An Anzellotti type pairing for divergence-measure fields and a notion of weakly super-$1$-harmonic functions

0.15 MB·

by  Christoph Scheven

An Anzellotti type pairing for divergence-measure fields
and a notion of weakly super-1-harmonic functions
7
Christoph Scheven∗ Thomas Schmidt†‡
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2

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Abstract
a
J
We study generalized products of divergence-measure fields and gradient measures of BV
0
functions. These products depend on the choice of a representative of the BV function, and
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herewesingleoutaspecificchoicewhichissuitableinordertodefineandinvestigateanotion

of weak supersolutions for the 1-Laplace operator.
]
P
A
1 Introduction
.
h
at For a positive integer n and an open set Ω in Rn, we consider the 1-Laplace equation
m
Du
[ div ≡0 on Ω. (1)
|Du|

1
v This equation formally arises as the Euler equation of the total variation and is naturally posed
6 for functions u: Ω → R of locally bounded variation whose gradient Du is merely a measure. In
5
order to make sense of (1) in this setting it has become standard[12, 9, 10, 5, 13, 14] to work with
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a generalized product, which has been studied systematically by Anzellotti [2, 3]. The product is
2
defined for u∈BV (Ω) and σ ∈L∞(Ω,Rn) with vanishing distributional divergence divσ as the
0 loc loc
. distribution
1 Jσ,DuK..=div(uσ)∈D′(Ω),
0
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and in fact the pairing Jσ,DuK turns out to be a signed Radon measure on Ω. By requiring
1
: kσkL∞(Ω,Rn) ≤1 and Jσ,DuK=|Du| one can now phrase precisely what it means that σ takes over
v the role of Du , and whenever there exists some σ with all these properties (including divσ ≡ 0),
i |Du|
X onecallsu∈BV (Ω)aBVsolutionof (1)oraweakly1-harmonicfunctiononΩ. Inasimilarvein,
loc
r variants of the pairing Jσ,DuK can be used to define BV solutions u of div Du = f for right-hand
a |Du|
sidesf ∈Ln (Ω)andtoexplainBV∩L∞ solutionsuofdiv Du =f evenforarbitraryf ∈L1 (Ω).
loc |Du| loc
In this note we deal with a notion of supersolutions of (1) or — this is essentially equivalent
— of solutions of div Du = −µ with a Radon measure µ on Ω. To this end, we first collect some
|Du|
preliminaries in Section 2. Then, in Section 3, we consider generalized pairings, which make sense
∗Fakult¨atfu¨rMathematik,Universit¨atDuisburg-Essen,Thea-Leymann-Str. 9,45127Essen,Germany.
Emailaddress: [email protected]. URL:http://www.uni-due.de/mathematik/ag scheven.
†FachbereichMathematik, Universit¨atHamburg,Bundesstr. 55,20146Hamburg,Germany.
Emailaddress: [email protected]. URL:http://www.math.uni-hamburg.de/home/schmidt.
‡Formerinstitution: DepartmentMathematik,Friedrich-Alexander-Universit¨atErlangen-Nu¨rnberg.
1
even for L∞ divergence-measurefields σ, but require precise evaluations of u∈BV (Ω)∩L∞(Ω)
loc loc
up to sets of zero (n−1)-dimensional Hausdorff measure Hn−1. We mainly investigate a pairing
Jσ,Du+K,whichisbuiltwithaspecificHn−1-a.e.definedrepresentativeu+ ofuandwhichdoesnot
seem to have been considered before (while a similar pairing with the mean-value representative
u∗ already occurred in [7, Theorem 3.2] and [14, Appendix A]). Adapting the approach in [4,
Section 5], we moreover deal with an up-to-the-boundary pairing Jσ,Du+K , which accounts for
u0
a boundary datum u . In Section 4 we employ the local pairing Jσ,Du+K in order to introduce a
0
notionofweaklysuper-1-harmonicfunctions,andweproveacompactnessstatementwhichcrucially
dependsonthechoiceoftherepresentativeu+. Finally,Section5isconcernedwitharefinednotion
of super-1-harmonicity which incorporates Dirichlet boundary values. This last notion is based on
(a modification of) the pairing Jσ,Du+K .
u0
We emphasize that the proofs, which are omitted in this announcement, can be found in the
companion paper [15], where we also provide a more detailed study of pairings and supersolu-
tions together with adaptions to the case of the minimal surface equation. Furthermore, in our
forthcoming work [16], we will discuss connections with obstacle problems and convex duality.
