Divergence measure fields and Cauchy's stress theorem

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Divergence Measure Fields and Cauchy's Stress Theorem.

M. SÏILHAVYÂ(*)

ABSTRACT - Divergence measure fields are integrable vector fields whose distributional divergence is a measure. Some versions are derived of the divergence theorem for divergence measure fields on sets of finite perimeter. Using these results, it is shown that Cauchy fluxes from the theory of Cauchy's stress theorem can be extended to a class of surfaces that includes singular surfaces of continuum mechanics (shock waves and phase boundaries). On the singular surfaces, the divergence of the stress has a surface delta type singularity, with tractions on a surface and its opposite different from each other.

1. Introduction.

Cauchy's stress theorem asserts that the forcef(S) exerted by one part of a continuous body on another part through a surface S of contact is expressed by

f(S)

S

TndA

wherenis the normal toS,dAis the element of area ofS, andT, the main object of the stress theorem, is the stress tensor (¹). Cauchy's derivation was heuristic, with unnecessary additional assumptions. Noll [19] raised the question of a rigorous derivation under minimal assumptions. Basic properties of interactions in a body were formalized in the concept of a Cauchy flux (²) in [13] and using this notion, the proof of the Cauchy stress

(*) Indirizzo dell'A.: Mathematical Institute of the AV CÏR, ZÏitnaÂ 25, 115 67 Prague 1, Czech Republic ± Department of Mathematics, University of Pisa, Via F.

Buonarroti, 2, 56127 Pisa, Italy.

2000 MSC74A99, 28A75

(¹) Similarly, the flux of heatF(S) through a surfaceSis given by the heat flux vectorqviaFS

SqndA.

(²) See Section 5.

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theorem was given under natural, albeit still restrictive assumptions (³).

These ideas were then adapted to the context of sets of finite perimeter [3, 29, 24, 14, 20] (⁴). The assumptions of [13] lead to bounded stress fields, and in [29] it was shown that the distributional divergence div T of T is a bounded (integrable) function; see also [1]. Clearly, there are many situations where the boundedness ofTand of divTare violated. Unbounded stress fields occur in fracture mechanics and in the existence theorems in nonlinear elasticity; div T has a (surface) d type singularity on singular surfaces (shock waves and propagating phase boundaries). Cauchy fluxes leading to unbounded stress fields and with divTan (unbounded) integrable function are treated in [24-25]. In [25] it was shown that at this generality the Cauchy flux can be defined only for «almost all» surfacesS. This covers unbounded stresses but excludes singular surfaces. The extension to stress fields arising in the presence of singular surfaces is in [7, 15-18].

These works consider Cauchy fluxes for which the resulting stress field may be unbounded with distributional divergence a Radon measure.

Following [7], such tensor/vector fields are called divergence measure fields in the subsequent treatment (⁵). However, the concept of almost every surface adopted in [7, 15-18] excludes surfaces where divT is not absolutely continuous with respect to Lebesgue's measure, in particular, the Cauchy flux is generally undefined on singular surfaces.

This paper (i) derives some new properties of the divergence measure fields including versions of the divergence theorem for them and (ii) uses (i) to extend Cauchy fluxes to a class of surfaces which includes the singular surfaces. To simplify the description, we switch from vector valued Cauchy fluxes (such as force) to scalar valued ones (such as the heat flux) (⁶); ac- cordingly, the stress tensorTchanges to the flux vectorq. The results may be extended to vector valued fluxes by components.

The divergence measure vector fields are locally integrable fields q:Rⁿ!Rⁿwhose distributional divergence divqis a Radon measure. The paper first briefly addresses the question of the nature of the measure divqin Section 3. It is shown that ifq2L^p_loc(Rⁿ)then for 1p<n=(n 1)

(³) See below.

(⁴) A variational approach to Cauchy's theorem is developed in [11], and an approach based on Whitney's geometric integration theory [28] is outlined in [21].

See also [23].

(⁵) The spaces of fields with integrable distributional divergence are treated in many works, e.g., [2, 12]; papers [4±6] consider divergence measure fields.

(⁶) See footnote (¹) above.

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any Radon measure can arise as div q, while ifn=(n 1)p 1; then the singularities of div qare not be arbitrary: div q vanishes on sets of Hausdorff dimension mn p=(p 1) if p is finite and of dimension m<n 1 ifp 1;moreover, in the latter case divqalso vanishes on each set ofn 1 dimensional Hausdorff measure 0.

Next, several versions of the divergence theorem are given for divergence measure vector fields and sets of finite perimeter. Ifqis a smooth vector field and W a smooth scalar field with compact support then the divergence theorem reads

@M

Wqn^MdHⁿ ¹

M

DWqdLⁿ

M

WqdivqdLⁿ 1:1

for any normalized set of finite perimeterMRⁿwith the measure theoretic boundary@M and the measure theoretic normaln^M(see Section 2 for definitions). For divergence measure vector fields the right hand side generalizes to

M

DWqdLⁿ

M

Wdivq

where the last integral is the integral of a continuous function with respect to the Radon measure divq, but the left hand side of (1.1) does not have an immediate meaning. It turns out that for divergence measure vector fields the expressionqn^M cannot be interpreted pointwise, i.e., the left hand side of (1.1) must be interpreted as a functional on scalar fields W on @M; [4-6]; this occurs even when the distributional divergence div qis an integrable function [26; Theorem 1.2, Chapter I]. It is shown that such a functional exists for every set of finite perimeter (Proposition 4.1, generalizing [6] to sets of finite perimeter); this functional is called the normal trace (⁷) ofq. Ifqis «bounded» near@M(see the definition of domination in Section 4), the normal trace has additional properties. Theorem 4.2 gives two conditions under which the normal trace is a measure; one of these guarantees a measure with support on the closure@Mof the measure theoretic boundary; the other guarantees a measure supported on @M but requiresq2L^p_loc(Rⁿ)with pn=(n 1) and regular boundary in some measure theoretic sense.

Finally, Theorems 4.4 and 4.6 give conditions which guarantee that the

(⁷) [6].

