Minimizing <span class="mathjax-formula">${L}^{\infty }$</span>-norm functional on divergence-free fields

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www.elsevier.com/locate/anihpc

Minimizing L ^∞ -norm functional on divergence-free fields

Baisheng Yan

Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA Received 15 November 2010; accepted 22 December 2010

Available online 18 February 2011

Abstract

In this paper, we study the minimization problem on theL^∞-norm functional over the divergence-free fields with given boundary normal component. We focus on the computation of the minimum value and the classification of certain special minimizers including the so-called absolute minimizers. In particular, several alternative approaches for computing the minimum value are given usingL^q-approximations and the sets of finite perimeter. For problems in two dimensions, we establish the existence of absolute minimizers using a similar technique for the absolute minimizers ofL^∞-functionals of gradient fields. In some special cases, precise characterizations of all minimizers and the absolute minimizers are also given based on equivalent descriptions of the absolutely minimizing Lipschitz extensions of boundary functions.

MSC:49J45; 49K30; 26B30; 35J92

Keywords:L^∞-norm functional; Divergence-free field; BV function; Power-law approximation; 1-Laplacian-type equation; Absolute minimizer

1. Introduction and main results

LetΩbe a bounded domain inRⁿ(n2) with Lipschitz continuous boundary∂Ωandσ be a positive continuous function onΩ¯. GivenH:Ω→Rⁿandβ:∂Ω→R, we study the value of the following minimization problem:

ρ(β, H )= min

G∈Sβ(Ω)

G+H σ

L^∞(Ω)

, (1.1)

whereSβ(Ω)is the set of all divergence-free fieldsGinΩ of fixed boundary normal-componentG·ν|∂Ω=β.

The motivation for studying such a problem is two-folds. First, in many variational problems, it is typical that finding the best constant for some inequalities to hold or the certain threshold condition for a problem to have some special solutions will eventually lead to computing optimal values involving theL^∞-norm of divergence-controlled quantities [4,8,12,13,22]. It is certainly desirable to find alternative ways to compute such values. Second, the study of theL^∞-norm of divergence-free fields is a special case of the study of the L^∞-functionals of general functions with certain A-quasiconvexity [9,15]. In working with the special problem (1.1) for divergence-free fields, we are hoping to further explore the similar ideas from the study of gradient fields, as in [6,7,19,23]. In particular, for our problem (1.1), we would like to study whether some special (hopefully unique) minimizers can be obtained through

E-mail address:yan@math.msu.edu.

doi:10.1016/j.anihpc.2011.02.004

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certain underlying selection principles similar to those for the viscosity solutions and the absolute minimizers in the gradient case.

We first address the issues concerning the alternative ways to compute the value ρ(β, H ). Motivated by some results in [10], we assume the following natural conditions forHandβ:

H∈L^∞ Ω;Rⁿ

, divH∈Lⁿ(Ω); β∈L^∞(∂Ω),

∂Ω

β dHⁿ⁻¹=0. (1.2)

The space of functionsH satisfying the conditions in (1.2) will be denoted byXn(Ω)and has been studied by many authors [5,8,11–13], even with divH being a Radon measure. In particular, ifH∈Xn(Ω), then anormal-component H·ν∈L^∞(∂Ω)can be definedHⁿ⁻¹-a.e. on∂Ωin such a way that the generalized divergence formula

Ω

ζdivH dx=

∂Ω

ζ (H·ν) dHⁿ⁻¹−

Ω

H· ∇ζ dx (1.3)

holds for allH ∈Xn(Ω)andζ ∈W^1,1(Ω); this formula can be extended toζ ∈BV(Ω),the space of functions of bounded variation inΩ.The admissible classSβ(Ω)is defined by

Sβ(Ω)=

G∈L^∞

Ω;Rⁿ divG=0, G·ν=β

(1.4) and is nonempty under the assumptions onβ in (1.2). ForG∈Sβ(Ω)andζ ∈W^1,1(Ω), by (1.3), it follows that

∂Ω

ζ (β+H·ν)−

Ω

ζdivH=

Ω

(G+H )· ∇ζ G+H

σ

L^∞(Ω)

Ω

σ|∇ζ| and hence we have that

ρ(β, H ) sup

ζ∈W^1,1(Ω) ζ=const

∂Ωζ (β+H·ν) dHⁿ⁻¹− _ΩζdivH dx

Ωσ|∇ζ|dx . (1.5)

One of the main motivations of the paper is that the equality holds in this relation:

Theorem 1.1.LetH, βsatisfy(1.2). Thenρ(β, H )=μ(β+H·ν,divH ),where the quantityμ(g, h)is defined by μ(g, h)= sup

ζ∈W^1,1(Ω) ζ=const

∂Ωζ g dHⁿ⁻¹− _Ωζ h dx

Ωσ (x)|∇ζ|dx (1.6)

for functionsgandhsatisfying the condition g∈L^∞(∂Ω), h∈Lⁿ(Ω),

∂Ω

g dHⁿ⁻¹=

Ω

h dx. (1.7)