2 Preliminaries
L∞ divergence-measure fields. We record two results related to the classes
DM∞(Ω,Rn)..={σ ∈L∞(Ω,Rn) : divσ exists as a signed Radon measure on Ω},
loc loc
DM∞(Ω,Rn)..={σ ∈L∞(Ω,Rn) : divσ exists as a finite signed Borel measure on Ω}.
Lemma 2.1 (absolute-continuity property for divergences of L∞ vector fields). Consider σ ∈
DM∞(Ω,Rn). Then, for every Borel set A⊂Ω with Hn−1(A)=0, we have |divσ|(A)=0.
loc
Lemma 2.1 has been proved by Chen & Frid [7, Proposition 3.1].
Lemma 2.2 (finiteness of divergences with a sign). If Ω is bounded with Hn−1(∂Ω) < ∞ and
σ ∈L∞(Ω,Rn)satisfiesdivσ ≤0inD′(Ω), thenwenecessarilyhaveσ ∈DM∞(Ω,Rn). Moreover,
there holds (−divσ)(Ω)≤ ωnnω−n1kσkL∞(Ω,Rn)Hn−1(∂Ω) with the volume ωn of the unit ball in Rn.
Indeed, it follows from the Riesz representation theorem that divσ in Lemma 2.2 is a Radon
measure. The finiteness of this measure will be established in [15] by a reasoning based on the
divergence theorem.
BV-functions. We mostly follow the terminology of [1], but briefly comment on additional con-
ventions and results. For u ∈ BV(Ω), we recall that Hn−1-a.e. point in Ω is either a Lebesgue
point (also called an approximate continuity point) or an approximate jump point of u; compare
[1, Sections 3.6,3.7]. We write u+ for the Hn−1-a.e. defined representative of u which takes the
Lebesgue values in the Lebesgue points and the larger of the two jump values in the approximate
jump points. Correspondingly, u− takes on the lesser jump values, and we set u∗ ..= 1(u++u−).
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Finally, if Ω has finite perimeter in Rn, we write uint for the Hn−1-a.e. defined interior trace of
∂∗Ω
u on the reduced boundary ∂∗Ω of Ω (compare [1, Sections 3.3,3.5,3.7]), and in the case that
Hn−1(∂Ω\∂∗Ω)=0 we also denote the same trace by uint.
∂Ω
The following two lemmas are crucial for our purposes. The first one extends [1, Proposition
3.62] and is obtained by essentially the same reasoning. The second one follows by combining [6,
Theorem 2.5] and [8, Lemma 1.5, Section 6]; compare also [11, Sections 4,10].
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Lemma2.3(BVextensionbyzero). IfwehaveHn−1(∂Ω)<∞,thenforeveryu∈BV(Ω)∩L∞(Ω)
we have 1Ωu∈BV(Rn) and |D(1Ωu)|(∂Ω)≤ ωnnω−n1kukL∞(Ω)Hn−1(∂Ω). In particular, Ω is a set of
finite perimeter in Rn, and uint is well-defined.
∂∗Ω
Lemma 2.4 (Hn−1-a.e. approximation of a BV function from above). For every u ∈ BV (Ω)∩
loc
L∞(Ω) there exist v ∈W1,1(Ω)∩L∞(Ω) such that v ≥v ≥u holds Ln-a.e. on Ω for every ℓ∈N
loc ℓ loc loc 1 ℓ
and such that v∗ converges Hn−1-a.e. on Ω to u+.
ℓ
∞
3 Anzellotti type pairings for L divergence-measure fields
We first introduce a local pairing of divergence-measure fields and gradient measures.
Definition 3.1 (local pairing). Consider u ∈ BV (Ω)∩L∞(Ω) and σ ∈ DM∞(Ω,Rn). Then
loc loc loc
— since Lemma 2.1 guarantees that u+ is |divσ|-a.e. defined — we can define the distribution
Jσ,Du+K..=div(uσ)−u+divσ ∈D′(Ω).
Written out this definition means
Jσ,Du+K(ϕ)=− uσ·Dϕdx− ϕu+d(divσ) for ϕ∈D(Ω). (2)
Z Z
Ω Ω
Nextwedefineaglobalpairing,whichincorporatesDirichletboundaryvaluesgivenbyafunction
u .