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normal trace is of the form

@M

Wq^MdHⁿ ¹ 1:2

where q^M is an Hⁿ ¹ integrable function. Theorem 4.4 deals with bounded vector fields (q2L¹(Rⁿ)) and shows that then the normal trace is as in (1.2) withq^M 2L¹(@M)for any set of finite perimeter; this has been proved in [2; Theorem 1.9] for open sets with Lipschitz boundary. Theorem 4.6 deals with a generalqand proves (1.2) for sets M whose boundaries do not intersect some exceptional set ofn 1 dimensional Hausdorff measure 0 where divqis too singular.

Using the divergence theorem 4.6, it is shown that every Cauchy flux that is defined for almost every surface in the sense of [7, 15-18] can be automatically extended to a class of surfaces where the measure divqhas a surface type singularity, thus including the singular surfaces. Clearly, this extended Cauchy flux reflects more fully the properties of the interaction.

However, generally the flux cannot be extended to all surfaces, for, firstly, the flux q vector has to satisfy the domination condition as mentioned above, and, secondly the surface cannot intersect the exceptional set where divqis too singular. Ifqis bounded then the domination condition is automatically satisfied, the exceptional set is void and the Cauchy flux can be extended to all surfaces.

After a brief recapitulation of the basic measure theoretic notions in Section 2, Section 3 states some properties of the divergence measure fields. The divergence theorems and Cauchy fluxes are discussed in Sections 4 and 5, respectively. The proofs are given in the rest of the paper.

2. Preliminaries.

For a setMRⁿwe denote byM^c:RⁿnMthe complement ofM. If x2Rⁿ and r>0 thenB(x;r)is the open ball of radiusrand center x.

Lⁿis the (outer)Lebesgue measureinRⁿ;ifARⁿthenjAj Lⁿ(A) is the Lebesgue measure ofA. If 0m<1; we denote byH^mthem dimensional Hausdorff measure [10; §§ 2.10.2-6]. Briefly, if ARⁿ then

H^m(A):lim

d!0H^m_d(A)

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where for anyd>0 thesizedapproximationH^m_d(A)is defined by H^m_d(A)aminf X¹

i1

(diamCi=2)^m:CiRⁿ; A[

i

Ci;diamCi<d

( )

; where diamCsup fjx yj:x;y2Cgis the diameter ofC,

a_m:G^m(1=2)=G(m=21);

and Gis the Euler gamma function. If mis an integer, then a_m is the volume of the unit ball inR^mandH^mcoincides with themdimensional area on mdimensional manifolds inRⁿ; in particularHⁿLⁿ.

IfMRⁿisLⁿmeasurable andx2Rⁿwe say thatMhas adensityat xif the following limit exists:

D(x;M):lim

r!0

M\B(x;r)

j j

anrⁿ ;

which is then called thedensity ofMat x. A pointx2Rⁿ is said to be a point of densityofMifD(x;M)1. For a measurable setMwe denote by M the set of all points of density of M. By Lebesgue's differentiation theorem [27; Theorem (7.2)],Mis a Borel set and the symmetric differ- ence ofM andM hasLⁿ measure 0. We define themeasure theoretic boundary@Mof a measurable setMby

@MRⁿn(M[(M^c));

cf. [10; § 4.5.12].@Mis a Borel set andj@Mj 0:We say that a measurable setMRⁿ isnormalizedifMM:

By a Borel measure in Rⁿ we mean any s additive function m:A![0;1] whose domainA is a salgebra which containsall Borel sets inRⁿ:Thus the restrictions ofLⁿ;H^mto their respective systems of measurable sets are Borel measures. By a (signed)Radon measureinRⁿ we mean any sadditive functionm:B !Rdefined on thesalgebraB of all Borel sets in Rⁿ. By a measurewe mean either a Borel or a Radon measure. If m is a Radon measure we denote by j jm the total variation measure,which is a nonnegative Radon measure. Thenk k m j j(Rm ⁿ)<1 denotes thetotal variationofm. We denote byM(Rⁿ)the set of all Radon measures on Rⁿ and by M(Rⁿ)the subset of nonnegative Radon measures onRⁿ.

IfARⁿ is a Borel set andma measure, we denote bym^j Athere- striction ofmtoA, i.e., a measure given by

m^j A(B)m(A\B)

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for anyBfrom the domain ofm. We say that a measuremissupportedon a Borel setAifmm^j A. We say that a measuremvanisheson a Borel set Nifm(A)0 for eachANfrom the domain ofm. Ifmis a measure and f:Rⁿ!Ra Borel function that is integrable with respect to m; then fm denotes the measure given by

(fm)(A)

A

f dm

for eachAfrom the domain ofm:The reader is referred to [10; Chapter 2]

for further details of the measure theory.

A measurable setMRⁿis said to be a set offinite perimeter(cf. [10;

Theorem 4.5.6]) if the distributional partial derivativesD_i1_M; 1in; of the characteristic function 1MofMare Radon measures.Mis a set of finite perimeter,Hⁿ ¹(@M)<1(cf. [10; Theorem 4.5.11]) ,there exists a Borel functionn^M:@M!Sⁿ ¹, whereSⁿ ¹ is the unit sphere inRⁿ such

that

M

DWdLⁿ

@M

Wn^MdHⁿ ¹ 2:1

for each W2C¹₀ (Rⁿ): Here C¹₀ (Rⁿ) is the set of all infinitely differ- entiable functions with compact support and DW is the gradient of W.

The functionn^M is determined by (2.1) uniquely to within a change on anHⁿ ¹ negligible subset of@M; and is called themeasure theoretic normal ofM.

3. Divergence measure fields.

Aq2L¹_loc(Rⁿ)(⁸) is said to be adivergence measure fieldif there exists am2M(Rⁿ)such that

Rⁿ

DWqdLⁿ

Rⁿ

Wdm 3:1

for everyW2C¹₀ (Rⁿ):One then writes divq:m:The space of all diver-

(⁸) Here L^p_loc(Rⁿ), 1p 1, stands for the set of all measurable maps q:Rⁿ!Rⁿthat are locally integrable with poverp.