The optimization problem (1.6) is similar to the problems appearing in several important studies, such as the dual variational principle for plasticity [12], the best constant for the Sobolev trace-embedding ofW^1,1(Ω)intoL¹(∂Ω) [4,24], the eigenvalue problem for 1-Laplacian operator [8,13], and the generalized Cheeger problems [2,18,20]. One readily verifies that

μ(g, h)= sup

ζ∈BV(Ω) ζ=const

∂Ωγ (ζ )g dHⁿ⁻¹− _Ωζ h dx

Ωσ d|Dζ| , (1.8)

whereγ:BV(Ω)→L¹(∂Ω)is the trace operator and|Dζ|is the total variation measure of the vector Radon mea- sureDζ. The formula (1.8) is different from the one used in the generalized Cheeger problem studied in [18] because we do not have any boundary condition onζ∈BV(Ω).However, as in [18], we will see that the numberμ(g, h)can also be characterized in terms of sets of finite perimeter instead of functions of bounded variation (see Theorem 3.3 below).

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We discuss another approach forμ(g, h)based on approximation by power-law functionals. LetB:BV(Ω)→R be defined by

B(ζ )=

∂Ω

γ (ζ )g dHⁿ⁻¹−

Ω

ζ h dx. (1.9)

Note thatB≡0 if and only ifg=h=0.AssumeB≡0.We define the following constrained minimization problems:

λ(g, h)= inf

ζ∈BV(Ω) B(ζ )=1

Ω

σ d|Dζ|, λ_p(g, h)= inf

ζ∈W^1,p(Ω) B(ζ )=1

Ω

σ^p|∇ζ|^p (1p <∞). (1.10) IfB≡0 (i.e.,g=h=0), defineλ(0,0)=λ_p(0,0)= +∞.

The following result provides another way to compute the quantityμ(g, h).

Theorem 1.2.It follows thatλ₁(g, h)=λ(g, h)=_μ(g,h)¹ >0and lim

p→1⁺

λ_p(g, h)=λ₁(g, h)=λ(g, h). (1.11)

AssumeB≡0.For each 1< p <∞, a standard direct method in the calculus of variations shows that there exists auniquefunctionu_p∈W^1,p(Ω)withB(u_p)=1 satisfying _Ωu_pdx=0 that minimizes the problem (1.10); that is,

Ωσ^p|∇u_p|^p=λ_p(g, h).We have the following result.

Theorem 1.3. There exist subsequence p_j →1⁺ as j → ∞, functions u¯∈BV(Ω) and F¯ ∈ L^∞(Ω;Rⁿ) with ¯FL^∞(Ω)1such that, asj→ ∞,

u_p_j u¯ inLⁿ⁻ⁿ¹(Ω), ∇u_p_j D^∗ u¯ as measures onΩ, (1.12)

|∇up_j|^p^j⁻²∇up_j F¯ inL^r(Ω)for anyr >1, (1.13)

div(σF )¯ =λh, σF¯·ν=λg, whereλ=λ(g, h). (1.14)

The functionu¯so determined is a minimizer forλ(g, h)in BV(Ω)if and only ifB(u)¯ =1.

Since the trace operatorγ:BV(Ω)→L¹(∂Ω)is not continuous under the weak-star convergence ofBV(Ω), one may not have B(u)¯ =1 for the function u¯ determined in the theorem; so u¯ may not be a minimizer forμ(g, h) inBV(Ω). But any such limitu¯ is a weak solution to a Neumann problem for a 1-Laplacian-type equation (see Remark 3.1). In general, the existence of a minimizer forλ(g, h) inBV(Ω)is unknown. However, under certain conditions (see Theorem 5.2 below), any such functionu¯will satisfyB(u)¯ =1 and hence is a minimizer forλ(g, h).

We now address the issues concerning the special (hopefully unique) minimizers forρ(β, H ) in problem (1.1).

Note that, withg=β+H·ν,h=divHandF¯ so determined in Theorem 1.3, the relationship G¯ = σF¯

λ(g, h)−H (1.15)

defines a minimizerG¯ forρ(β, H ); all such minimizersG¯ can also be characterized by minimizing theL^q-norm as q→ ∞(see Proposition 4.2 below) in much similar way as for theL^∞-functionals of gradients of scalar functions [6,7,19]. TheΓ-convergence of the general power-law functionals of divergence-free fields as power tends to infinity has been studied in [9]. However, unlike the gradient case, viscosity and comparison principles seem intractable for our problem (1.1) with divergence-free vector-fields. Instead, we focus on the principle of absolute minimizers. In a natural analogy to the absolute minimizers forL^∞-functionals of gradients, we make the following definition.

Definition 1.1.A minimizerG¯ ∈Sβ(Ω)forρ(β, H )is called anabsolute minimizerprovided that^G^¯⁺_σ^HL^∞(E) ^G⁺_σ^HL^∞(E)holds for all open setsEΩwithconnectedΩ\Eand all fieldsG∈Sβ(Ω)satisfying

E

G· ∇ζ dx=

E

G¯· ∇ζ dx ∀ζ∈C₀^∞ Rⁿ

. (1.16)

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If E is a set of finite perimeter inΩ, then there is a local characterization of the condition (1.16) in terms of the interior normal-components relative toΩ on∂^∗E; see Remark 2.2(b). The requirement of connectedness onΩ\E seems necessary as seen in the two dimensional case.