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Definition 3.2 (up-to-the-boundary pairing). Consider u ∈ W1,1(Ω) ∩L∞(Ω), u ∈ BV(Ω) ∩
0
L∞(Ω), and σ ∈DM∞(Ω,Rn). Then we define the distribution Jσ,Du+K ∈D′(Rn) by setting
u0
Jσ,Du+K (ϕ)..=− (u−u )σ·Dϕdx− ϕ(u+−u∗)d(divσ)+ ϕσ·Du dx (3)
u0 Z 0 Z 0 Z 0
Ω Ω Ω
for ϕ∈D(Rn).
We emphasize that the pairings in Definitions 3.1 and 3.2 coincide on ϕ with compact support
in Ω (since an integration-by-parts then eliminates u in (3)). However, the up-to-the-boundary
0
pairing stays well-defined even if ϕ does not vanish on ∂Ω. In addition, we remark that both
pairings can be explained analogously with other representatives of u.
In some of the following statements we impose a mild regularity assumption on ∂Ω, namely we
require
Hn−1(∂Ω)=P(Ω)<∞, (4)
where P stands for the perimeter. We remark that the condition (4) is equivalent to having
P(Ω) < ∞ and Hn−1(∂Ω\∂∗Ω) = 0 and also to having 1 ∈ BV(Rn) and |D1 | = Hn−1 ∂Ω.
Ω Ω
For a more refined discussionwe refer to [17], where the relevance of (4) for certain approximation
results is pointed out.
Two vital properties of the pairing are recordedin the next statements. The proofs will appear
in [15].
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Lemma 3.3 (the pairing trivializes on W1,1-functions).
• (local statement) For u∈W1,1(Ω)∩L∞(Ω), σ ∈DM∞(Ω,Rn), and ϕ∈D(Ω), we have
loc loc loc
Jσ,Du+K(ϕ)= ϕσ·Dudx.
Z
Ω
• (global statement with traces) If Ω is bounded with (4), then for every σ ∈ DM∞(Ω,Rn)
there exists a uniquely determined normal trace σ∗ ∈L∞(∂Ω;Hn−1) with
n
kσn∗kL∞(∂Ω;Hn−1) ≤kσkL∞(Ω,Rn)
such that for all u,u ∈W1,1(Ω)∩L∞(Ω) and ϕ∈D(Rn) there holds
0
Jσ,Du+K (ϕ)= ϕ(σ·Du)dx+ ϕ(u−u )intσ∗dHn−1.
u0 Z Z 0 ∂Ω n
Ω ∂Ω
Next we focus on bounded σ with divσ ≤ 0. By Lemma 2.2 the pairings stay well-defined in
this case.
Proposition3.4(thepairingisaboundedmeasure). Fixσ ∈L∞(Ω,Rn)withdivσ ≤0inD′(Ω).
• (local estimate) For u ∈ BV (Ω) ∩ L∞(Ω), the distribution Jσ,Du+K is a signed Radon
loc loc
measure on Ω with
|Jσ,Du+K|≤kσkL∞(Ω,Rn)|Du| on Ω.
• (globalestimate withequalityatthe boundary) If Ω is bounded with (4), for u ∈W1,1(Ω)∩
0
L∞(Ω) and u ∈ BV(Ω)∩L∞(Ω) the pairing Jσ,Du+K is a finite signed Borel measure on
u0
Rn with
Jσ,Du+Ku0 −(u−u0)i∂nΩtσn∗Hn−1 ∂Ω ≤kσkL∞(Ω,Rn)|Du| Ω. (5)
(cid:12) (cid:12)
(cid:12) (cid:12)
4 Weakly super-1-harmonic functions
We now give a definition of super-1-harmonic functions, which employs the convenient notation
S∞(Ω,Rn)..={σ ∈L∞(Ω,Rn) : |σ|≤1 holds Ln-a.e. on Ω}.
Definition 4.1 (weakly super-1-harmonic). We call u ∈ BV (Ω) ∩ L∞(Ω) weakly super-1-
loc loc
harmonic on Ω if there exists some σ ∈ S∞(Ω,Rn) with divσ ≤ 0 in D′(Ω) and Jσ,Du+K = |Du|
on Ω.