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gence measure fields is denoted by L¹_locM^div(Rⁿ) and we write L^p_locM^div(Rⁿ):L¹_locM^div(Rⁿ)\L^p_loc(Rⁿ)whenever 1p 1.

Forn1 the spaceL¹_locM^div(Rⁿ)reduces to the space of functions of bounded variation onR¹;throughout the rest of the paper we assume that n>1: The following are archetypical examples of divergence measure fields:

EXAMPLES3.1 ± (i)(Transversal fields). Letf 2L¹_loc(Rⁿ ¹)and define q2L¹_loc(Rⁿ) byq(x₁;. . .; x_n)(0;. . .;0; f(x₁;. . .; x_n ₁)); x2Rⁿ:Then q2L¹_locM^div(Rⁿ) anddivq0.

(ii)(Singular surface). Lete₁ be the coordinate vector in the x₁ di- rection and q(x)e1 if x1>0 and q(x)0 if x10; x2Rⁿ. Then q2L¹_locM^div(Rⁿ) anddivqHⁿ ¹^j fx2Rⁿ:x10g.

Ifmis a Radon measure and 0m<1; we say thatmisH^m absolutely continuous if j j(B)m 0 for every Borel set BRⁿ with H^m(B)0:We say that a Borel setBhassfiniteH^mmeasureifBis a union of countably many Borel sets of finiteH^mmeasure.

THEOREM3.2. Let n=(n 1)p 1; q2L^p_locM^div(Rⁿ); and set d: n p=(p 1) if p<1,

n 1 if p= 1.

(

(i)If p<1; thenjdivqj(B)0for everyBorel set B ofsfiniteH^d measure;

(ii)if p 1thendivqisHⁿ ¹ absolutelycontinuous.

For the given range ofp, the value ofdchanges monotonically from 0 to n 1: The theorem imposes a restriction on the dimensionality of the measure divq: ifq2L^p_loc(Rⁿ); then divqcannot be concentrated on sets of dimension d:Thus, e.g., the Dirac dcannot occur as the divergence of someq2L^p_loc(Rⁿ)withpn=(n 1);the improved integrability ofqim- plies improved regularity of divq. In particular, for bounded vector fields divq is absolutely continuous with respect to the n 1 dimensional Hausdorff measure. The bound d is optimal: if 1p<n=(n 1); then essentially every measure can occur as a divergence of someq2L^p_loc(Rⁿ) while if n=(n 1)p 1then there are vector fields q2L^p_loc(Rⁿ)with divergences concentrated on sets of dimension s higher than but arbi- trarily close tod:

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EXAMPLE 3.3. ± (i) If 1p<n=(n 1) then for anysigned Radon measuremwith compact support there exists aq2L^p_locM^div(Rⁿ)such that divqm:

(ii) If n=(n 1)p 1 then for anys>d there exists a q2L^p_locM^div(Rⁿ)such thatdivqis notH^s absolutelycontinuous.

The vector fieldqis constructed as the Newton force (the gradient of the Newton potential) for the uniform mass distribution on a compact setK with 0<H^m(K)<1; see Proposition 6.1.

4. The divergence theorem.

Proposition 4.1 and Theorems 4.2, 4.4 and 4.6 discuss the divergence theorem for normalized sets of finite perimeterMandq2L¹_locM^div(Rⁿ) in decreasing generality but with improving properties of the boundary term (see the discussion in Introduction). It is first noted in Proposition 4.1 that the boundary term is always a linear functional on the space Lip₀(@M) of Lipschitz continuous functions with compact support on

@M. For any set WRⁿ we denote Lip₀(W)the set of all real valued

Lipschitz continuous functions W on W such that fx2W:W(x)60g is bounded, with norm

W

k k_Lip₀_(W)Lip(W)k kW _C(W) where Lip(W)is the Lipschitz constant ofWand

W

k k_C(W):supfjW(x)j:x2Wg:

PROPOSITION4.1 (Normal trace as a functional). If M is a normalized set with finite perimeter andq2L¹_locM^div(Rⁿ)then there exists a linear functionalN^M(q;):Lip₀(@M)!Rsuch that

N^M(q;Wj_@M)

M

DWqdLⁿ

M

Wdivq 4:1

for everyW2Lip₀(Rⁿ):If N:(M^c)then

N^M(q;)N^N(q;) divq^j @M:

4:2

If M or M^c is bounded then N^M(q;) is continuous with respect to k k_Lip₀_(@M):

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HereWj@Mis the restriction ofWto@Mand the assertion is that the right hand side of (4.1) depends only on the boundary values of W: N^M(q;)is called the normal trace of q on@M: Although the frameworks are not strictly comparable, (4.1) may be considered to be a generalization of the relevant results of [4-6]. Equation (4.2) says that the normal traces from the two sides of@Mare different if divqis concentrated on@M;see (4.11), below, for a more concrete form, cf. also Example 3.1(ii).

Next we are concerned with specific forms of the normal trace. IfMis a normalized set of finite perimeter andq2L¹_locM^div(Rⁿ)we say that the normal traceN^M(q;)is

(i) ameasureif there exists an^Mn^M(q)2M(@M)such that N^M(q;W)

Rⁿ

Wdn^M

for everyW2Lip₀(@M);

(ii) an integrable function if there exists q^Mq^M(q)2 2L¹(@M;Hⁿ ¹)such that

N^M(q;W)

@M

Wq^MdHⁿ ¹

for everyW2Lip₀(@M).

The following theorem discusses conditions under which the normal trace is a measure. IfMis a normalized set of finite perimeter, we say that aq2L¹_loc(Rⁿ)is

(i) weakly dominated on @M if there exists a sequence r_j>0;

r_j !0; and a constantC<1such that 1

a_nrⁿ_j

@M

B(x;r_j)

q(y)n^M(x)

dLⁿ(y)dHⁿ ¹(x)C 4:3

for everyj2N;

(ii) dominatedon @M if there exists a functiong2L¹(@M;Hⁿ ¹) and a sequencer_j>0; r_j!0; such that

1 a_nrⁿ_j

B(x;r_j)

q(y)n^M(x)

dLⁿ(y)g(x) 4:4

forHⁿ ¹a.e.x2@Mand everyj2N:

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Clearly, ifqis dominated on@Mthen it is weakly dominated on@M:If q2L¹(Rⁿ) then q is dominated on any @M; if q2L¹_loc(Rⁿ) then q is dominated on@MifMis bounded.

