and (7) of Theorem 4.1 with respect to Ec,Ed and {Ci}. Thus the extended ∇u and ψ, together with Ec,Ed and {Ci}, satisfy all assumptions of Theorem 4.1. Integrating by parts twice yields u ψ = u div (∇ψ)=− ∇u ·∇ψ U U U = − ψ div (∇u)= ψ u =0, U U which means that u is the distributional solution of the Laplace equation u =0. Since the Laplace equation is elliptic, the theorem follows from [15, Corollary of Theorem 8.12]. 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MR0924157 (88k:00002) [15] , Functional Analysis, McGraw-Hill, New York, 1991. MR1157815 (92k:46001) Departement´ de mathematiques,´ Universite´ Catholique de Louvain, 2 chemin du cy- clotron, B-1348 Louvain-la-Neuve, Belgium E-mail address: [email protected] Department of Mathematics, University of California, Davis, California 95616 E-mail address: [email protected]; [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use