The existence of an absolute minimizer is unknown in general. However, in dimensionn=2, using the fact that a divergence-free field is a rotated gradient and some results about absolute minimizers in the gradient case, we are able to show that any minimizer G¯ obtained through (1.15) from a vector-field F¯ determined in Theorem 1.3 is an absolute minimizer. With given Dirichlet boundary conditions, the similar result in the gradient case would follow from theΓ-convergence of the power-law energies to theL^∞-energy [7,9]. However, some care need to be taken here because the Dirichlet boundary conditions are not uniquely determined from the normal trace of the divergence-free fields, especially when∂Ω consists of disjoint closed curves.

We summarize the results for the two-dimensional problem in the following theorem, which provides a concrete procedure of finding the absolute minimizers for ρ(β, H ) in the special case and also indicates that the absolute minimizers may not be unique; the convex setΣand the Lipschitz continuous functionsα_cgiven in the theorem will be specified later.

Theorem 1.4. Letn=2. Then any minimizerG¯ determined by a function F¯ in Theorem1.3through(1.15)is an absolute minimizer forρ(β, H ).

In the special case when σ =1,H=0 and ∂Ω consists of k+1 disjoint Lipschitz Jordan curves, there exist nonempty compact convex setΣ⊂R^kand certain given Lipschitz continuous functionsαcon∂Ω,distinct for different c∈Σ,such that any absolute minimizerG¯ ∈Sβ(Ω)is representable asG¯ =(ϕ¯_x₂,− ¯ϕ_x₁), whereϕ¯ is the absolute minimizing Lipschitz extension ofα_cfor somec∈Σ.

The paper is organized as follows. In Section 2, we collect some notation and preliminary results on functions of bounded variation and sets of finite perimeter, mostly from [3,17], and on the normal-components for functions inXn(Ω)and we define the measures(F, Dv)for functionsF ∈Xn(Ω)andv∈BV(Ω)in a slightly different way from those used in [4,5,8,11–13]. In Section 3, we prove Theorems 1.2 and 1.3 by giving several characterizations of μ(g, h).In Section 4, we present two proofs of Theorem 1.1, one based on Theorem 1.3 and the other on a natural direct approach analogous to the approach forL^∞-functionals of gradient fields given in [7,9] using the limit ofp-power functionals asp→ ∞.In Section 5, we provide a sufficient condition for the existence of minimizers forλ(g, h)in BV(Ω)and maximizing sets ofμ(g, h). A highly non-trivial interesting example is also given (see Example 5.1). In Section 6, we study two-dimensional problems and we prove Theorem 1.4 as two separate theorems (Theorems 6.2 and 6.5). The proof of Theorem 6.5 relies on several equivalent descriptions of the absolutely minimizing Lipschitz extension as the viscosity solution to the infinity Laplacian equation as given in [6,19,23].

2. Notation and preliminaries

LetUbe an open set inRⁿ. LetL^p(U )andW^1,p(U )be the usual Lebesgue and Sobolev spaces [1]. A function u∈L¹(U )is said to havebounded variationinUif|Du|(U )= _U|Du|<∞,where

U

|Du| =sup

U

udivϕ dxϕ∈C₀¹ U;Rⁿ

,ϕL^∞(U )1

. (2.1)

We denote byBV(U )the space of all functions inL¹(U )having bounded variation inU; this is a Banach space with normuBV(Ω)= uL¹(Ω)+ |Du|(Ω).It is well-known that u∈BV(U )if and only if u∈L¹(U ) and, for each i=1,2, . . . , n, the distributional derivativeux_i is a measure μi of finite total variation in the spaceM(U ) of all Radon measures onU.Hence, the distributional gradient ofuis a vector measureDu=(μ₁, μ₂, . . . , μ_n).

Let E be a Borel set inRⁿ. The perimeter P (E, U )of E inU is defined to be P (E, U )= _U|Dχ_E|;write P (E)=P (E,Rⁿ).We say thatEis a set offinite perimeterinU ifP (E, U ) <∞.A setEis called aCaccioppoli setifP (E, U ) <∞for every bounded open setU inRⁿ.For a Caccioppoli setE, a pointx∈Rⁿis said to be in the reduced boundary∂^∗EofEif _B

(x)|Dχ_E|>0 for all >0 and the vector limitν_E(x)=lim→0 ^{B (x)} Dχ_E

B (x)|Dχ_E| exists and satisfies|ν_E(x)| =1.(Note that we use the same notation as [17] for the reduced boundary, which is different

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from the notation used in [3], whereFEis used.) This unit vectorν_E(x)is called thegeneralized inner normaltoE atx∈∂^∗E.By Theorem 3.59 in [3] or Theorem 4.4 in [17], we know that, as Radon measures inM(Rⁿ),

Dχ_E=ν_E|Dχ_E|, |Dχ_E| =Hⁿ⁻¹ ∂^∗E. (2.2)

Given a measurable setEinRⁿand a numbert∈ [0,1], the setE^t of all points whereEhas densitytis defined by

E^t=

x∈Rⁿlim

→0

|E∩B(x)|

|B(x)| =t

.