THEOREM4.2. (Normal trace as a measure).Letq2L¹_locM^div(Rⁿ)and let M be a normalized set of finite perimeter. Then

(i) ifqis weaklydominated on@M; the normal trace is a measure supported on@M;

(ii) if q2L^p_locM^div(Rⁿ), where n=(n 1)p<1; d is as in Theorem3.2,qis dominated on@M, and

@sM has as finiteH^dmeasure 4:5

where@_sM fx2@M:D(x;M)does not existg; then the normal trace is a measure supported on@M:

Assertion (i) guarantees a measure supported on @M, which may be large, while (ii) guarantees a measure supported on@M; which is a set with Hⁿ ¹(@M)<1:Condition (4.5) requires that the «singular part»@sM of the boundary be small. We note thatD(x;M)exists and is equal to 1/2 for Hⁿ ¹ a.e. x2@M; however, since d<n 1; Condition (4.5) requires more. Theorem 4.2 holds also ifp 1; in which case (4.5) can be omitted, but in this special case the normal trace is an integrable function, cf.

Theorem 4.4, below.

Finally we consider situations when the normal trace is an integrable function. Let firstq2L¹_locM^div(Rⁿ)\L¹(Rⁿ); it will turn out that then the normal trace is an integrable function for every normalized set of finite perimeter; moreover, it is given by a functionq⁰(x;n)which we shall now introduce. If x2Rⁿ; n2Sⁿ ¹; and r>0; let B(x; n;r):B(x;r)\

\ fy2Rⁿ:(y x)n<0g:Ifq2L¹_loc(Rⁿ); we define a functionq⁰:Rⁿ Sⁿ ¹!Rby

q⁰(x;n) limr!0

n an 1rⁿ

B(x;n;r)

q(y) x y x y

j jdLⁿ(y) if the limit exists and is finite;

0 if the limit either does not exist or is infinite;

8>

>>

<

>>

>:

x2Rⁿ;n2Sⁿ ¹:The functionq⁰(x;n)is a generalization of the expression q(x)n:

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REMARK4.3. Ifq2L¹_loc(Rⁿ)andx2Rⁿis a Lebesgue point ofqthen q⁰(x;n)q(x)n

for everyn2Sⁿ ¹:

Hereqis any representation of the classq. In Example 3.1(ii), q⁰(x;n) e₁n if e₁n>0;

0 else;

(

for anyxwithx10. Thusq⁰is bound to be nonlinear in the presence of discontinuities in the normal component ofq.

THEOREM4.4. (Normal trace as a function, special case).

(i) If q2L¹_locM^div(Rⁿ)\L¹(Rⁿ) then for everynormalized set of finite perimeter M the normal trace is a bounded function, i.e., there exists a q^M2L¹(@M;Hⁿ ¹)such that

@M

Wq^MdHⁿ ¹

M

DWqdLⁿ

M

Wdivq 4:6

for everyW2Lip₀(Rⁿ);moreover, q^Mis given by q^M(x)q⁰(x;n^M(x)) 4:7

forHⁿ ¹a.e.x2@M; andkq^Mk_L1(@M;Hⁿ ¹) kqk_L1(Rⁿ):

(ii)If, more generally,q2L¹_locM^div(Rⁿ)andqis dominated on@M;

then the normal trace is an integrable function q^Mwhich satisfies(4.7)for Hⁿ ¹a.e.x2@M.

The existence of the normal trace as in (i) has been proved in [2;

Theorem 1.9] for open sets with Lipschitz boundary. In addition, (4.7) shows that the normal trace depends on the shape of@Monly through the normal n^M. In the context of Cauchy fluxes, assertions of this type are called Cauchy's postulate.

To proceed to the generalp, we need to isolate the part of the measure divqwhich is singular with respect toHⁿ ¹:The following proposition, a direct generalization of Lebesgue's decomposition, is a basis for that. If 0m<1; we say that anh2M(Rⁿ)isH^msingularif it is supported on a Borel setBwithH^m(B)0:

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PROPOSITION4.5. If h2M(Rⁿ)and0mn then there exists a unique decomposition ofhas

hh_<mh_mh_>m

4:8

whereh_<m;h_m;h_>m2M(Rⁿ)have the properties

(i) h_<misH^msingular;

(ii) h_misH^mabsolutelycontinuous and supported on a set ofsfinite H^mmeasure;

(iii) h_>m(B)0for everyBorel set B ofsfiniteH^mmeasure.

We may say that the dimensions ofh_<m;h_m;h_>mare less than, equal to, and bigger thanm, respectively. The Lebesgue decomposition is the case mn by noting h_>n 0 by (iii). By the Radon Nikodym theorem [27;

Theorem (10.39)] we have

h_m(A)fH^m^j S0

4:9

for some Borel setS₀Rⁿof s finiteH^mmeasure and some nonnegative Borel functionf:Rⁿ!Rwithf 2L¹(S₀;H^m); f(x)>0 for everyx2S₀ and f(x)0 for every x2=S₀(⁹). Then S₀; f are determined to within a change on aH^mnull set. For aq2L¹_locM^div(Rⁿ)we apply Proposition 4.5 withmn 1 to obtain

jdivqj jdivqj_<n ₁ jdivqj_n ₁ jdivqj_>n ₁: Thenjdivqj_n ₁fHⁿ ¹^j S₀wheref; S₀are as above; moreover,

divq^j S₀JHⁿ ¹^j S₀

for some Borel function J:Rⁿ!R such that J2L¹(S₀;Hⁿ ¹) and J(x)0 for x2=S0: The set S0 is the (analog of the) singular set of continuum mechanics and J is related to the jump of the normal trace of q across@M; see (4.11), below.