We next provide a compactness result for super-1-harmonic functions. We emphasize that this
result does not hold anymore if one replaces u+ by any other representative of u in the definition.
We also remark that the assumed type of convergence is very natural, and indeed the statement
applies to every increasing sequence of super-1-harmonic functions which is bounded in BV (Ω)
loc
and L∞(Ω).
loc
Theorem 4.2 (convergence from below preserves super-1-harmonicity). Consider a sequence of
weakly super-1-harmonic functions u on Ω. If u locally weak∗ converges to a limit u both in
k k
BV (Ω) and L∞(Ω) and if u ≤ u holds on Ω for all k ∈ N, then u is weakly super-1-harmonic
loc loc k
on Ω.
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Proof. In view of Definition 4.1 there exist σ ∈ S∞(Ω,Rn) with divσ ≤ 0 in D′(Ω) and
k k
Jσ ,Du+K=|Du | onΩ. Possiblypassingto asubsequence,we assumethat σ weak∗convergesin
k k k k
L∞(Ω,Rn) to σ ∈S∞(Ω,Rn) with divσ ≤0 in D′(Ω), andas before we regarddivσ anddivσ as
k
non-positive measures on Ω. We fix a non-negative ϕ∈D(Ω) and approximations v of u with the
ℓ
properties of Lemma 2.4. Relying on(2), Lemma 2.1, and the dominated convergencetheorem,we
then infer
Jσ,Du+K(ϕ)=− uσ·Dϕdx+ ϕu+d(−divσ)
Z Z
Ω Ω
= lim − v σ·Dϕdx+ ϕv∗d(−divσ) = lim Jσ,Dv+K(ϕ).
ℓ→∞(cid:20) ZΩ ℓ ZΩ ℓ (cid:21) ℓ→∞ ℓ
Since the pairing trivializes on v ∈ W1,1(Ω), we can exploit the local weak∗ convergence of σ in
ℓ loc k
L∞(Ω,Rn) and the inequalities v∗ ≥u+ ≥u+ to arrive at
loc ℓ k
Jσ,Dv+K(ϕ)= lim Jσ ,Dv+K(ϕ)=− lim v σ ·Dϕdx+ lim ϕv∗d(−divσ )
ℓ k→∞ k ℓ k→∞ZΩ ℓ k k→∞ZΩ ℓ k
≥− v σ·Dϕdx+liminf ϕu+d(−divσ ).
ZΩ ℓ k→∞ ZΩ k k
Next we rely in turn on the dominated convergence theorem, on the observation that u σ locally
k k
weakly convergesto uσ in L1 (Ω,Rn), onthe definition in (2), on the coupling Jσ ,Du+K=|Du |,
loc k k k
and finally on a lower semicontinuity property of the total variation. In this way, we deduce
lim − v σ·Dϕdx+liminf ϕu+d(−divσ )
ℓ→∞(cid:20) ZΩ ℓ k→∞ ZΩ k k (cid:21)
=− uσ·Dϕdx+liminf ϕu+d(−divσ )
ZΩ k→∞ ZΩ k k
=liminf − u σ ·Dϕdx+ ϕu+d(−divσ )
k→∞ (cid:20) ZΩ k k ZΩ k k (cid:21)
=liminfJσ ,Du+K(ϕ)=liminf ϕd|Du |
k→∞ k k k→∞ ZΩ k
≥ ϕd|Du|.
Z
Ω
Collecting the estimates, we arrive at the inequality Jσ,Du+K ≥ |Du| of measures on Ω. Since the
oppositeinequalityisgenerallyvalidbyProposition3.4,weinferthatuisweaklysuper-1-harmonic
on Ω.
5 Super-1-harmonic functions with respect to Dirichlet data
Finally,weintroduceaconceptofsuper-1-harmonicfunctionswithrespecttoageneralizedDirichlet
boundarydatum. In[16]wewillshowthatthisnotionisusefulinconnectionwithobstacleproblems.
Concretely, consider a bounded Ω with (4), u ∈ W1,1(Ω)∩L∞(Ω), and u ∈ BV(Ω)∩L∞(Ω).
0
WethenextendthemeasureDuonΩtoameasureD uonΩwhichtakesintoaccountthepossible
u0
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deviation of uint from the boundary datum (u )int. To this end, writing ν for the inward unit
∂Ω 0 ∂Ω Ω
normal of Ω, we set
D u..=Du Ω+(u−u )intν Hn−1 ∂Ω.
u0 0 ∂Ω Ω
Now, for σ ∈ S∞(Ω,Rn) with divσ ≤ 0 in D′(Ω) — which is meant to potentially satisfy a
coupling like Jσ,Du+K = |D u| on Ω — we adopt the viewpoint that σ∗ should typically equal
u0 u0 n
the constant 1. If this is not the case, we compensate for this defect by extending (−divσ) to a
measure on Ω with
(−divσ) ∂Ω..=(1−σ∗)Hn−1 ∂Ω. (6)
n
Then we define a modified pairing Jσ,Du+K∗ by interpreting u+ as max{uint,(u )int} on ∂Ω and
u0 ∂Ω 0 ∂Ω
extendingthe(divσ)-integralin(3)fromΩtoΩ. Inotherwords,wedefinethemeasureJσ,Du+K∗
u0
by setting
Jσ,Du+K∗ ..=Jσ,Du+K + (u−u )int (−divσ) ∂Ω. (7)
u0 u0 0 ∂Ω +
(cid:2) (cid:3)
With these conventions, we now complement Definition 4.1 as follows.
Definition 5.1 (super-1-harmonic function with respect to a Dirichlet datum). For bounded Ω
with (4) and u ∈W1,1(Ω)∩L∞(Ω), we say that u ∈BV(Ω)∩L∞(Ω) is weakly super-1-harmonic
0
on Ω with respect to u if there exists some σ ∈ S∞(Ω,Rn) with divσ ≤ 0 in D′(Ω) such that the
0
equality of measures Jσ,Du+K∗ =|D u| holds on Ω.
u0 u0
For σ ∈ S∞(Ω,Rn) with divσ ≤ 0 in D′(Ω) and u ∈W1,1(Ω)∩L∞(Ω), u ∈BV(Ω)∩L∞(Ω),
0
we get from (5), (6), (7)
Jσ,Du+K∗ ∂Ω= (u−u )int − (u−u )int σ∗ Hn−1 ∂Ω.
u0 0 ∂Ω + 0 ∂Ω − n
(cid:16)(cid:2) (cid:3) (cid:2) (cid:3) (cid:17)
Thus, the boundary condition in Definition 5.1 is equivalent to the Hn−1-a.e. equality σ∗ ≡−1 on
n
theboundaryportion{uint <(u )int}∩∂Ω,whilenorequirementismadeon{uint ≥(u )int}∩∂Ω.
∂Ω 0 ∂Ω ∂Ω 0 ∂Ω
Webelievethatthisisverynatural,inparticularinthecasen=1,wheresuper-1-harmonicityofu
onaninterval[a,b]just meansthatu is increasingupto acertainpointanddecreasingafterwards,
andwhereσ∗ cantakethevalue−1atmostatone endpointandonlyifuismonotoneontheopen
n
interval (a,b).
Another indication that Definition 5.1 is meaningful is provided by the next statement, which
will also be proved in [15]. We emphasize that the statement does not hold anymore (not even for
n=1,u =u ≡0,andu ∈W1,1(Ω))ifonereplacesJσ,Du+K∗ withJσ,Du+K inthedefinition.
0;k 0 k 0 u0 u0
Theorem5.2. SupposethatΩisboundedwith (4),andconsiderweaklysuper-1-harmonicfunctions
u ∈BV(Ω)∩L∞(Ω) on Ω with respect toboundary data u ∈W1,1(Ω)∩L∞(Ω). If u converges
k 0;k 0;k
strongly in W1,1(Ω) and weakly∗ in L∞(Ω) to some u , and if u weak∗ converges to a limit u in
0 k
BV(Ω) and L∞(Ω) such that u ≤ u holds on Ω for all k ∈ N, then u is weakly super-1-harmonic
k
on Ω with respect to u .
0
Remark. In the situation of the theorem, it follows from the previously recorded reformulation of
the boundary condition that u is also weakly super-1-harmonic on Ω with respect to every u ∈
0
W1,1(Rn)∩L∞(Ω) such that Hn−1({u∗ ≤ uint < u∗}∩∂Ω) = 0. Roughly speaking, this means
0 ∂Ω 0
that the boundary values can always be decreased and that they can even be increased as long aes the
trace of u is not traversed. In view of the 1-dimensieonal case, we believe that this behavior is very
reasonable.
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