OBSTACLE PROBLEMS WITH DOUBLE BOUNDARY CONDITION FOR LEAST GRADIENT FUNCTIONS IN METRIC MEASURE SPACES

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OBSTACLE PROBLEMS WITH DOUBLE BOUNDARY

CONDITION FOR LEAST GRADIENT FUNCTIONS IN

METRIC MEASURE SPACES

JOSH KLINE

Abstract. In the setting of a metric space equipped with a doubling

measure supporting a (1,1)-Poincaré inequality, we study the problem of

minimizing the BV-energy in a bounded domain Ωof functions bounded

between two obstacle functions inside Ω, and whose trace lies between

two prescribed functions on the boundary of Ω.If the class of candi-

date functions is nonempty, we show that solutions exist for continuous

obstacles and continuous boundary data when Ωis a uniform domain

whose boundary is of positive mean curvature in the sense of Lahti,

Malý, Shanmugalingam, and Speight (2019). While such solutions are

not unique in general, we show the existence of unique minimal solu-

tions. Our existence results generalize those of Ziemer and Zumbrun

(1999), who studied this problem in the Euclidean setting with a single

obstacle and single boundary condition.

1. Introduction

Given some function fon the boundary of a domain Ωand a function

ψon Ω, the obstacle problem for least gradient functions is the problem of

minimizing the BV-energy in Ωover all functions u∈BV (Ω) whose trace

agrees with falmost everywhere on ∂Ωand such that u≥ψalmost ev-

erwhere in Ω.In the Euclidean setting, this problem was ﬁrst studied by

Ziemer and Zumbrun in [52], where they showed that a continuous solution

exists for ψ∈C(Ω) and f∈C(∂Ω) such that f≥ψon ∂Ω,provided

the boundary of Ωhas nonnegative mean curvature and is not locally area-

minimizing. Their work generalized some of the earlier results of Sternberg,

Williams, and Ziemer [51], who introduced and studied the Dirichlet problem

for least gradient functions (without obstacle) in the Euclidean setting. Re-

cently existence, uniqueness, and regularity of an anisotropic formulation of

the obstacle problem for least gradients in the Euclidean setting was studied

in [12]. For the relationship between the obstacle problem for least gradients

and dual maximization problems in the Euclidean setting, see [48].

Date: October 19, 2022.

2020 Mathematics Subject Classiﬁcation. Primary 46E36; Secondary 26A45, 49Q20,

31E05.

Key words and phrases. Metric measure space, bounded variation, least gradient, ob-

stacle problem, double boundary condition.

arXiv:2210.10845v1 [math.AP] 19 Oct 2022

2 JOSH KLINE

In [35], existence and regularity of solutions to the obstacle problem for

least gradients were studied in the setting of a metric space equipped with a

doubling measure and supporting a Poincaré inequality. Here solutions were

not required to attain the boundary condition in the sense of traces; com-

peting functions need only satisfy the obstacle condition inside the domain

and satisfy the boundary condition outside the domain.

In this paper, we continue the study of obstacle problems for least gradient

functions in the metric setting. In contrast to [35] however, we insist that

solutions address the boundary condition in the sense of traces, following [52]

in the Euclidean setting. Furthermore, we include a second obstacle function

and consider a double boundary condition. That is, for ψ1, ψ2: Ω →Rand

f, g ∈L1(∂Ω),we consider the problem

min{kDuk(Ω) : u∈BV (Ω), ψ1≤u≤ψ2, f ≤T u ≤g},

where the inequalities are in the almost everywhere sense. By deﬁning

Kψ1,ψ2,f,g(Ω) to be the class of BV-functions in Ωsatisfying the obstacle

and boundary conditions, we refer to solutions to this problem as strong

solutions to the Kψ1,ψ2,f,g-obstacle problem (see Section 2.3 below for the

precise deﬁnitions). We adopt the following standing assumptions:

•(X, d, µ)is a complete metric measure space supporting a (1,1)-

Poincaré inequality, with µa doubling Borel regular measure,

•Ω⊂Xis a bounded domain with µ(X\Ω) >0,

• H(∂Ω) <∞,H|∂Ωis doubling, and H|∂Ωis lower codimension 1

Ahlfors regular, see (2.3),

• H({z}) = 0 for all z∈∂Ω.

where His the codimension 1 Hausdorﬀ measure on ∂Ω, see (2.2). The

assumption that singletons on the boundary of the domain are H-negligible

is necessary to obtain Lemma 2.17, see Example 2.20. Our main result is

the following:

Theorem 1.1. Let Ωbe a uniform domain with boundary of positive mean

curvature as in Deﬁnition 2.11. Let f, g ∈C(∂Ω) and ψ1, ψ2∈C(Ω) be such

that Kψ1,ψ2,f,g(Ω) 6=∅.Then there exists a strong solution to the Kψ1,ψ2,f,g-

obstacle problem.