THEOREM 4.6 (Normal trace as a function, general case). Let 1p 1; q2L^p_locM^div(Rⁿ)and let M be a normalized set M of finite perimeter for whichqis dominated on@M and

jdivqj_<n ₁(@M)0:

4:10

(⁹) On the contrary, the measure h_>m, despite of being H^m absolutely continuous, cannot be expressed as in (4.9), because it is supported on a set of nonfiniteH^mmeasure.

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Then the normal trace of q on @M is an integrable function q^M2L¹(@M;Hⁿ ¹). If N:(M^c)then also the normal trace ofqon@N is an integrable function and

q^M(x)q^N(x) J(x) 4:11

forHⁿ ¹a.e. x2@M@N:

Equation (4.10) ensures that @M does not intersect the region where div q is too singular. In contrast to the situation q2L¹_locM^div(Rⁿ)\

\L¹_loc(Rⁿ); in the present context the normal traceq^Mdoes not seem to be given by (4.7) generally. However, we have the following weak form of the local dependence ofq^Mon the shape of@M:

REMARK 4.7. If Mi; i1;2; are two normalized sets of finite perimeter which satisfythe hypothesis of Theorem4.6and if S: fx2@M1\

\@M₂:n^M¹(x)n^M²(x)gthen the normal traces q^Mⁱsatisfy q^M¹(x)q^M²(x) forHⁿ ¹ a:e: x2S:

5. Cauchy fluxes.

LetP be the set of all bounded normalized sets of finite perimeter. An oriented surfaceis a pairS(S;^ n^S)such that is a Borelsubset of@Mofsome M2P andn^S(x)n^M(x)forHⁿ ¹a.e.x2S:^ LetS betheset of alloriented surfaces. We say that the oriented surfacesS(^S;n^S)andT(T;^ n^T)are compatibleif there exists an oriented surfaceU(U;^ n^U)2S such that

S^[T^ U^ and n^U(x) n^S(x) if x2S;^ n^T(x) if x2T^ (

forHⁿ ¹a.e.x2U:^ We then writeUS[T:IfM2P we interpret@Mas the oriented surface@M(@M;n^M):

We denote by G the set of all Borel functions (not classes of equivalence)h:Rⁿ![0;1] such thath2L¹_loc(Rⁿ). Ifh2G andh2M(Rⁿ)we denote [7]

P_hh: fMRⁿ:M2P;

@M

h dHⁿ ¹<1;h(@M)0g;

S _hh: fS2S :S@M;M2P_hhg:

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We say that NS is a null set if N S nS hh for some h2G; h2M(Rⁿ). We say that a setD S contains almost all ofS ifS nD is a null subset ofS :Ifpis a property associated with all surfacesS2S ; we say thatpholds for a.e.S2S if there areh2G andh2M(Rⁿ)such that p(S)is true for allS2S _hh:

ACauchy fluxis any mappingF:D !R; where D S ; such that for someD0 D that contains almost all ofS we have

(i) if S, T2D₀ are disjoint compatible material surfaces then S[T2D₀and

F(S[T)F(S)F(T);

(ii) there existsh2G such that jF(S)j

S

h dHⁿ ¹ 5:1

for everyS2D0;

(iii) there existsh2M(Rⁿ)such that jF(@P)j h(P) for anyM2P with@M2D0:

We say thatFis aCauchy flux of classL^p_loc; 1p 1;if the function has in (5.1) can be chosen inL^p_loc:

The above definition is equivalent to the one given in [7, 15-18]; the papers [13, 29] deal withh2L¹_loc(Rⁿ),hcLⁿ; c2R; and [24-25] with h2L^p_loc(Rⁿ),hfLⁿ; f 2L^p_loc(Rⁿ); 1p 1:

THEOREM5.1. F is a Cauchyflux if and onlyif there exists a vector fieldq2L¹_locM^div(Rⁿ)such that, for anyrepresentationqof the classq,

F(S)

S

qn^SdHⁿ ¹ 5:2

for a.e. S2D:The correspondence F$q is one to one if one identifies Cauchyfluxes that differ onlyon null subsets ofS and interpretsqas Lebesgue classes of equivalence. Moreover, F is of class L^p_locif and onlyif q2L^p_locM^div(Rⁿ), 1p 1:

The fieldqis called the flux vectorcorresponding toF. Theorem 5.1 follows from the results of [7]; previous special cases are [13, 29, 24-25].

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For a given Cauchy fluxFwith the flux vectorqdefine

P: fM2P: qis dominated on@M andjdivqj_<n ₁(@M)0g;

S : fS2S :S@M for someM2Pg;

andF:S !Rby

F(S)

S

q^MdHⁿ ¹ 5:3

for anyS2S whereM2P; S@Mandq^Mis the normal trace ofqon

@M; which exists by Theorem 4.6. The function F is well defined by Remark 4.7.

THEOREM 5.2. If F is a Cauchyflux then F is a Cauchyflux and F(S)F(S)for a.e. S2S ;if F is a Cauchyflux of class L¹_locthen F is defined onS S :

The fluxFis definednaturallyas the densities from (5.3) satisfy the divergence theorem. Its domainScontains singular surfacesS(surfaces with divq^j S60) providedqis dominated onS.By (4.11),

F(S)F( S) divq(S) where Sis the surfaceSwith the opposite orientation.