The setsE⁰andE¹ can be considered as the measure-theoretic interior and exterior ofE. So the set∂^mE=Rⁿ\ (E⁰∪E¹)is defined to be themeasure-theoretic boundaryofE(or theessential boundaryofE); clearly∂^mE⊂∂E.

(Note again that our notation for the essential boundary is different from that used in [3].) A well-known theorem (cf., [3, Theorem 3.61]) states that ifEas finite perimeter inRⁿ,then

∂^∗E⊂E¹² ⊂∂^mE, Hⁿ⁻¹

∂^mE\∂^∗E

=0. (2.3)

In particular,Ehas density either 0,¹₂ or 1 atHⁿ⁻¹-a.e.x∈Rⁿ.

In what follows, we assumeΩis a bounded domain with Lipschitz boundary∂Ω inRⁿand define the family P(Ω)=

E⊂Ω0< P (E, Ω) <∞

. (2.4)

The trace operatoru|∂Ω can be extended as a linear bounded operatorγ=γΩ:BV(Ω)→L¹(∂Ω)(see [3, Theo- rem 3.87] and [17, Theorem 2.10]) so that, for eachu∈BV(Ω),

lim→0

1 ⁿ

B(a)∩Ω

u(x)−γ (u)(a)dx=0 (2.5)

forHⁿ⁻¹-almost everya∈∂Ω;moreover, for allζ ∈C¹(Rⁿ;Rⁿ),

Ω

udivζ dx= −

Ω

ζ·d(Du)+

∂Ω

γ (u)ζ·ν dHⁿ⁻¹. (2.6)

The trace operatorγΩ isontofromW^1,1(Ω)toL¹(∂Ω)(see, e.g., [5, Lemma 5.5]).

It is well-known that (see [5, Lemma 5.2] and [17, Remark 2.12]), for eachu∈BV(Ω), there exists a sequence u_j∈C^∞(Ω)∩W^1,1(Ω)such that

(a) u_j→u inLⁿ−ⁿ1(Ω) (b)

Ω

|∇u_j|dx→

Ω

|Du| (c) γ (u_j)=γ (u). (2.7) By [3, Corollary 3.49 and Remark 3.50], we have the following Poincaré inequality: there exists a constantC such that, for allu∈BV(Ω),

u−(u)_Ω

Lⁿⁿ−1(Ω)C

Ω

|Du|, (2.8)

where(u)Ω=_|_Ω¹_| _Ωu dxis the average ofuonΩ.

Letu∈BV(Ω).By [3, Theorem 3.40], for almost everyt ∈R, the set{u > t} = {x∈Ω |u(x) > t}has finite perimeter inΩand thecoarea formula

|Du|(B)= ∞

−∞

|Dχ_{_u>t_}|(B) dt, Du(B)= ∞

−∞

Dχ_{_u>t_}(B) dt (2.9)

holds for any Borel setB⊂Ω.Ifu∈W^1,1(Ω)and ispreciselyrepresented inΩ, say,u∈C(Ω)∩W^1,1(Ω)(see [21]), then for all Borel functionsψ:Ω→R

Ω

ψ|∇u| = ∞

−∞ u⁻¹(t )

ψ dHⁿ⁻¹

dt= ∞

−∞ Ω

ψ d|Dχ_{_u>t_}|

dt. (2.10)

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We gather some useful convergence results inBV(Ω)in the following proposition; see, e.g., [3, Propositions 3.6, 3.13 and Theorem 3.88] and [17, Theorem 2.11].

Proposition 2.1.Letu, u_j∈BV(Ω),|Du_j|(Ω)Mandu_j→uinL¹(Ω)asj→ ∞.Then

u_j u inLⁿ−ⁿ1(Ω), (2.11)

u_j→u inL^q(Ω)for each1q < n

n−1, (2.12)

|Du|(A)lim inf

j→∞ |Du_j|(A) ∀A⊂Ωopen, (2.13)

Ω

φ d|Du|lim inf

j→∞

Ω

φ d|Duj|, (2.14)

whereφis any nonnegative lower semi-continuous function inΩ. If, in addition, we assume

jlim→∞

Ω

|Du_j| =

Ω

|Du|, (2.15)

then we have

γ (u_j)→γ (u) inL¹(∂Ω), (2.16)

jlim→∞

Ω

ψ d|Duj| =

Ω

ψ d|Du| (2.17)

for all bounded continuous functionsψ∈C(Ω).In particular,|Du_j|^∗ |Du|inM(Ω).

We prove the following result providing a formula for the traces of certain functions.

Proposition 2.2.For eachu∈BV(Ω), γ (u)(a)=lim

→0

2

|B(a)|

B(a)

χ_Ω(x)u(x) dx

forHⁿ⁻¹-a.e.a∈∂Ω;for eachE∈P(Ω),γ (χE)=χ∂Ω∩∂^∗EinL¹(∂Ω).

Proof. Note that Ω has finite perimeter in Rⁿ (see [3, Proposition 3.62]) and that ∂^∗Ω ⊂∂Ω =∂^mΩ. Hence Hⁿ⁻¹(∂Ω\Ω¹²)=0.By (2.5), forHⁿ⁻¹-a.e.a∈∂Ω,

lim→0

1 ⁿ

B(a)∩Ω

u(x)−γ (u)(a)lim

→0

1 ⁿ

B(a)∩Ω

u(x)−γ (u)(a)=0,

from which we have γ (u)(a)lim

→0

|B(a)∩Ω|

|B(a)| =lim

→0

1

|B(a)|

B(a)

χ_Ω(x)u(x) dx.

SinceHⁿ⁻¹(∂Ω\Ω¹²)=0,forHⁿ⁻¹-a.e.a∈∂Ω, it follows that lim→0|B(a)∩Ω|

|B(a)| =¹₂;the first statement is proved.

To prove the second statement, letφ=γ (χE)∈L¹(∂Ω)be the trace function. Then φ(a)=2 lim

→0

|B(a)∩E|

|B(a)| . (2.18)

We write∂Ω=(∂Ω∩E⁰)∪(∂Ω∩E¹)∪(∂Ω∩∂^∗E)∪[∂Ω∩(∂^mE\∂^∗E)].First, sinceE⊂Ω,Ω¹²∩E¹= ∅; thus Hⁿ⁻¹(∂Ω∩E¹)=0.Ifa∈∂Ω∩E⁰, thenφ(a)=0. Ifa∈∂Ω∩∂^∗Ethen, by (2.3),a∈E¹² and henceφ(a)=1.

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Finally, by Lemma 2.3,Hⁿ⁻¹(∂^mE\∂^∗E)=0, and henceφ(a)=χ_∂Ω_∩_∂∗E(a)forHⁿ⁻¹-a.e.a∈∂Ω,which proves γ (χ_E)=χ_∂Ω_∩_∂∗EinL¹(∂Ω). 2

In the rest of this section, we review the spaceXn(Ω)and normal-components on∂Ω.The normal-component operator (calledinterior normal trace) can be defined on the reduced boundary of a set of finite perimeter for the so- calleddivergence-measurevector fieldsF ∈L^∞(Ω;Rⁿ)with divF being a Radon measure [11]. However, for our purpose, we only consider the vector fields ofXn(Ω)withLⁿintegrable divergence. The following result is proved in [5].

Theorem 2.3. (See Theorem 1.2 in [5].) There exists a linear (outward) normal-component operator δ = δΩ:Xn(Ω)→L^∞(∂Ω)such that, for allF∈Xn(Ω),

δ(F )

L∞(∂Ω)FL^∞(Ω), (2.19)

δ(ζ|Ω)=ζ (x)·ν(x) ∀ζ∈C¹ Rⁿ;Rⁿ

, (2.20)

Ω

vdivF +

Ω

F · ∇v=

∂Ω

γ (v)δ(F ) dHⁿ⁻¹ ∀v∈W^1,1(Ω). (2.21)

Moreover, if F_j, F ∈Xn(Ω)satisfyF_j F^∗ inL^∞(Ω;Rⁿ)anddivF_jdivF inLⁿ(Ω),thenδ(F_j) δ(F )^∗ in L^∞(∂Ω).

We extend the functionF · ∇vin (2.21) from v∈W^1,1(Ω)tov∈BV(Ω)by defining the pairing(F, Dv) as a measure forF ∈Xn(Ω)andv∈BV(Ω). We do this in a slightly different way from [4,5,8,11–13] by making(F, Dv) a Radon measure on wholeRⁿ.To do so, givenv∈BV(Ω),F ∈Xn(Ω), define a distributionL:C₀^∞(Rⁿ)→Rby

L(ϕ)=

∂Ω

ϕγ (v)δ(F ) dHⁿ⁻¹−

Ω

ϕvdivF+(∇ϕ·F )v

dx ∀ϕ∈C₀^∞ Rⁿ

. (2.22)

Theorem 2.4.There exists a unique Radon measureωinM(Rⁿ), denoted byω=(F, Dv), such thatL(ϕ)= _Rnϕ dω for allϕ∈C₀^∞(Rⁿ).Moreover, for any positive bounded continuous functionσ onΩand any Borel setBinRⁿ,

B

dω

B

d|ω| F

σ

L^∞(U∩Ω)

B∩Ω

σ d|Dv| (2.23)

whereU is any open set containingB.In particular, the measureω=(F, Dv)is concentrated onΩ and absolutely continuous with respect to|Dv|Ω.

Proof. Letv_j be an approximation sequence as determined in (2.7) forv∈BV(Ω).Using (2.21) withv=ϕv_j, we have

L(ϕ)= lim

j→∞

Ω

(F· ∇v_j)ϕ dx ∀ϕ∈C₀^∞ Rⁿ

. (2.24)

Hence

L(ϕ)FL^∞ϕL^∞ lim

j→∞

Ω

|∇v_j|dx= FL^∞ϕL^∞

Ω

|Dv|

for allϕ∈C₀^∞(Rⁿ).Since C₀^∞(Rⁿ)is dense in C₀(Rⁿ),L can be uniquely extended as a linear functional L˜ on C₀(Rⁿ)that still satisfies

L(ϕ)˜ FL^∞(Ω)ϕL^∞(Ω)

Ω

|Dv| ∀ϕ∈C₀ Rⁿ

.

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By Riesz’s theorem, there exists a unique finite Radon measureωonRⁿsuch thatL(ϕ)˜ = _Rnϕ dωfor allϕ∈C₀(Rⁿ).

We now prove (2.23). LetBbe any Borel set inRⁿand letK⊂B⊂U, whereKis compact andU is open. For any >0, by the outer regularity of measureσ d|Dv|, there exists an open setV ⊂UcontainingKsuch that

V∩Ω

σ d|Dv|<

K∩Ω

σ d|Dv| +.

Given anyϕ∈C_c(V )withϕL^∞(V )1, letϕ_k∈C^∞_c (V )be such thatϕ_k−ϕL^∞(V )→0 ask→ ∞.LetU_k= suppϕkbe the compact support ofϕkinV. Let us consider the measuresμj=σ|∇vj|dxandλ=σ d|Dv|inM(Ω).

Since _Ω|∇vj|dx→ _Ω|Dv|,by Proposition 2.1, we haveμj(Ω)→λ(Ω)andλ(A)lim infj→∞μj(A)for all open setsA⊂Ω. This impliesλ(Ω\A)lim sup_j_→∞μj(Ω\A)for all open setsA⊂Ω.TakingA=Uk∩Ω, we have, for eachk=1,2, . . . ,

lim sup

j→∞

U_k∩Ω

σ|∇vj|dx

U_k∩Ω

σ d|Dv|.

Therefore, by (2.24), L(ϕ)˜ = lim

k→∞L(ϕ_k)= lim

k→∞

lim

j→∞

Uk∩Ω

(F· ∇v_j)ϕ_kdx lim sup

k→∞

F σ

L^∞(U∩Ω)

ϕ_kL^∞

lim sup

j→∞

Uk∩Ω

σ|∇v_j|dx

F σ

L^∞(U∩Ω)

lim sup

k→∞

Uk∩Ω

σ d|Dv|

F σ

L^∞(U∩Ω)

V∩Ω

σ d|Dv|.

Since|ω|(V )=sup{ ˜L(ϕ)|ϕ∈C_c(V ),ϕL^∞(V )1}, we have

|ω|(V ) F

σ

L^∞(U∩Ω)

V∩Ω

σ d|Dv|.

Therefore

|ω|(K)|ω|(V ) F

σ

L^∞(U∩Ω) K∩Ω

σ d|Dv| +

,

for all >0.This proves

|ω|(K) F

σ

L^∞(U∩Ω)

K∩Ω

σ d|Dv|

for allK⊂B⊂U,Kcompact andUopen. Hence (2.23) follows by the inner regularity of|ω|andσ d|Dv|Ω. 2 Remark 2.1.Sinceω=(F, Dv)is concentrated onΩ, we can write _Rnϕ dω= _Ωϕ dω.Therefore, by (2.22) we obtain a more general divergence formula

Ω

ϕ d(F, Dv)=

∂Ω

ϕγ (v)δ(F ) dHⁿ⁻¹−

Ω

ϕvdivF+(∇ϕ·F )v

dx (2.25)

for allF ∈Xn(Ω),v∈BV(Ω)and all Lipschitz functionsϕ∈W^1,^∞(Ω).

We have the following compensated compactness result; see also [5, Theorems 4.1 and 4.2].

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Proposition 2.5.Letu_j, u∈BV(Ω)satisfy, for some1q_n₋ⁿ₁,u_j→uinL^q(Ω)and letF_j, F∈Xn(Ω)satisfy F_j F^∗ inL^∞(Ω),divF_jdivF inL^q(Ω), whereq=_q₋^q₁.Then(F_j, Du_j) (F, Du)^∗ as Radon measures in M(Ω).Furthermore, if, in addition,

jlim→∞

Ω

|Duj| =

Ω

|Du|, (2.26)

then(F_j, Du_j) (F, Du)^∗ inM(Rⁿ).

Proof. Given anyϕ∈C₀¹(Rⁿ), by (2.25), we have, for eachj=1,2, . . . ,

Ω

ϕ d(F_j, Du_j)=

∂Ω

ϕγ (u_j)δ(F_j) dHⁿ⁻¹−

Ω

ϕu_jdivFj+(∇ϕ·F_j)u_j .

First, ifϕ∈C₀¹(Ω), then there vanishes the boundary term and hence we have

jlim→∞

Ω

ϕ d(F_j, Du_j)=

Ω

ϕ d(F, Du) (2.27)

for allϕ∈C₀¹(Ω). This proves the weak-star convergence inM(Ω). Now, assume (2.26). Then by Theorem 2.3, δ(F_j) δ(F )^∗ inL^∞(∂Ω),and by (2.16) in Proposition 2.1,γ (u_j)→γ (u)inL¹(∂Ω).Hence, (2.27) holds for all ϕ∈C₀¹(Rⁿ),which proves the weak-star convergence inM(Rⁿ). 2

The following result defines a boundary normal-component for functions inXn(Ω)on sets of finite perimeter; the similar definition has been given in [11, Theorem 5.2] for a broader class of functions.

Proposition 2.6. Given any F ∈Xn(Ω) and any set E⊂Ω of finite perimeter, there exists a function θ˜_E(F )∈ L^∞(∂^∗E;dHⁿ⁻¹), called theinterior normal-component ofF on∂^∗E, such that

E

(ϕdivF + ∇ϕ·F ) dx= −

∂∗E

ϕθ˜_E(F ) dHⁿ⁻¹ ∀ϕ∈C¹ Rⁿ

. (2.28)

Moreover, ifEis an open set with Lipschitz boundary, thenθ˜E(F )= −δE(F ),whereδE(F )∈L^∞(∂E)is the normal- component ofF ∈Xn(E)on∂Edefined above.

Proof. Sinceχ_E∈BV(Ω), the measure(F, Dχ_E)is well-defined above as a Radon measure inRⁿconcentrated onΩ and absolutely continuous relative to|Dχ_E|Ω=Hⁿ⁻¹ (Ω∩∂^∗E). Hence, by the Radon–Nikodym theorem, there exists a functionθ_E(F )∈L^∞(Ω∩∂^∗E;dHⁿ⁻¹)withθ_E(F )L^∞FL^∞(U ),U⊂Ω any open set containingE, such that

d(F, DχE)=θE(F ) dHⁿ⁻¹

Ω∩∂^∗E

onRⁿ. (2.29)

(The functionθ_E(F )has been called theinterior normal-trace relative toEofF on∂^∗Ein [11, Theorem 5.2] for a broader class of functionsF.) Let us define

˜

θ_E(F )(x)=

θ_E(F )(x) x∈Ω∩∂^∗E,

−δ(F )(x) x∈∂Ω∩∂^∗E, (2.30)

whereδ(F )=δ_Ω(F )is the normal-component ofF on∂Ω defined in Theorem 2.3. Combining formula (2.25) with (2.29) and Proposition 2.2, we have the following general Gauss–Green formula:

E

(ϕdivF + ∇ϕ·F ) dx= −

∂^∗E

ϕθ˜E(F ) dHⁿ⁻¹ ∀ϕ∈C¹ Rⁿ

.

This proves (2.28). IfEis a Lipschitz domain itself, then by (2.21) we easily see thatθ˜E(F )= −δE(F ). 2

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Remark 2.2.(a) The formula (2.28) generalizes a case of the Gauss–Green formula in [11, Theorem 5.3] since we allowϕ∈C¹(Ω)¯ and∂E∩∂Ω= ∅;in [11, Theorem 5.3], only setsEΩ are considered. Furthermore, we have θ˜_E(F )=F ·ν_Eon∂^∗EifF ∈C¹(Rⁿ;Rⁿ).

(b) LetG,G¯ ∈Sβ(Ω)andE⊂Ω be a set of finite perimeter inΩ. Then the condition (1.16) in Definition 1.1 is equivalent to the local condition:θ˜_E(G)= ˜θ_E(G)¯ inL^∞(∂^∗E;dHⁿ⁻¹).

3. Characterization ofμ(g, h)and proof of Theorems 1.2 and 1.3

Given any functionuonΩ, denote byE_t(u)the upper-level set{u > t} = {x∈Ω|u(x) > t}for eacht∈R.We first prove the following useful result.

Lemma 3.1.Letu∈BV(Ω).Then

u(x)= ∞ 0

χ_E_t_(u)(x) dt− 0

−∞

1−χ_E_t_(u)(x) dt

Lⁿ-a.e.x∈Ω ,

γ (u)(a)= ∞ 0

γ (χE_t(u))(a) dt− 0

−∞

1−γ (χE_t(u))(a)

dt

Hⁿ⁻¹-a.e.a∈∂Ω .

Proof. By writingu=u⁺−u⁻, without loss of generality, we assumeu0, so in the two identities there are only integral terms from 0 to∞.The first identity is easy; so we only prove the second identity. We proceed to prove

γ _∞

0

χ_E_t_(u)(x) dt

(a)= ∞ 0

γ (χ_E_t_(u))(a) dt (3.1)

for Hⁿ⁻¹-a.e.a∈∂Ω.Note that, for almost every t∈R, the set E_t(u) has finite perimeter in Ω and soχ_E_t_(u)∈ BV(Ω). Let{t_j},j=1,2, . . . ,be a dense sequence of(0,∞)such that eachχ_E_t_(u)∈BV(Ω).By Proposition 2.2, we have that, forHⁿ⁻¹-a.e.a∈∂Ω,

γ (χ_E_tj_(u))(a)=lim

→0

2

|B(a)|

B(a)

χ_E_tj_(u)(x) dx ∀j =1,2, . . . , (3.2) and for eachN=1,2, . . . ,

γ N

0

χ_E_t_(u)(x) dt

(a)=lim

→0

2

|B(a)|

B(a)

N 0

χ_E_t_(u)(x) dt

dx.

For such ana∈∂Ω, the functionr(t )=γ (χ_E_t_(u))(a)is a non-increasing function int∈(0,∞)and hence is continuous almost everywhere. At any continuity pointt₀of this function, by (3.2), it follows that

γ (χ_E_t

0(u))(a)=lim

→0

2

|B(a)|

B(a)

χ_E_t

0(u)(x) dx. (3.3)

Hence, by Fubini’s theorem, γ

N 0

χ_E_t_(u)(x) dt

(a)=lim

→0

N 0

2

|B(a)|

B(a)

χ_E_t_(u)(x) dx dt

= N 0

lim→0

2

|B(a)|

B(a)

χ_E_t_(u)(x) dx dt= N 0

γ (χ_E_t_(u))(a) dt,

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where the change of order of the limit into the integral is justified by the dominated convergence theorem. Finally note that, by [16, Proposition 6, p. 340],

N 0

χE_t(u)(x) dt=min

u(x), N

→u(x)= ∞ 0

χE_t(u)(x) dt

in the norm topology ofBV(Ω)asN→ ∞. Hence, by the continuity of the trace-operator, γ

N 0

χE_t(u)dt

→γ _∞

0

χE_t(u)dt

inL¹(∂Ω)asN→ ∞.This proves (3.1) forHⁿ⁻¹-a.e.a∈∂Ω. 2

In what follows, we assumeσ is the function given in the introduction; that is,σ is continuous function inΩ and satisfies, for two positive constantsσ₀andM₀, that

0< σ0σ (x)M0<∞ ∀x∈Ω. (3.4)

Letg∈L^∞(∂Ω)andh∈Lⁿ(Ω)satisfy (1.7) above. Define the following functionals onBV(Ω):

N (u)=

Ω

σ d|Du|, B(u)=

∂Ω

γ (u)g dHⁿ⁻¹−

Ω

uh dx. (3.5)

Proposition 3.2.Given0< m <∞,the following statements are equivalent:

(a) mN (ζ )B(ζ ) ∀ζ∈C^∞(Ω)∩W^1,1(Ω).

(b) mN (u)B(u) ∀u∈BV(Ω).

(c) mN (χ_E)B(χ_E) ∀E∈P(Ω).

(d) mN (χ_E)B(χ_E) ∀E∈P(Ω)open.

Proof. That (a) implies (b) follows by the approximation (2.7) and the convergence result Proposition 2.1. That (b) implies (c) is immediate as χ_E∈BV(Ω)for allE∈P(Ω).To prove that (c) implies (d), note that ifE∈P(Ω) thenΩ\E∈P(Ω)andDχ_Ω_\_E= −Dχ_E and∂^∗(Ω\E)=∂^∗E; hence, by condition (1.7),N (χ_Ω_\_E)=N (χ_E)and B(χ_Ω_\_E)= −B(χ_E). This proves that (c) implies (d). Finally we prove that (d) implies (a). LetL^±(u)=mN (u)± B(u).Then, by (d),

L^±(χ_E)0 ∀E∈P(Ω)open.

Given anyζ ∈C^∞(Ω)∩W^1,1(Ω), we writeζ =ζ⁺−ζ⁻,whereζ^±(x)=max{±ζ (x),0}. Thenζ^±∈C(Ω)∩ W^1,1(Ω)are nonnegative and

γ (ζ )=γ ζ⁺

−γ ζ⁻

, |∇ζ| =∇ζ⁺+∇ζ⁻.

HencemN (ζ )−B(ζ )=L⁻(ζ )=L⁻(ζ⁺−ζ⁻)=L⁻(ζ⁺)+L⁺(ζ⁻).We claim thatL⁻(ζ⁺)0 andL⁺(ζ⁻)0 and hencemN (ζ )B(ζ )follows, as desired of (a). Since the argument is similar, we only proveL⁻(ζ⁺)0. Note that, by Lemma 3.1 and Fubini’s theorem,

Ω

h(x)ζ⁺(x) dx=

Ω

∞ 0

χ_{_ζ+>t}(x)h(x) dt dx= ∞

0 Ω

χ_{_ζ+>t}(x)h(x) dx

dt; (3.6)

∂Ω

gγ ζ⁺

dHⁿ⁻¹= ∞

0 ∂Ω

γ (χ_{_ζ+>t})g dHⁿ⁻¹

dt. (3.7)

Minimizing <span class="mathjax-formula">${L}^{\infty }$</span>-norm functional on divergence-free fields

Minimizing L ∞ -norm functional on divergence-free fields

Baisheng Yan

Minimizing L ^∞ -norm functional on divergence-free fields