By ψ1, ψ2∈C(Ω),we mean that these functions are extended real val-

ued functions, and continuous with respect to the standard topology on the

extended real line. As such we can consider continuous obstacle functions

ψ1≡ −∞ and ψ2≡ ∞.However throughout this paper, we do insist that the

boundary functions fand gare real-valued, so as to utilize certain extension

results from [37].

Strong solutions to this problem may fail to be unique (see [37] and the

discussion below), and so a comparison theorem for strong solutions will not

hold in general. However, in Proposition 4.4 and Remark 4.5, we show that

OBSTACLE PROBLEMS FOR LEAST GRADIENTS 3

unique minimal and maximal strong solutions exist for continuous obstacle

and boundary functions, and we obtain the following comparison-type result:

Theorem 1.2. Let f1, f2, g1, g2∈C(∂Ω) such that f1≤f2and g1≤g2

H-a.e. on ∂Ω.Let ψ1, ψ2, ϕ1, ϕ2∈C(Ω) be such that ψ1≤ψ2and ϕ1≤ϕ2

µ-a.e. in Ω.Suppose that

K1(Ω) := Kψ1,ϕ1,f1,g1(Ω) 6=∅6=Kψ2,ϕ2,f2,g2(Ω) =: K2(Ω).

If u1is the minimal strong solutions to the K1-obstacle problem and u2is

a strong solution (not necessarily minimal) to the K2-obstacle problem, then

u1≤u2µ-a.e. in Ω.Likewise, if v1is a strong solution to the K1-obstacle

problem and v2is the maximal strong solution to the K2-obstacle problem,

then v1≤v2µ-a.e. in Ω.

Problems involving related double boundary conditions have been stud-

ied in a variety of diﬀerent contexts. In stochastic analysis in particular,

double-boundary (non-)crossing problems, where one tries to determine the

probability that a stochastic process remains between two prescribed bound-

aries, have been studied extensively and have many statistical applications.

For a sampling, see [8, 10, 13, 39] and references therein. Related notions

of double boundary layers also appear in ﬂuid dynamics and perturbation

theory, see [9, 30] for example. As such, it seems natural to consider such a

double boundary condition in the context of BV-energy minimizers.

By setting ψ2≡ ∞,and f=g, we recover the classical obstacle problem

for least gradients. If in addition we set ψ1≡ −∞,then we recover the

Dirichlet problem for least gradient functions (also referred to as the least

gradient problem). As mentioned above, the least gradient problem was

ﬁrst studied in the Euclidean setting by Sternberg, Williams, and Ziemer

[51], who showed that unique solutions exist for continuous boundary data,

provided that the boundary of the domain has nonnegative mean curvature

and is not locally area-minimizing.

Since its introduction in [51], existence, uniqueness, and regularity of the

least gradient problem above have been studied extensively in the Euclidean

setting; for a sampling, see [15–17, 20, 26, 29, 41, 43–47, 50, 53] and the refer-

ences therein. In particular, weighted versions of this problem have applica-

tions to current density impedance imaging, see for example [43–45].

In recent decades, analysis in metric spaces has become a ﬁeld of active

study, in particular when the space is equipped with a doubling measure and

supports a Poincaré inequality, see for example [3, 6, 23,25]. A deﬁnition of

BV functions and sets of ﬁnite perimeter was extended to this setting by

Miranda Jr. in [42], and consequently, a theory of least gradient functions

in metric spaces has been developed in recent years, see for example [18, 19,

22, 27, 29, 32, 37, 38, 40]. In [37], Lahti, Malý, Shanmugalingam, and Speight

studied the Dirichlet problem for least gradients, originally introduced in

[51], in the metric setting. To do so, they introduced a deﬁnition of positive

mean curvature in the metric setting (Deﬁnition 2.11 below) and showed

4 JOSH KLINE

that solutions exist for continuous boundary data if the boundary of the

domain satisﬁes this condition. It is this curvature condition that we assume

in Theorem 1.1.

In [37] it was also shown that in the weighted unit disc, uniqueness and

continuity of solutions may fail even for Lipschitz boundary data. As the

problem we study in this paper is a generalization of this Dirichlet prob-

lem, we cannot guarantee uniqueness or continuity of strong solutions to

the Kψ1,ψ2,f,g-obstacle problem. For this reason, our comparison-type result,

Theorem 1.2, is stated in terms of minimal and maximal strong solutions.

For more on existence, uniqueness, and regularity of solutions to the weighted

least gradient problem, see [17, 26, 53].

To construct our solution, we adapt the program ﬁrst implemented by

Sternberg, Williams, and Ziemer to obtain solutions to the Dirichlet problem

for least gradients in the Euclidean setting in [51]. There they constructed

a least gradient solution by ﬁrst solving the Dirichlet problem for boundary

data consisting of superlevel sets of the original boundary data. This method

relies upon the fact, ﬁrst discovered by Bombieri, De Giorgi, and Giusti [7],

that characteristic functions of superlevel sets of least gradient functions are

themselves of least gradient. The construction of solutions to the obstacle

problem for least gradients in [52] is an adaptation of the program from [51].