6. Proof of Theorem 3.2 and Example 3.3.

PROOF OFTHEOREM3.2. (i): It suffices to prove thatjdivqj(B)0 for each Borel setBwithH^d(B)<1:LetBbe a Borel set withH^d(B)<1:

From the Hahn decomposition [27; Theorem 10.36] we deduce that there exist Borel sets BB with B\B ;; BBB such that divq^j B0: Our goal is to prove that divq(B)0: It suffices to prove only divq(B)0 for which in turn by [10; § 2.2.5] it suffices to prove that divq(K)0 for any compact subsetK of B: Thus letKB be compact. LetW:Rⁿ ![0;1] be given by

W(x)

1 if j jx <1;

2 j jx if 1j j x 2;

0 if j jx >2 8>

<

>:

and note thatWis a Lipschitz continuous function withjDWj 1 forLⁿa.e.

x2Rⁿ:Lete>0. Sinced<nandH^d(K)<1, we haveLⁿ(K)0:Using q2L^p_loc(Rⁿ);p<1; we deduce that there exists a bounded open setUwith

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KUandkqk_Lp(U)<e:Furthermore, by [10; Theorem 2.2.2(2)] this setU may be chosen so as to satisfy divq(UnK)<e:SinceKis compact, there exists and>0 such that for any ballB(x;r)withr<dandK\B(x;r)6 ;, we haveB(x;2r)U:From the definition ofH^d;there exists a covering of K by a finite system of balls B(x_i;r_i); i2I; with r_i<d and adP

i r_i^dH^d(B)1 (¹⁰). Thus S

i2IB(xi;2ri)U:LetW_i(x):W(r_i¹(x xi));

i2I; x2Rⁿ, and let

v(x)max fW_i(x):i2Ig

for everyx2Rⁿ:Then 0v1; vis Lipschitz continuous, andv1 on K. Since the support ofvis inU, we obtain

divq(K)

K

vdivq

U

DvqdLⁿ

UnK

vdivq 6:1

directly from the definition of divq(see (3.1)). We now estimate the right hand side. Since 0v1 and divq(UnK)e; we have

UnK

vdivq2e:

6:2

Further,

U

DvqdLⁿL^1=qkqk_Lp(U) L^1=qe 6:3

where q:p=(p 1) is the conjugate HoÈlder exponent and L

UjDvj^qdLⁿ:One easily finds that forLⁿa.e.x2Rⁿthere exists at least onei2Isuch thatDv(x)DW_i(x)and hence

L

U

jDvj^qdLⁿX

i2I

Rⁿ

jDW_ij^qdLⁿ

X

i2I

B(xi;2ri)

jDW_ij^qdLⁿ 2ⁿa_nX

i2I

r^{n q}_i 2ⁿa_n=a_d(H^d(B)1)

(¹⁰) In this proof we switch without change in notation from the Hausdorff measureH^dto the spherical measureS ^d, [10; §§ 2.10.2±6], which is possible by the inequalities H^dS ^d[(2n=(n1]^d=2H^d, see [10; § 2.10.6], which show that H^dandS^dhave the same system of null sets and sets of finite measure.

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where we note thatn qdandjDW_ij r_i¹onB(xi;2ri):Thus 0divq(K)(2ⁿan=ad(H^d(B)1))^1=qe2e

by (6.1), (6.2) and (6.3). The arbitrariness ofe>0 gives divq(K)0. Thus divq

j j(B)0.

(ii): We have to prove that ifBRⁿ is a Borel set withHⁿ ¹(B)0 then divj qj(B)0. Thus letHⁿ ¹(B)0; letBhave the same meaning as in the proof of part (i), letKbe a compact subset ofB, and prove that divq(K)0: Let e>0; let U be an open set with KU such that divq(UnK)<e;corresponding to thisUwe find and>0 as in the proof of part (i). Since Hⁿ ¹(K)0; there exists a covering of K by a finite system of ballsB(x_i;r_i); i2I; withr_i<dandP

i r_iⁿ ¹<e:Ifvis as above, then (6.2) holds while (6.3) is replaced by

U

DvqdLⁿLk kq _L1(U)

6:4

whereL

UjDvjdLⁿ:As above, LX

i2I

Rⁿ

DW_i

j jdLⁿX

i2I

B(xi;2ri)

DW_i

j jdLⁿ 2ⁿan

X

i2I

r_iⁿ ¹2ⁿane:

Thus we have

0divq(K)2ⁿanek kq _L1(U)2e

by (6.1), (6.2) and (6.4). Hence divq(K)0 and consequently

divq(B)0: p

PROPOSITION6.1. Letmis a signed Radon measure onRⁿ with compact support and

q(x): 1 na_n

Rⁿ

(x y)dm(y) x y j jⁿ 6:5

for everyx2Rⁿfor which

Rⁿ

x y

j j¹ ⁿdj j(y)m <1:Then (i) q2L¹_locM^div(Rⁿ)and

divqm;

6:6

(ii) if1p<n=(n 1)thenq2L^p_loc(Rⁿ);

(iii) if n=(n 1)p 1 then q2L^p_loc(Rⁿ) provided j j(B(x;m r)) cr^m for all x2Rⁿ and all 0<r<a, where m>d; a>0, c>0 are constants and d is as in Theorem3.2.

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PROOF. Writeh:j j;m let f(x) :

Rⁿ

x y

j j¹ ⁿdh(y)

for every x2Rⁿ so that f:Rⁿ![0;1] and prove that f2L^p_loc(Rⁿ) for everyp satisfying 1p<n=(n 1):By HoÈlder's inequality, with q the conjugate exponent,

f^p(x)k kh ^p=q

Rⁿ

jx yj ^p(n ¹⁾dh(y):

Therefore, ifz2Rⁿandr>0,

B(z;r)

f^p(x)dLⁿ(x)k kh ^p=q

Rⁿ

B(z;r)

x y

j j ^p(n ¹⁾dLⁿ(x)dh(y):

For anyy2Rⁿwe haveB(z;r)B(y;jz yj r)and therefore

B(z;r)

jx yj ^p(n ¹⁾dLⁿ(x)

B(y;jz yjr)

x y

j j p(n 1)dLⁿ(x)

C(jz yj)r)^{n p(n} ¹⁾ whereC:na_n=(n p(n 1)):Therefore

B(z;r)

f^p(x)dLⁿ(x)Ck kh ^p=q

Rⁿ

(jz yj r)^{n p(n} ¹⁾dh(y):

The last integrand is a bounded function ofyon the compact support ofh and thus the integral is finite. Hencef2L^p_loc(Rⁿ); thus in particular,fis finite for Lⁿ a.e. x2Rⁿ and henceq is defined Lⁿ a.e. onRⁿ. (i): By

q(x)

j j n ¹a_n¹(x) we see that q is locally integrable. Equation (6.6) is standard by noting thatqis the derivative of the Newton potential corresponding to the mass distributionm:(ii): Has been proved above. (iii): Letm satisfy the hypothesis of (iii) and letsbe any number such thatd<s<m:

Assume first thatp<1and denote byqthe conjugate exponent. Writing x y

j j¹ ⁿjx yj ^s=qjx yj^s=q1 ⁿ; we obtain by HoÈlder's inequality f^p(x)

x y

j j ^sdh(y)_p=q x y

j j^{p(s=q n1)}dh(y):

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Prove that there exists aC<1such that

x y

j j ^sdh(y)<C 6:7

for everyx2Rⁿ:Letx2Rⁿand writeM(r)h(B(x;r))forr>0 so that M(r)cr^m. Then, standardly,

x y

j j ^sdh(y)

B(x;a)

x y

j j ^sdh(y)

B(x;a)^c

x y j j ^sdh(y)

^a

0

r ^sdM(r)

B(x;a)^c

a ^sdh(y)

r ^sM(r)j^a₀s ^a

0

r ^s ¹M(r)dra ^skhk

cma^{m s}

m s a ^skhk :C which proves (6.7). Hence,

f^p(x)C^p=q

x y

j j^{p(s=q n1)}dh(y):

Thus ifz2Rⁿandr>0; we have, usingB(z;r)B(y;jz yj r);

B(z;r)

f^p(x)dLⁿ(x)C^p=q

Rⁿ

B(y;jz yjr)

x y

j j^{p(s=q n1)}dLⁿ(x)dh(y):

6:8

Bys>dwe havenp(s=q n1)>0;thus the inner integral is finite and equal to nan(np(s=q n1)) ¹(jz yj r)np(s=q n1); which is a bounded function of y on the compact support ofh:Thus the right hand side of (6.8) is finite. The casep 1is similar. p PROOF OFEXAMPLE3.3. (i): This follows from Proposition 6.1 (i), (ii). Proof of (ii): If 0mn then there exists a compact set K such that 0<H^m(K)<1and for some constantc,

H^m(K\B(x;r))cr^m

for all x2Rⁿ and r>0 [9; Corollary 4.12]. Choose any m such that d<m<s:Let m:H^m^j K and let q be given by (6.5). By Item (i) of Proposition 6.1, q2L¹_locM^div(Rⁿ)and divqm: The measuremsatisfies

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the hypothesis of Item (iii) of Proposition 6.1 and thusq2L^p_loc(Rⁿ):On the other hand, sincem<sandH^m(K)<1; we haveH^s(K)0: p

7. Proof of Proposition 4.1 and Theorem 4.2.

Let j be a radially symmetric mollifier on Rⁿ; for any r>0 let j_r(z)r ⁿj(z=r); z2Rⁿ:For anyf 2L¹_loc(Rⁿ)denote byfr2C¹(Rⁿ)the rmollification off,

f_r(x)

Rⁿ

j_r(x y)f(y)dLⁿ(y);

x2Rⁿ:Ifq2L¹_locM^div(Rⁿ)then divq_r(x)

Rⁿ

j_r(x y)divq(y)

for every x2Rⁿ: For anyMRⁿ let 1_M be the characteristic function of M.

LEMMA 7.1. Let M be a normalized set of finite perimeter and W2Lip₀(Rⁿ) such that W0 on @M. Then the function u:1_MW is in Lip₀(Rⁿ)and Du1_MDWforLⁿa.e. point inRⁿ:

PROOF. Letv2C₀¹(Rⁿ); by the Gauss-Green theorem for the setM and the functionvW;

Rⁿ

uDvdLⁿ

M

WDvdLⁿ

M

vDWdLⁿ

Rⁿ

v1_MDWdLⁿ

since vW0 on @M: Thus the weak derivative of u is Du1_MDW and satisfies j jDu Lip(W) forLⁿ a.e. point of Rⁿ:The mollifications u_r of u satisfy

Dur(x)

Rⁿ

j_r(x y)Du(y)dLⁿ;

henceDu_r(x)Lip(W)and consequentlyu_r(x) u_r(y)Lip(W)jx yj for everyx,y2Rⁿ:Furthermore,ur(x)!u(x)for everyx2Rⁿ:Indeed, if x2M it suffices to use that u is continuous on M, vanishes onM^c; and D(x;M)1; ifx2(M^c) thenu vanishes onM^c; is bounded onM, and

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D(x;M^c)1: If x2@M then j j u Lip(W)r on B(x;r): The limit in ur(x) ur(y)

Lip(W)jx yjgivesju(x) u(y)j Lip(W)jx yjfor each

x,y2Rⁿ: p

PROOF OFPROPOSITION4.1. Let Mbe a normalized set of finite perimeter. Firstly, note that (3.1) holds for every W2Lip₀(Rⁿ). Indeed, it suffices to apply (3.1) to the sequence of mollificationsW_rofWand letr!0:

Secondly, note that ifW2Lip₀(Rⁿ)andW0 on@Mthen

M

DWqdLⁿ

M

Wdivq0:

7:1

Indeed, the function u1MW is Lipschitz continuous by Lemma 7.1; the application of (3.1) tougives (7.1). For anyW2Lip₀(@M); defineN^M(q;W) by

N^M(q;W)

M

D~WqdLⁿ

M

~ Wdivq 7:2

where~Wis any Lipschitz extension ofWtoRⁿ with compact support. The existenceW~is easily deduced from the existence of an extension with the same Lipschitz constant ([10; Theorem 2.10.43]) and the fact that fx2@M;W(x)60g is bounded. We note that the value of the right hand side of (7.2) is independent of the extensionW~by (7.1). Moreover,N^M(q;)is linear since ifl2R;one can choose the extension corresponding tolWto be l~Wand similarly for the sum. This completes the proof of the existence of N^M(q;):To prove (4.2), it suffices to write (4.1) forMand forN, to add the results, and subtract (3.1). Next, letMbe bounded and prove that

jN^M(q;W)j Ck kW _Lip₀_(@M) 7:3

for some Cand allW2Lip₀(@M):Using a suitable cutoff function that is equal to 1 on the (bounded) closure ofM, one can show that the extension~W ofWcan be chosen as to satisfy

~ W

k k_Lip₀_(Rⁿ₎Dk kW _Lip₀_(@M) 7:4

whereDis a constant independent ofW:But then from (7.2) and (7.4) we obtain (7.3) whereCD(

Mj jdLq ⁿkdivqk):Finally, ifM^c is bounded, thenN^N(q;)is continuous and (4.2) establishes the continuity ofN^M(q;):

p LEMMA 7.2. Letq2L¹_locM^div(Rⁿ) and let M be a normalized set of finite perimeter.