In the metric setting, the construction of solutions to the Dirichlet problem

for least gradients given in [37] is also inspired by the method established in

[51]. However both [51] and [52] utilize smoothness properties and tangent

cones for the boundaries of certain solution sets, tools which are not available

in the metric setting. Thus the construction in [37] is a further modiﬁcation

of the method from [51]. As we are studying the double obstacle problem

with double boundary condition in the metric setting, our construction is

inspired by that of [37].

In [37], the authors deﬁned weak solutions to the Dirichlet problem for

least gradients (Deﬁnition 2.7 below) which are easily obtained via the direct

method of calculus of variations for a large class of boundary data. They

then used these weak solutions to construct their strong solution. Since we

have introduced a double boundary condition, and thus competitor functions

are not ﬁxed outside the domain, it is diﬃcult to obtain weak solutions of

this form by the direct method. For this reason, we deﬁne a family of ε-

weak solutions (Deﬁnition 2.12 below), which consider the BV-energy in

slight enlargements of the domain Ω.By controlling these ε-weak solutions

as ε→0,we obtain the proper building blocks from which to construct

our strong solution to the original problem in the manner of [37]. Since our

argument involves enlargements of Ω, we assume that Ωis a uniform domain

in order to use the strong BV extension results obtained in [33].

The structure of our paper is as follows: in Section 2, we introduce the

basic deﬁnitions, notations, and assumptions used throughout the paper. In

Section 3, we prove preliminary results regarding solutions to the double

OBSTACLE PROBLEMS FOR LEAST GRADIENTS 5

obstacle, double boundary problem when the obstacle and boundary func-

tions are characteristic functions of certain open sets. In Section 4, we use

these preliminary results to prove Theorem 1.1 and Theorem 1.2. In Sec-

tion 5, we provide examples illustrating how, in the absence of obstacles,

strong solutions to the double boundary problem may not be solutions to

the Dirichlet problems for either boundary condition. In this section, we

also study how solutions of the double obstacle, double boundary problem

converge to solutions of the double obstacle, single boundary problem as the

double boundary data converge to a single datum (see Theorem 5.5 below).

Acknowledgments. The author was partially supported by the NSF Grant

#DMS-2054960 and the Taft Research Center Graduate Enrichment Award.

The author would like to thank Nageswari Shanmugalingam for her kind

encouragement and many helpful discussions regarding this project.

2. Background

2.1. General metric measure spaces. Throughout this paper, we assume

that (X, d, µ)is a complete metric measure space, with µa doubling Borel

regular measure supporting a (1,1)-Poincaré inequality (deﬁned below). By

doubling, we mean that there exists a constant C≥1such that

0< µ(B(x, 2r)) ≤Cµ(B(x, r)) <∞

for all x∈Xand r > 0.By iterating this condition, there exist constants

CD≥1and Q > 1such that

(2.1) µ(B(y, r))

µ(B(x, R)) ≥C−1

Dr

RQ

for every 0< r ≤R,x∈X, and y∈B(x, R).In this paper we let Cdenote

constants, depending, unless otherwise stated, only on Ωand the doubling

and Poincaré inequality constants (see below), whose precise value is not

needed. The value of Cmay diﬀer even within the same line.

Complete metric spaces equipped with doubling measures are necessarily

proper, i.e. closed and bounded sets are compact. For any open set Ω⊂X,

we deﬁne L1

loc (Ω) as the space of functions that are in L1(Ω0)for every

Ω0bΩ,that is, for every open set Ω0such that Ω0is a compact subset of Ω.

Other local function spaces are deﬁned analogously. For a ball B=B(x, r)

and λ > 0, we often denote λB := B(x, λr).If A, B ⊂X, then by A@B,

we mean that µ(A\B) = 0.By a domain, we mean a nonempty connected

open set in X.

We say that Xis A-quasiconvex if A≥1and for all points x, y ∈X, there

exists a curve γconnecting xand ysuch that

`(γ)≤Ad(x, y).

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OBSTACLEPROBLEMSWITHDOUBLEBOUNDARYCONDITIONFORLEASTGRADIENTFUNCTIONSINMETRICMEASURESPACESJOSHKLINEAbstract.Inthesettingofametricspaceequippedwithadoublingmeasuresupportinga(1;1)-Poincaréinequality,westudytheproblemofminimizingtheBV-energyinaboundeddomainoffunctionsboundedbetweentwoobstaclefunctionsi...

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OBSTACLE PROBLEMS WITH DOUBLE BOUNDARY CONDITION FOR LEAST GRADIENT FUNCTIONS IN METRIC MEASURE SPACES

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