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(i) If q is weaklydominated on @M then the normal trace is represented bya measuren^Mp^M s^Mwhere

p^Mw lim

r!0 q_r@Mn^MHⁿ ¹ ^j @M inM(Rⁿ);

7:5

s^Mw lim

r!0 m_r in M(Rⁿ);

7:6

whereq_ris the mollification ofq,m_r2M(Rⁿ)is given by hm_r;Wi

@M

M

W(y)j_r(y x)dLⁿ(y)ddivq(x)

for everyW2C₀(Rⁿ); and w* denotes the weak* convergence, understood along an appropriate sequence ofrtending to0;the support ofn^Mis in@M.

(ii) Ifqis dominated on@M thenp^M q₀Hⁿ ¹^j @M where q₀w lim

r!0 q_r_@Mn^M in L¹(@M;Hⁿ ¹) where w denotes the weak convergence.

PROOF(i): Sinceqis dominated on@M; there exists a sequencer_j!0 and aC<1such that (4.3) holds. Since for eachx2@Mandr>0;

q_r(x)n^M(x)

D

B(x;r)

q(y)n^M(x)

dLⁿ(y)

whereDis the maximum ofj; we deduce from (4.3) that

@M

q_r(x)n^M(x)

dHⁿ ¹(x)E

for some E<1 and all rr_j; j2N: Thus the total variation of q_rj_@Mn^MHⁿ ¹^j @Mis bounded and hence, for some subsequence ofr_j; still denoted byr; the limit in (7.5) exists. Further, one easily finds that jhm_r;Wij divqj(@M)k kW _L1(Rⁿ) for each W2C0(Rⁿ) and each r; i.e., km_rk divqj(@M):Thus for some subsequence ofr_j; still denoted byr; the limit in (7.5) exists. LetW2Lip₀(Rⁿ):The divergence theorem for smooth vector fields and sets of finite perimeter reads

@M

Wq_rn^MdHⁿ ¹

M

DWq_rdLⁿ

M

Wdivq_rdLⁿ; 7:7

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and

M

DWq_rdLⁿ!

M

DWqdLⁿ;

@M

Wq_rn^MdHⁿ ¹!

Rⁿ

Wdp^M

sinceq_r!qinL¹_loc(Rⁿ)and by (7.5). Next note that

M

Wdivq_rdLⁿ

Rⁿ

W_rdivq

where

W_r(x)

M

W(y)j_r(x y)dLⁿ(y);

7:8

and write

Rⁿ

W_rdivq

M

W_rdivq

(M^c)

W_rdivq

@M

W_rdivq:

7:9

The three terms on the right hand side of (7.9) converge, respectively, to

M

Wdivq; 0;

Rⁿ

Wds^M:

The first two limits follow from (¹¹)

W_r(x)! W(x) for everyx2M;

0 for everyx2(M^c)

by the dominated convergence theorem, while the last limit is (7.6) by observing that

@MW_rdivq hm_r;Wi:To summarize, the limit in (7.7) gives

Rⁿ

Wdp^M

M

DWqdLⁿ

Rⁿ

Wds^M

M

Wdivq:

Thus the normal trace N^M(q;) is represented by a measure n^M: The support ofn^M is in@M. Indeed, the support ofq_rj_@_Mn^MHⁿ ¹^j @Mis in @M and to show that the support of s^M is in @M, we note that for each r<r; the support ofm_r is infx2Rⁿ:dist(x; @M)rg:

(¹¹) Here we use thatMis anormalizedset of finite perimeter.

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(ii): If q is dominated on @M; there exists a sequence r_j!0 and a g2L¹(@M;Hⁿ ¹)such that (4.4) holds forHⁿ ¹ a.e.x2@M:IfDis the maximum ofj; then

jq_r(x)n^M(x)j Dg(x)

forHⁿ ¹a.e.x2@Mand everyrr_j; j2N:The sequence of functions q_rn^M is thus Hⁿ ¹ equiintegrable on @M and hence [8; Corollary 11,

§ IV.8] there exists a subsequence of r_j; still denoted r; and a q₀2 2L¹(@M;Hⁿ ¹)such that

q_rn^M*q0 in L¹(@M;Hⁿ ¹)

The limit in (7.5) isp^Mq0Hⁿ ¹^j @M: p

PROOF OFTHEOREM4.2. (i): Follows from Lemma 7.2(i).

(ii): We only have to prove that the measuren^M of (i) is supported on

@M: By Lemma 7.2(ii) we have p^Mq₀Hⁿ ¹^j @M: The measure s^M satisfies

hs^M;Wi lim

r!0

@M

W_rdivq

whereW_ris given by (7.8). At everyx2@MwhereD(x;M)exists we have W_r(x)!D(x;M)W(x):By (4.5) and Theorem 3.2,jdivqj(@_sM)0:Thus we have thatW_r!D(;M)Wforjdivqj^j @Ma.e.x2Rⁿ:Hence

limr!0

@M

W_rdivq!

@M

W(x)D(x;M)divq(x):

Thuss^MD(;M)divq^j @Mand consequently

n^Mq0Hⁿ ¹^j @M D(;M)divq^j @M;

which is a measure supported on@M: p

8. Proof of Theorems 4.4 and 4.6.

Ifh2M(Rⁿ)and 0m<1; themdimensional upper density ofhby u^m(x;h)lim

r!0sup h(B)

a_mr^m:Ban open ball of radiusrr; x2B

; x2Rⁿ: