1.1. Metric random walk spaces

https://doi.org/10.1051/cocv/2020087 www.esaim-cocv.org

LEAST GRADIENT FUNCTIONS IN METRIC RANDOM WALK SPACES

Wojciech G´ orny

and Jos´ e M. Maz´ on

Abstract. In this paper we study least gradient functions in metric random walk spaces, which include as particular cases the least gradient functions on locally finite weighted connected graphs and nonlocal least gradient functions on R^N. Assuming that a Poincar´e inequality is satisfied, we study the Euler-Lagrange equation associated with the least gradient problem. We also prove the Poincar´e inequality in a few settings.

Mathematics Subject Classification.05C81, 35R02, 26A45, 05C21, 45C99.

Received January 3, 2020. Accepted November 30, 2020.

1. Introduction and preliminaries

A metric random walk space [X, d, m] is a Polish metric space (X, d) together with a familym= (m_x)_x∈X of probability measures that encode the jumps of a Markov chain. Important examples of metric random walk spaces are: locally finite weighted connected graphs, finite Markov chains and [R^N, d, m^J] withdthe Euclidean distance and

m^J_x(A) :=

Z

A

J(x−y)dL^N(y) for every Borel setA⊂R^N ,

where J :R^N →[0,+∞[ is a measurable, nonnegative and radially symmetric function withR

R^NJdx= 1.

In the Euclidean space, least gradient problems are closely related to the study of minimal surfaces. They first appeared in the pioneering work of Bombieriet al.[5], where the authors show that the boundaries of superlevel sets of a function of least gradient are area-minimizing in the sense that the characteristic functions of those sets are also functions of least gradient (see also [19] for stability results on functions of least gradient). Conversely, Sternberg, Williams, and Ziemer proved in [25] (see also [26]) existence of a function of least gradient with a given continuous trace by explicitly constructing each of its superlevel sets in such a way that they are area- minimizing and reflect the boundary condition. Due to their important applications in conductivity imaging, such problems (including anisotropic cases) have received extensive attention in the past decade (see for instance:

[7, 8, 11, 14, 21–23]). In particular, in [14] the authors give a formulation of the problem of Euler-Lagrange

Keywords and phrases:Random walk, least gradient functions, total variation flow, functions of bounded variation.

1 Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland.

2 Departamento de An`alisis Matem`atico, Universitat de Val`encia, Dr. Moliner 50, 46100 Burjassot, Spain.

* Corresponding author:[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2021

type, where the structure of the solution is governed by a single divergence-measure vector field. We will utilise a similar approach in the nonlocal setting.

To our knowledge, the only results on least gradient problems for nonlocal operators are to one obtained in [13]. The aim of this paper is to study least gradient functions in the general setting of metric random walk spaces. As a particular case, we obtain results in the nonlocal least gradient problem on graphs. Moreover, some results obtained in this paper are new also in the context of nonlocality defined by a continous nonnegative kernel on a Euclidean space.

The paper is organized as follows: in the first part of the paper, we provide the Euler-Lagrange formulation associated with the least gradient problem in general metric random walk spaces. Under the assumption that a nonlocal Poincar´e type inequality is satified, we prove equivalence of the Euler-Lagrange equations with the variational formulation and study some properties of solutions to the least gradient problem in this setting. In the second part of the paper, we prove that such Poincar´e inequality holds in many important examples of metric random walk spaces, such as the -step random walk in weighted Euclidean spaces or Carnot-Carath´eodory spaces and on locally finite graphs. Moreover, we show that if the space does not satisfy the Poincar´e inequality, then solutions in the Euler-Lagrange sense might not exist.

1.1. Metric random walk spaces

Let (X, d) be a Polish metric space equipped with its Borelσ-algebra. A random walk mon X is a family of probability measures m_x onX, x∈X, satisfying the two technical conditions: (i) the measuresm_xdepend measurably on the point x∈X, i.e., for any Borelian A of X and any Borelian B of R, the set {x∈X : mx(A)∈B}is Borelian; (ii) each measure mxhas finite first moment,i.e.for some (hence any)z∈X, and for any x∈X one hasR

Xd(z, y)dmx(y)<+∞(see [24]).

A metric random walk space[X, d, m] is a Polish metric space (X, d) with a random walkm.

Let [X, d, m] be a metric random walk space. A Radon measure ν onX is invariant for the random walk m= (mx) if

dν(x) = Z

y∈X

dν(y)dm_y(x),

that is, for any ν-measurable set A, it holds that A is mx-measurable for ν-almost all x∈X, x7→mx(A) is ν-measurable, and

ν(A) = Z

X

mx(A)dν(x).

Hence, for anyu∈L¹(X, ν), it holds thatu∈L¹(X, mx) forν-a.e.x∈X,x7→

Z

X

u(y)dmx(y) isν-measurable, and

Z

X

u(x)dν(x) = Z

X

Z

X

u(y)dmx(y)

dν(x).

The measureν is said to bereversibleif, moreover, a more detailed balance condition dmx(y)dν(x) = dmy(x)dν(y)

holds true. Under suitable assumptions on the metric random walk space [X, d, m], such an invariant and reversible measure ν exists and is unique, as we will see below in the examples. Note that the reversibility condition implies the invariance condition.

We will assume that the metric measure space (X, d, ν) isσ-finite.

Example 1.1. (1) Consider (R^N, d,L^N), with d the Euclidean distance and L^N the Lebesgue measure. Let J :R^N →[0,+∞[ be a measurable, nonnegative and radially symmetric function verifyingR

R^NJ(z)dz= 1. In (R^N, d,L^N) we have the following random walk, starting atx,

m^J_x(A) :=

Z

A

J(x−y)dL^N(y) ∀A⊂R^N Borelian.

Applying Fubini’s Theorem it is easy to see that the Lebesgue measureL^N is an invariant and reversible measure for this random walk. Hence, [X, d, m^J] is a random walk space.

(2) LetK:X×X →Rbe a Markov kernel on a countable spaceX, i.e.

K(x, y)≥0, ∀x, y∈X, X

y∈X

K(x, y) = 1 ∀x∈X.

Then, for

m^K_x(A) :=X

y∈A

K(x, y),

[X, d, m^K] is a metric random walk space for any metric d on X. Basic Markov chain theory guarantees the existence of a unique stationary probability measure (also called steady state)πonX, that is,

X

x∈X

π(x) = 1 and π(y) = X

x∈X

π(x)K(x, y) ∀y∈X.

We say thatπis reversible forK if the following detailed balance equation K(x, y)π(x) =K(y, x)π(y)

holds true forx, y∈X.

(3) Consider a locally finite weighted discrete graph G= (V(G), E(G)), where each edge (x, y)∈E(G) (we will write x∼y if (x, y)∈E(G)) has a positive weightwxy=wyx assigned. Suppose further that wxy = 0 if (x, y)6∈E(G). The graph is equipped with the standard shortest path graph distancedG, that is, dG(x, y) is the minimal number of edges connecting xandy. We will assume that any two points are connected,i.e., that the graph is connected. Forx∈V(G) we define the weight at the vertexxas

d_x:=X

y∼x

w_xy = X

y∈V(G)

w_xy,

and the neighbourhood NG(x) :={y ∈ V(G) : x∼y}. The graph is locally finite, i.e. the sets NG(x) are assumed to be finite. When all the weights are 1,dxcoincides with the degree of the vertex xin a graph, that is, the number of edges containing vertex x.

For each x∈V(G) we define the following probability measure m^G_x = 1

dx

X

y∼x

w_xyδ_y.

We have that [V(G), d_G, m^G] is a metric random walk space. It is not difficult to see that the measureν_Gdefined as

νG(A) :=X

x∈A

dx, A⊂V(G)

is an invariant and reversible measure for this random walk.

(4) From a metric measure space (X, d, µ) we can obtain a metric random walk space with the so called-step random walk associated to µ, as follows. Assume that balls in X have finite measure and that Supp(µ) =X.

Given >0, the -step random walk on X, starting at point x, consists in randomly jumping in the ball of radiusaroundx, with probability proportional toµ; namely

m^µ,_x :=µ B(x, ) µ(B(x, )).

Note that, if µ(B(x, )) = µ(B(y, )) for any x, y∈X, then by Fubini-Tonelli theorem for any nonnegative functiong measurable with respect toµ⊗µwe have

Z

X

Z

X

g(x, y) dm^µ,_x (y)

dµ(x) = 1

µ(B(x, )) Z

X

Z

B(x,)

g(x, y) dµ(y)

! dµ(x)

= 1

µ(B(x, )) Z

X×X

g(x, y)χ{(x,y)∈X×X: d(x,y)<}(x, y) dµ(y)dµ(x)

= 1

µ(B(y, )) Z

X

Z

B(y,)

g(x, y) dµ(x)

!

dµ(y) = Z

X

Z

X

g(x, y) dm^µ,_x (x)

dµ(y), soµis an invariant and reversible measure for the metric random walk [X, d, m^µ,]. The condition that the balls have the same measure is satisfied for instance on Euclidean spaces and Carnot groups, or more generally when X is an Ahlfors regular metric measure space withµ(B(x, )) =C^swiths >0.

(5) Given a metric random walk space [X, d, m] with invariant and reversible measure ν for m, and given a ν-measurable set Ω⊂X with ν(Ω)>0, if we define, forx∈Ω,

m^Ω_x(A) :=

Z

A

dm_x(y) + Z

X\Ω

dm_x(y)

!

δ_x(A) ∀A⊂Ω Borelian,

we have that [Ω, d, m^Ω] is a metric random walk space and it easy to see ([17]) thatν Ω is reversible form^Ω. In the case that Ω is a closed bounded subset ofR^N, we obtain in this way the metric random walk [Ω, d, m^J,Ω], withm^J,Ω= (m^J)^Ω, that is

m^J,Ω_x (A) :=

Z

A

J(x−y)dy+ Z

Rⁿ\Ω

J(x−z)dz

!

dδ_x ∀A⊂Ω Borelian.

From now on, we will assume that [X, d, m] is a metric random walk space with an invariant and reversible measure ν.

Let us recall some of the results about them-Perimeter and them-Total Variation given in [18].

1.2. m-Perimeter

In this context, them-interactionbetween twoν-measurable subsetsAandB ofX is defined as Lm(A, B) :=

Z

A

Z

B

dmx(y)dν(x).

By the reversibility assumption onν with respect tom, we have L_m(A, B) =L_m(B, A).

We define the concept of m-perimeter of aν-measurable subsetE⊂X as Pm(E) =Lm(E, X\E) =

Z

E

Z

X\E

dmx(y)dν(x).

It is easy to see that

Pm(E) = 1 2 Z

X

Z

X

|χ_E(y)−χ_E(x)|dmx(y)dν(x).

Moreover, ifE isν-integrable, we have

P_m(E) =ν(E)− Z

E

Z

E

dm_x(y)dν(x).

Example 1.2. (1) Let [R^N, d, m^J] be the metric random walk space given in Example 1.1(1) with invariant measure L^N. Then,

P_mJ(E) = 1 2

Z

R^N

Z

R^N

|χ_E(y)−χ_E(x)|J(x−y)dydx,

which coincides with the concept ofJ-perimeter introduced in [15] (see also [16]). On the other hand, P_mJ,Ω(E) =1

2 Z

Ω

Z

Ω

|χ_E(y)−χ_E(x)|J(x−y)dydx.

Note that, in general,P_mJ,Ω(E)6=P_mJ(E).

Moreover, ifL^N(E)<∞,

P_mJ,Ω(E) =L^N(E)− Z

E

Z

E

dm^J,Ω_x (y)dx

=L^N(E)− Z

E

Z

E

J(x−y)dydx− Z

E

Z

R^N\Ω

J(x−z)dz

! dx.

Therefore,

P_mJ,Ω(E) =P_mJ(E)− Z

E

Z

R^N\Ω

J(x−z)dz

!

dx, ∀E⊂Ω.

(2) In the particular case of a graph [V(G), d_G, m^G], given A, B⊂V(G), Cut(A, B) is defined as Cut(A, B) := X

x∈A,y∈B

w_xy =L_mG(A, B),

and the perimeter of a set E⊂V(G) is given by

|∂E|:= Cut(E, E^c) = X

x∈E,y∈V\E

wxy.

Then, we have that

|∂E|=P_mG(E) for allE⊂V(G).

1.3. m-Total Variation

Associated to the random walkm= (mx) andν, we define the space BV_m(X) :=

u:X →R ν-measurable : Z

X

Z

X

|u(y)−u(x)|dm_x(y)dν(x)<∞

.

We have L¹(X, ν)⊂BV_m(X). Foru∈BV_m(X), we define itsm-total variation as T V_m(u) := 1

2 Z

X

Z

X

|u(y)−u(x)|dm_x(y)dν(x).

Note that

P_m(E) =T V_m(χ_E).

Recall the definition of the generalized product measureν⊗m_x (see, for instance, [2]), which is defined as the measure inX×X given by

ν⊗m_x(U) :=

Z

X

Z

X

χ_U(x, y)dm_x(y)dν(x) forU ∈ B(X×X),

where one needs the map x7→mx(E) to be ν-measurable for any Borel set E∈ B(X). By the reversibility of ν, it holds that

Z

X×X

gd(ν⊗mx) = Z

X

Z

X

g(x, y)dmx(y)dν(x)=

Z

X

Z

X

g(x, y)dmy(x)dν(y).

for every g∈L¹(X×X, ν⊗mx). Therefore, we can write T Vm(u) = 1

2 Z

X

Z

X

|u(y)−u(x)|d(ν⊗mx)(x, y).

Example 1.3. Let [V(G), d_G, m^G] be the metric random walk given in Example1.1(3) with the invariant and reversible measureνG. Then

T V_mG(u) = 1 2

Z

V(G)

Z

V(G)

|u(y)−u(x)|dm^G_x(y)dνG(x)

= 1 2 Z

V(G)

1 dx



 X

y∈V(G)

|u(y)−u(x)|wxy



dνG(x)

= 1 2

X

x∈V(G)

dx



 1 dx

X

y∈V(G)

|u(y)−u(x)|wxy





=1 2

X

x∈V(G)

X

y∈V(G)

|u(y)−u(x)|wxy.

Note thatT V_mG(u) coincides with the anisotropic total variation defined in [6].

Let us now recall some properties of them-total variation given in [18].

Proposition 1.4. Ifφ:R→Ris Lipschitz continuous then, for everyu∈BVm(X),φ(u)∈BVm(X) and T Vm(φ(u))≤ kφkLipT Vm(u).

Proposition 1.5. T V_m is convex and continuous in L¹(X, ν) and lower semi-continuous respect to the weak convergence in L^q(X, ν),q= 1,2.

Let us remark that Proposition1.5still works even if the functions do not belong toL¹(X, ν), we only need that (u_n−u)→0 inL¹(X, ν).

As in the local and nonlocal case, we have the following coarea formula, given in Theorem 2.7 of [18], relating the total variation of a function with the perimeter of its superlevel sets. While the result in [18] is stated for u∈L¹(X, ν), this assumption is never used, and the coarea formula holds for allu∈BV_m(X).

Theorem 1.6 (Coarea formula). For anyu∈BVm(X), letEt(u) :={x∈X : u(x)> t}. Then, T V_m(u) =

Z +∞

−∞

P_m(E_t(u)) dt.

For a functionu:X →Rwe define its nonlocal gradient∇u:X×X →Ras

∇u(x, y) :=u(y)−u(x) ∀x, y∈X.

For a functionz:X×X →R, itsm-divergencediv_mz:X→Ris defined as (div_mz)(x) := 1

2 Z

X

(z(x, y)−z(y, x))dm_x(y).

Forp≥1, we define the space X_m^p(X) :=

z∈L^∞(X×X, ν⊗m_x) : div_mz∈L^p(X, ν)

.

For u∈BV_m(X)∩L^p0(X, ν) and z∈X_m^p(X), 1≤p≤ ∞, having in mind that ν is reversible, we have the following Green’s formula

Z

X

u(x)(div_mz)(x)dν(x) =−1 2

Z

X×X

∇u(x, y)z(x, y)dν⊗dm_x.

In the next result we characterizeT V_mand them-perimeter using them-divergence operator. Let us denote by sign₀(r) the usual sign function and by sign(r) the multivalued sign function

sign(u)(x) :=







1 if u(x)>0,

−1 if u(x)<0, [−1,1] if u= 0.

Proposition 1.7. Let1≤p≤ ∞. For u∈BVm(X)∩L^p0(X, ν), we have T Vm(u) = sup

Z

X

u(x)(divmz)(x)dν(x) : z∈X_m^p(X), kzk_∞≤1

.

In particular, for anyν-measurable set E⊂X, we have

Pm(E) = sup Z

E

(divmz)(x)dν(x) : z∈X_m¹(X), kzk_∞≤1

.

Let us recall the concept of ergodicity (see, for example, [10]).

Definition 1.8. Let [X, d, m] be a metric random walk space with invariant finite measure ν. A Borel set B ⊂X is said to beinvariant with respect to the random walkmifm_x(B) = 1 wheneverxis in B.

The invariant finite measure ν is said to be ergodic ifν(B) = 0 orν(B) =ν(X) for every invariant set B with respect to the random walkm.

The following result was obtained in [17].

Theorem 1.9. Let [X, d, m] be a metric random walk with invariant finite measure ν. Then, the following assertions are equivalent:

(i) ν is ergodic.

(ii) IfA, B ⊂X are disjoint sets such that0< ν(A), ν(B)< ν(X)andA∪B =X, thenLm(A, B)6= 0.

In particular, the Theorem above gives an important family of ergodic measures: on a length space, a doubling measure ν is ergodic with respect to the random walkm^ν,ε. This is formalised in the following Corollary.

Corollary 1.10. Let [X, d, m^ν,ε] be a metric random walk space, with a finite measure ν which is invariant with respect to the random walk m^ν,ε. Suppose that X is a length space and that every ball in X has positive ν-measure. Then,ν is ergodic.

Proof. LetA, B be the sets in condition (ii) of Theorem1.9. First, notice that there exists a ball B(x0, δ) with δ≤^ε₄ such that

0< ν(A∩B(x0, δ)), ν(B∩B(x0, δ))< ν(B(x0, δ)). (1.1) Suppose to the contrary that there is no such ball. Then, for every pointx∈X we have either havex∈Aeor x∈B, wheree

Ae={x∈X : for everyδ≤ε

4 we have ν(A∩B(x, δ)) =ν(B(x, δ))}

and

Be={x∈X : for everyδ≤ε

4 we have ν(B∩B(x, δ)) =ν(B(x, δ))}.

Notice that sinceXis a length space, we haved(A,e B) = 0; if not,e i.e.d(A,e B)e > c >0, then the assumption that X is a length space implies that there exists a point with positive distance to bothAeandB, which contradictse the fact that Ae∪Be =X. Now, take x0 ∈Aeandy0∈Be such thatd(x0, y0)< ^ε₈. Then, by definition ofAewe have ν(A∩B(x₀,^ε₄)) =ν(B(x₀,^ε₄)); similarly, by definition ofBe we haveν(B∩B(y₀,₈^ε)) =ν(B(y₀,^ε₈)), which yields a contradiction becauseAandBare disjoint. Hence, there exists a ballB(x0, δ) withδ≤ ^ε₄which satisfies condition (1.1).

Now, we letB(x0, δ) be such a ball and compute L_m(A, B) =

Z

A

Z

B

dm^ν,ε_x (y) dν(x)≥ Z

A∩B(x0,δ)

Z

B

dm^ν,ε_x (y) dν(x)

= Z

A∩B(x₀,δ)

Z

B

1

ν(B(x, ε))χ{y: d(x,y)<ε}(y) dν(y) dν(x)

≥ Z

A∩B(x₀,δ)

Z

B∩B(x₀,δ)

1

ν(B(x, ε))dν(y) dν(x)

≥ Z

A∩B(x0,δ)

Z

B∩B(x0,δ)

1

ν(B(x₀,2ε))dν(y) dν(x) =ν(A∩B(x0, δ))ν(B∩B(x0, δ)) ν(B(x₀,2ε)) >0,

so condition (ii) in Theorem1.9is satisfied and the measureν is ergodic with respect to the random walkm^ν,ε.

1.4. BV-functions in metric measure spaces

Let (X, d, ν) be a metric measure space. For functions in L¹(X, ν) we have the concept of total variation introduced by Miranda in [20] (see also [1]). To introduce this concept recall that for a functionu:X→R, its slope (also called local Lipschitz constant) is defined by

|∇u|(x) := lim sup

y→x

|u(y)−u(x)|

d(x, y) ,

with the convention that|∇u|(x) = 0 ifxis an isolated point.

A function u∈L¹(X, ν) is said to be a BV-function if there exist locally Lipschitz functionsun converging to uin L¹(X, ν) and such that

sup

n∈N

Z

X

|∇un|dν(x)<∞.

We shall denote the space of all BV-functions byBV(X, d, ν). Foru∈BV(X, d, ν) the total variation|Du|ν on an open set A⊂X is defined as:

|Du|ν(A) := inf

lim inf

n→∞

Z

A

|∇un|(x)dν(x) : un∈Liploc(A), un →uin L¹(A, ν)

.

We say that a measureνon a metric spaceX is doubling, if there exists a constantCd≥1 such that following condition holds:

0< ν(B(x,2r))≤C_dν(B(x, r))<∞

for all x∈ X and r >0. The constant C_d is called the doubling constant of X. By iterating the doubling condition, we see that a classical estimate holds (see for instance [9]):

Proposition 1.11. Let y∈B(x, R)and setr∈(0, R). Then ν(B(y, r))

ν(B(x, R)) ≥4^−s r

R s

for all s≥log₂(Cd). The number log₂(Cd)is called the homogenous dimension ofX.

The metric measure space (X, d, ν) is said to support a local 1-Poincar´e inequality if there exist constants c >0 andλ≥1 such that, for anyu∈Lip(X, d), the inequality

Z

B(x,r)

|u(y)−u_B(x,r)|dν(y)≤cr Z

B(x,λr)

|∇u|(y)dν(y)

holds, where

u_B(x,r):= 1 ν(B(x, r))

Z

B(x,r)

u(y)dν(y).

We recall the following result proved in Theorem 2.22 of [18] (see also [12], Thm. 3.1), which links the total variation defined above and the nonlocal total variation of the type presented in Example1.1(4):

Theorem 1.12. Let (X, d, ν) be a metric measure space with doubling measure ν and supporting a local 1- Poincar´e inequality. Givenu∈L¹(X, ν), we have thatu∈BV(X, d, ν)if and only if

lim inf

ε→0

1

εT Vm^ν,ε(u)<∞.

2. Least gradient functions in metric random walk spaces

From now on, we will assume that [X, d, m] is a metric random walk space with an invariant and reversible measure ν, withν(X)<∞.

Given Ω⊂X aν-measurable set, we define itsm-boundary as

∂mΩ :={x∈X\Ω : mx(Ω)>0}.

We set Ωm:= Ω∪∂mΩ.

2.1. Nonlocal least gradient problem

We will deal with Ω⊂X such that 0< ν(Ω)< ν(X). Then, assuming thatν is ergodic, by Theorem1.9, we have that

ν(∂mΩ)>0.

From now on we will assume that ν is ergodic.

Given a function ψ∈L¹(∂_mΩ) we considerm-least gradient problemas the variational problem

min{T V_m(u) : u∈BV_m(Ω_m), such thatu=ψ ν−a.e. on∂_mΩ}. (2.1) We may equivalently state the problem on the whole space X and not only on Ω_m; suppose that instead of defining the boundary condition only on∂_mΩ, we takeψ∈L¹(X, ν) and we extenduto the whole spaceX by settingu=ψ ν-a.e. onX\Ω. Let us denote byT V_m(u, A) the total variation ofu Ain a Borel setA⊂X. By the reversibility of ν and the definition of∂_mΩ, we have

T Vm(u, X) =T Vm(u,Ωm) + Z

∂mΩ

Z

X\Ωm

|ψ(y)−ψ(x)|dmx(y)dν(x) +T Vm(ψ, X\Ωm),

hence the total variations ofuonX and on Ω_mdiffer by a constant, so they have the same minimizers. Finally, let us remark that the assumption that ν(X)<∞ is used a few times to simplify the proofs, but it is not strictly necessary: for instance, we could instead assume that 0< ν(∂_mΩ)<∞andψ has compact support in X, so that we can restrict our considerations to a subset ofX with finite measure. This would be the case for instance for the random walkm^J from Example 1.1(1).

Using the direct method of calculus of variation we have the following existence result.

Theorem 2.1. Ifψ∈L^∞(∂_mΩ)the m-least gradient problem(2.1)has a solution.

Proof. Let

τ= inf{T Vm(u) : u∈BV_m(Ω_m), such thatu=ψon∂_mΩ},

andun∈BVm(Ωm) a sequence of admissible functions such thatT Vm(un)→τ. Firstly, we correct the functions u_n, we set

ufn(x) :=











kψk∞ if un(x)>kψk∞

u_n(x) if u_n(x)∈[−kψk∞,kψk∞]

−kψk_∞ if u_n(x)<−kψk_∞.

By Proposition1.4, we haveT Vm(ufn)≤T Vm(un), hence the functionsufnare admissible. Moreover, the sequence {ufn} is uniformly bounded, then by the Dunford-Pettis Theorem, we can assume, taking a subsequence if

necessary, thatufn * u∈BVm(Ωm) weakly inL¹(X, ν). Then, by Proposition1.5, we have T V_m(u)≤lim inf

n→∞ T V_m(uf_n) =τ.

Finally, since uis the weak limit of the sequence{ufn}, we have lim inf

n→∞ ufn(x)≤u(x)≤lim sup

n→∞ ufn(x) ν−a.e. x∈Ωm.

Thus,u=ψ ν−a.e. on∂_mΩ, souis a solution of them-least gradient problem (2.1).

Now, we define m-least gradient functions and prove nonlocal analogues of Miranda’s Theorem, see [19], and a theorem by Bombieri, de Giorgi and Giusti linking a m-least gradient function to the minimality of its superlevel sets, see [5].

Definition 2.2. We say that u∈BV_m(X) is am-least gradient function on Ω, if for every g∈BV_m(X) such that g≡0 ν-a.e. onX\Ω we have

T Vm(u)≤T Vm(u+g).

Proposition 2.3. Suppose that u_n ∈BV_m(X)is a sequence of m-least gradient functions in Ω convergent in L¹(X, ν)tou∈BV_m(X). Thenuis a function ofm-least gradient in Ω.

Proof. By Proposition 1.5, m-total variation is continuous with respect to convergence in L¹(X, ν). Fix g∈ BVm(X) with support in Ω. Hence

T V_m(u)←T V_m(u_n)≤T V_m(u_n+g)→T V_m(u+g),

henceuis a function ofm-least gradient.

Theorem 2.4. Suppose thatu∈BV_m(X). Thenuis a function of m-least gradient inΩif and only if χ_E

t(u)

is a function of m-least gradient for all (equivalently - almost all)t∈R.

Proof. Fixt∈Rand letu1= min(u, t) andu2=u−u1= max(u−t,0). By the coarea formula, which may be applied even ifu /∈L¹(X, ν), we have

T Vm(u) = Z +∞

−∞

Pm(Et(u))dt= Z t

−∞

Pm(Et(u))dt+ Z +∞

t

Pm(Et(u))dt

=T Vm(u1) +T Vm(u2).

This implies thatu1andu2 are functions ofm-least gradient in Ω. In fact, for anyg∈BVm(X) such thatg≡0 ν-a.e. onX\Ω, we have

T V_m(u₁) +T V_m(u₂) =T V_m(u)≤T V_m(u+g)≤T V_m(u₁+g) +T V_m(u₂), henceu1 is a function ofm-least gradient. We make a similar argument foru2. Now, we set

uε,t=1

εmin(ε,max(u−t,0))

and notice that by the above argument it is a function ofm-least gradient (it is a rescaled minimum ofu2and a constant).

Now, we prove thatuε,t→χ_E

t(u) inL¹(X, ν). Indeed, Z

X

|uε,t−χ_E

t(u)|dν= Z

X\Et(u)

|uε,t−χ_E

t(u)|dν+ Z

Et(u)\Et+ε(u)

|uε,t−χ_E

t(u)|dν

+ Z

E_t+ε(u)

|uε,t−χ_E

t(u)|dν= Z

E_t(u)\Et+ε(u)

|uε,t−χ_E

t(u)|dν,

because the first and last integrals are equal to zero. But Z

Et(u)\Et+ε(u)

|uε,t−χ_E

t(u)|dν ≤ν(Et(u)\Et+ε(u)), which goes to zero as ε→0, because E_t(u) =S

t⁰>tE_t⁰(u) andν(X) is finite. Hence, Proposition 2.3implies that χ_E

t(u)is a function ofm-least gradient.

The implication in the other direction follows directly from the coarea formula.

Let Ω ⊂R^N be a bounded open set. In [14] it is proved that the Dirichlet problem for the 1–Laplacian operator







−div Du

|Du|

= 0, in Ω, u=h , on∂Ω,

(2.2)

has a solution u∈BV(Ω) for everyh∈L¹(∂Ω). The relaxed energy functional associated to problem (2.2) is the functional Φh:L^N−1N (Ω)→(−∞,+∞] defined by

Φ_h(u) =





 Z

Ω

|Du|+ Z

∂Ω

|u−h|dH^N−1 ifu∈BV(Ω),

+∞ ifu∈L^N−1N (Ω)\BV(Ω).

(2.3)

In [14] it is showed that the minimizer of the functional Φhcoincides with the solution of problem (2.2) and with the function of least gradient on Ω that coincides with hon the boundary. A nonlocal version of the above first statement was obtained in [13], more precisely for the random walkm^J given in Example1.1(1). We will see now that using an adaptation of the method developed in [13], we can obtain similar results for general metric random walk spaces. Although some of the proofs are similar, we will give them for the sake of completeness.

Given a function ψ∈L¹(∂_mΩ) andu∈BV_m(Ω), we define the function

uψ(x) :=







u(x) if x∈Ω ψ(x) if x∈∂mΩ.

Consider the relaxed energy functionalJψ:L¹(Ω, ν)→[0,+∞[ given by Jψ(u) :=T Vm(uψ) = 1

2 Z

Ωm

Z

Ωm

|uψ(y)−uψ(x)|dmx(y)dν(x). (2.4)

This functional Jψ is the nonlocal version of the energy functional Φh defined by (2.3).

Since ifv∈BVm(Ωm), such thatv=ψ ν−a.e. on∂mΩ, then (v_|Ω)ψ=v, we have the following result.

Proposition 2.5. Given ψ∈L¹(∂mΩ), For u∈BVm(Ωm), such that u=ψ ν−a.e. on ∂mΩ, the following are equivalent:

(i) uis a solution of them-least gradient problem(2.1).

(ii) u_|Ω is a minimizer ofJψ.

In [18], the authors study them-1-Laplacian operator ∆^m₁, which formally is the operator

∆^m₁u(x) = Z

Ω_m

uψ(y)−u(x)

|uψ(y)−u(x)|dm_x(y) forx∈Ω.

Consider the nonlocal1-Laplace problemwith Dirichlet boundary conditionψ:







−∆^m₁u(x) = 0, x∈Ω, u(x) =ψ(x), x∈∂mΩ.

(2.5)

Definition 2.6. Letψ∈L¹(∂mΩ). We say thatu∈BVm(Ω) is a solution to (2.5) if there existsg∈L^∞(Ωm× Ωm) withkgk_∞≤1 verifying

g(x, y) =−g(y, x) for (ν⊗dmx)-a.e (x, y) in Ωm×Ωm, (2.6)

g(x, y)∈sign(uψ(y)−uψ(x)) for (ν⊗dmx)-a.e (x, y) in Ωm×Ωm, (2.7) and

− Z

Ωm

g(x, y) dmx(y) = 0 forν−a.ex∈Ω. (2.8) Definition 2.7. Letp≥1. We say that (m, ν) satisfies ap-Poincar´e Inequality in Ω if there existsλ >0 such that

λ Z

Ω

|u(x)|^p dν(x)≤ Z

Ω

Z

Ωm

|uψ(y)−u(x)|^pdmx(y) dν(x) + Z

∂mΩ

|ψ(y)|^pdν(y) for allu∈L^p(Ω, ν) andψ∈L^p(∂_mΩ).

We have the following result.

Theorem 2.8. Letψ∈L^∞(∂mΩ). If(m, ν)satisfies ap-Poincar´e inequality inΩfor allp >1, thenu∈L¹(Ω) is a solution to problem (2.5)if and only if it is a minimizer of the functionalJψ given by the formula (2.4).

In fact, it is enough to assume that there exist solutions to (2.5); hence, we will start by proving the existence of solutions to (2.5). We prove this existence result by taking limits as pgoes to 1 in the following Dirichlet problem







− Z

Ω_m

|(up)ψ(y)−up(x)|^p−2((up)ψ(y)−up(x))dmx(y) = 0, x∈Ω,

up=ψ, x∈∂mΩ.

(2.9)

Theorem 2.9. Given ψ∈L^∞(∂mΩ) and p >1. If (m, ν) satisfies a p-Poincar´e inequality in Ω, then there exists a solution up of problem(2.9). Moreover,

kupk∞≤ kψk∞. (2.10)

Proof. Let us consider the functional Fp(u) := 1

2p Z

Ω_m

Z

Ω_m

|uψ(y)−uψ(x)|^pdmx(y)dν(x), u∈L^p(Ω).

Set

θ:= inf

u∈L^p(Ω,ν)

Fp(u), and let{un} be a minimizing sequence. Then,

θ= lim

n→∞Fp(un) and K:= sup

n∈N

Fp(un)<+∞.

Since (m, ν) satisfies a p-Poincar´e inequality, λ

Z

Ω

|un(x)|^p dν(x)≤ Z

Ω

Z

Ω_m

|uψ(y)−u(x)|^pdmx(y) dν(x) + Z

∂_mΩ

|ψ(y)|^pdν(y)

= 2pFp(un) + Z

∂mΩ

|ψ(y)|^pdν(y)≤2pK+ Z

∂mΩ

|ψ(y)|^pdν(y).

Therefore, we obtain that

Z

Ω

|u_n(x)|^pdν(x)≤C ∀n∈N. Hence, up to a subsequence, we have

u_n* u_p in L^p(Ω).

Furthermore, using the weak lower semi-continuity of the functionalFp, we get Fp(up) = inf

u∈L^p(Ω)Fp(u).

Thus, givenλ >0 andw∈L^p(Ω) (we extend it to∂mΩ by zero), we have 0≤ Fp(up+λw)− Fp(up)

λ ,

or equivalently, 0≤1

2 Z

Ω_m

Z

Ω_m

|(up)ψ(y) +λw(y)−((up)ψ(x) +λw(x))|^p− |(up)ψ(y)−(up)ψ(x)|^p pλ

dmx(y)dν(x).

Now, sincep >1, we pass to the limit asλ↓0 to deduce 0≤1

2 Z

Ωm

Z

Ωm

|(up)ψ(y)−(up)ψ(x)|^p−2((up)ψ(y)−(up)ψ(x))×(w(y)−w(x)) dmx(y) dν(x).

Takingλ <0 and proceeding as above we obtain the reverse inequality. Consequently, we conclude that 0 = 1

2 Z

Ωm

Z

Ωm

|(up)ψ(y)−(up)ψ(x)|^p−2((up)ψ(y)−(up)ψ(x))×(w(y)−w(x))dmx(y)dν(x)

=− Z

Ω_m

Z

Ω_m

|(up)_ψ(y)−(u_p)_ψ(x)|^p−2((u_p)_ψ(y)−(u_p)_ψ(x))dm_x(y)w(x)dν(x).

In particular, since w= 0 in∂mΩ, it follows that 0 =−

Z

Ω

Z

Ω_m

|(up)ψ(y)−up(x)|^p−2((up)ψ(y)−up(x))dmx(y)w(x)dν(x), which shows thatup is a solution of (2.9). To finish let us see that (2.10) holds.

SetM :=kψk∞, multiply quation (2.9) by (up−M)⁺ and integrate over Ω to obtain 0 =−

Z

Ωm

Z

Ωm

|(up)ψ(y)−(up)ψ(x)|^p−2((up)ψ(y)−(up)ψ(x))dmx(y)((up)ψ−M)⁺(x)dν(x), or equivalently

0 = 1 2

Z

Ωm

Z

Ωm

|(up)ψ(y)−(up)ψ(x)|^p−2((up)ψ(y)−(up)ψ(x))

× ((u_p)_ψ−M)⁺(y)−((u_p)_ψ−M)⁺(x)

dm_x(y)dν(x).

In addition, since

|r−s|^p−2(r−s)(r⁺−s⁺)≥ |r⁺−s⁺|^p, it holds that

Z

Ωm

Z

Ωm

((up)ψ−M)⁺(y)−((up)ψ−M)⁺(x)

pdmx(y)dν(x)≤0.

Then, using again thep-Poincar´e inequality we get Z

Ω

|(u_p−M)⁺(x)|^pdν(x) = 0.

This shows thatup≤M ν-a.e. in Ω for anyp >1. Analogously, we can verify thatup≥ −M ν-a.e. in Ω. Thus

kupk_∞≤M for everyp >1.

We are now ready to prove the existence of solutions to problem (2.5).

Theorem 2.10. Givenψ∈L^∞(∂mΩ). If (m, ν)satisfies ap-Poincar´e inequality in Ωfor allp >1, then there exists a solution to problem (2.5).

Proof. By Theorem2.9, there exists a subsequencepn→1, still denoted byp, such that u_p* u weakly inL¹(Ω)

and

|(up)ψ(y)−(up)ψ(x)|^p−2((up)ψ(y)−(up)ψ(x))* g(x, y) weakly inL¹(Ωm×Ωm).

The function gisL^∞-bounded by 1, satisfies

− Z

Ω_m

g(x, y) dm_x(y) = 0 forν−a.ex∈Ω, and, moreover, it is antisymmetric. In order to see that

g(x, y)∈sign(uψ(y)−uψ(x)) for (ν⊗dmx)-a.e (x, y) in Ωm×Ωm, sincekgk_∞≤1, we need to prove that

− Z

Ωm

Z

Ωm

g(x, y) dm_x(y)u_ψ(x) dν(x)≥1 2

Z

Ωm

Z

Ωm

|u_ψ(y)−u_ψ(x)|dm_x(y)dν(x). (2.11) In fact, it holds that

1 2 Z

Ω_m

Z

Ω_m

|(up)ψ(y)−(up)ψ(x)|^pdmx(y)dν(x)

=− Z

Ω_m

Z

Ω_m

|(u_p)_ψ(y)−(u_p)_ψ(x)|^p−2((u_p)_ψ(y)−(u_p)_ψ(x)) dm_x(y)(u_p)_ψ(x)dν(x)

=− Z

∂mΩ

Z

Ωm

|up(y)−up(x)|^p−2(up(y)−up(x)) dmx(y)ψ(x) dν(x).

Therefore,

p→1lim 1 2 Z

Ω_m

Z

Ω_m

|up(y)−u_p(x)|^pdm_x(y)dν(x) =− Z

∂_mΩ

Z

Ω_m

g(x, y) dm_x(y)ψ(x) dν(x)

=− Z

Ωm

Z

Ωm

g(x, y) dmx(y)uψ(x) dν(x).

Then, 1 2 Z

Ω_m

Z

Ω_m

|u_ψ(y)−u_ψ(x)|dm_x(y)dν(x)≤lim inf

p→1

1 2

Z

Ω_m

Z

Ω_m

|u_p(y)−u_p(x)|^pdm_x(y)dν(x)

=− Z

Ωm

Z

Ωm

g(x, y) dmx(y)uψ(x) dν(x),

so equation (2.11) holds. Hence, g satisfies all the conditions in Definition 2.6, so u is a solution to problem

(2.5).

Proof of Theorem 2.8. Letube a solution of problem (2.5). Then, there existsg∈L^∞(Ω_m×Ωm) withkgk∞≤1 verifying (2.6), (2.7) and (2.8).

Given w ∈ L¹(Ω, ν), multiplying (2.8) by w(x)−u(x), integrating, and having in mind (2.7) and the antisymmetry of g, (2.6), we get

0 =− Z

Ωm

Z

Ωm

g(x, y)dmx(y)(wψ(x)−uψ(x))dν(x)

= 1 2 Z

Ω_m

Z

Ω_m

g(x, y)[(w_ψ(y)−w_ψ(x))−(u_ψ(y)−u_ψ(x))]dm_x(y)dν(x)

≤ 1 2 Z

Ωm

Z

Ωm

|w_ψ(y)−w_ψ(x)|dm_x(y)dν(x)−1 2

Z

Ωm

Z

Ωm

|u_ψ(y)−u_ψ(x)|dm_x(y)dν(x)

=Jψ(w)− Jψ(u).

Therefore,uis a minimizer ofJ_ψ.

Assume now thatuminimizes the functionalJ_ψ. Theorem2.10shows the existence of a solutionuof (2.5).

Namely, there existsg: Ω_m×Ω_m→Rsuch thatg∈L^∞(Ω_m×Ω_m),kgk_∞≤1,g(x, y) =−g(y, x) (ν⊗dm_x)-a.e (x, y) in Ωm×Ωm,

g(x, y)∈sign(uψ(y)−uψ(x)) (ν⊗dmx)−a.e (x, y)∈Ωm×Ωm, (2.12) and

− Z

Ωm

g(x, y) dmx(y) = 0 ν−a.e x∈Ω. (2.13) Sinceuis a minimizer ofJψ, we have

Jψ(u)− Jψ(u) = 0.

On the other hand, arguing as in the other implication, we obtain that 0 =−

Z

Ω_m

Z

Ω_m

g(x, y) dm_x(y)(u_ψ(x)−u_ψ(x))dν(x)

=1 2

Z

Ωm

Z

Ωm

g(x, y)[(uψ(y)−uψ(x))−(uψ(y)−uψ(x))]dmx(y)dν(x)

=1 2

Z

Ωm

Z

Ωm

|uψ(y)−uψ(x)|dmx(y)dν(x)

−1 2

Z

Ω_m

Z

Ω_m

g(x, y)(u_ψ(y)−u_ψ(x))dm_x(y)dν(x)

=Jψ(u)−1 2

Z

Ωm

Z

Ωm

g(x, y)(uψ(y)−uψ(x))dmx(y)dν(x).

Therefore, 1 2 Z

Ωm

Z

Ωm

g(x, y)(uψ(y)−uψ(x))dmx(y)dν(x) =1 2

Z

Ωm

Z

Ωm

|uψ(y)−uψ(x)|dmx(y)dν(x).

Hence,

g(x, y)∈sign(uψ(y)−uψ(x)) (ν⊗dmx)−a.e (x, y)∈Ωm×Ωm,

which jointly with (2.12) and (2.13) imply thatuis a solution to problem (2.5).

The proof of Theorem2.8sparks a natural question whether the solution to problem (2.5) is unique. It turns out that uniqueness of solutions fails even in the simplest settings. Consider a graphGwith three vertices and two edges, namelyV(G) ={−1,0,1}andE(G) ={(−1,0),(0,1)}, with weights of both edges equal to one. It is endowed with graph distance and the measuresν_G andm^G as described in Example1.1(3). Take Ω ={0}and set ψ(x) =x. The solution is defined only on the single point{0} and wheneveru(0)∈[−1,1],uis a solution to problem (2.5) with the function g(x, y) = sign₀(y−x). A similar example can also be constructed in the continuous case, see Example 3.1 of [13].

Also the vector field g is not uniquely defined. It can be easily seen if we take the space from the previous paragraph and set Ω ={0} and ψ≡0. Then, if we take u(0) = 0, we obtain a solution to problem (2.5) with the function g(x, y)≡ 0. However, it is also a solution to problem (2.5) with the function g(x, y) ≡0 and g(x, y) = sign₀(y−x), so the choice ofgis not unique. Nonetheless, let us stress that in the second part of the proof of Theorem 2.8 we take a function g corresponding to a single solution and show that it works for all solutions; in particular, a single functiong determines the structure of all solutions to (2.5).

Theorem 2.11. Assume that (m, ν) satisfies a p-Poincar´e inequality in Ω for all p >1. Let ψ∈L^∞(X\Ω) andu∈BVm(X)such thatu=ψ ν-a.e. on X\Ω. Then, the following are equivalent:

(i) u_|Ωis a minimizer of Jψ.

(ii) u_|Ω is a solution of to problem(2.5).

(iii) uis a function ofm-least gradient in Ω.

(iv) u_|Ωm is a solution of them-least gradient problem(2.1).

Proof. By Proposition2.5and Theorem2.8, we already know that the conditions (i), (ii) and (iv) are equivalent.

(i) implies (iii): Let g∈BV_m(X) such thatg≡0ν-a.e. onX\Ω. Then

T Vm(u) =Jψ(u_|Ω)≤ Jψ(u_|Ω+g_|Ω) =T Vm(u+g).

Therefore,uis a function ofm-least gradient in Ω.

(iii) implies (i): Fixedv∈BVm(Ω), we have to see thatJψ(u_|Ω)≤ Jψ(v). Now, ifg:=vψ−u∈BVm(Ωm) and g≡0ν-a.e. onX\Ω, we have

Jψ(u_|Ω) =T Vm(u)≤T Vm(u+g) =T Vm(vψ) =Jψ(v).

Finally, we note that Theorem 1.12implies that in the case of the random walkm^ν,ε, if the boundary data are regular enough, we have a uniform estimate on the nonlocal gradient on some subsequence (still denoted by u_ε).

Proposition 2.12. Let(X, d, ν)be a metric measure space with doubling measureν and supporting a 1-Poincar´e inequality. Let u_ε∈BV_mν,ε(X) be a sequence of solutions to the nonlocal least gradient problem on Ω⊂X, where 0< ν(Ω)< ν(X), for boundary data ψ ∈BV(X, d, ν)∩L^∞(X, ν). Let ε→0. Then on a subsequence (still denoted by u_ε) we have

Z

X

− Z

B(x,ε)

|(uε)ψ(y)−(uε)ψ(x)|dν(y) dν(x)≤M ε.

Proof. Notice that the left hand side of the desired inequality is 2T Vm^ν,ε(uε). Asuεis a minimizer of the total variationT Vm^ν,ε with boundary dataψ, we have

2T V_mν,ε(u_ε)≤2T V_mν,ε(ψ).

Asψ∈BV(X), by Theorem1.12on a subsequence (still denoted byε) forεsufficiently close to 0 we have 2T Vm^ν,ε(ψ)≤4ε·lim inf

ε→0

1

εT Vm^ν,ε(ψ) =M ε,

where M = 4 lim inf_ε→0¹_εT V_mν,ε(ψ).

This uniform estimate, which holds as long as the boundary datumψ∈L^∞(∂mΩ) admits a bounded extension toBV(X, d, ν), suggests that possibly the sequenceuεconverges on a subsequence; this is exactly what happens in the case of the random walkm^Jfrom Example1.1(1) and an analogue of Proposition2.12is a crucial estimate in that proof, see [13]. This motivates the following Remark.

Remark 2.13. Let [R^N, d, m^J] be the metric random walk space of Example1.1 (1) and assume alsoJ(x)≥ J(y) if |x| ≤ |y|. Let Ω be a smooth bounded domain inR^N and ˜ψ∈L^∞(∂Ω). In [13] it is proved that if we take a function ψ∈W^1,1(∂_mJΩ)∩L^∞(∂_mJΩ) such thatψ|_∂Ω= ˜ψandu is a solution of the problem







−∆^m₁^Ju(x) = 0, x∈Ω, u(x) =ψ(x), x∈∂_mJΩ, forJε(x) := _ε¹NJ ^x_ε

. Then, up to a subsequence,

u→u inL¹(Ω), where uis a solution to (2.2) withh= ˜ψ.

In particular, if we take

J(x) := 1

L^N(B(0,1))χ_B(0,1)(x), then

J(x) = 1

L^N(B(0, ))χ_B(0,)(x).

Hence,

m^L_x^N,=m^J_x, and, consequently, if u is a solution of the problem







−∆^m₁^{LN ,}u(x) = 0, x∈Ω, u(x) =ψ(x), x∈∂_mLN ,Ω,

up to a subsequence,

u→u inL¹(Ω), where uis a solution to (2.2) withh= ˜ψ.

Therefore, it is natural to pose the following problem: Let (X, d, µ) be a metric measure space and let m^µ, be the-step random walk associated toµ, that is,

m^µ,_x :=µ B(x, ) µ(B(x, )).

Given f ∈L^∞(∂Ω,|Dχ_Ω|µ) we consider the functionalTf :L²(Ω, µ)→]− ∞,+∞] defined by

Tf(u) :=











|Du|µ(Ω) + Z

∂Ω

|u−f|d|Dχ_Ω|µ ifu∈BV(Ω, d, µ)∩L²(Ω, µ),

+∞ ifu∈L²(Ω, µ)\BV(Ω, d, µ).

We define the (multivalued) operator−∆_1,µ:=∂T_f.

Letf ∈L^∞(∂Ω,|Dχ_Ω|µ) such that there existsψ∈W^1,1(∂m^µ,Ω)∩L^∞(∂m^µ,Ω), withψ|∂Ω=f. It is natural to ask the following question: are there metric measure spaces, for which if u is a solution of the problem







−∆^m₁^µ,u(x) = 0, x∈Ω, u(x) =ψ(x), x∈∂m^µ,Ω, then up to a subsequence

u→u in L¹(Ω, µ), where uis a solution to the problem







−∆1,µu(x) = 0, x∈Ω, u(x) =f(x), x∈∂Ω.

In a forthcoming paper we will study this problem.

2.2. Nonlocal median value property

Let us introduce some notation for this subsection. Given aν-measurable functionu: Ω→R, we decompose the spaceX as

E₊^x ={y∈X :u_ψ(y)> u_ψ(x)}, E₋^x ={y∈X:u_ψ(y)< u_ψ(x)},

E₀^x={y∈X :u_ψ(y) =u_ψ(x)}.

Asmxis a probability measure, for any x∈Ω we have

mx(E₊^x) +mx(E^x₋) +mx(E₀^x) = 1,

so the following two conditions are equivalent:

−mx(E₀^x)≤mx(E₋^x)−mx(E₊^x)≤mx(E₀^x) (2.14) and

mx(E₊^x∪E^x₀)≥1

2 and mx(E₋^x ∪E₀^x)≥ 1

2. (2.15)

Definition 2.14. We say that a ν-measurable function u: Ω→ R satisfies the m-median value property if either of the conditions (2.14) or (2.15) is satisfied forν-a.e. x∈Ω.

In the next Theorem, we prove that solutions to (2.5) satisfy the nonlocal median value property. In the other direction, it is not necessarily the case; examples to the contrary exist even in Euclidean spaces, see for instance ([13], Exam. 3.2). Nonetheless, a partial converse still holds.

Theorem 2.15. Suppose thatu∈BVm(Ω)is a solution to(2.5). Then it satisfies them-median value property.

Moreover, if u∈ BVm(Ω) satisfies the m-median value property, then it satisfies Definition 2.6 except for antisymmetry of the function g.

Proof. Suppose that u ∈BVm(Ω) is a solution to (2.5). Take the antisymmetric function g ∈L^∞(X×X) associated to ugiven by Definition 2.6; in particular, forν-a.e.x∈Ω it satisfies

Z

X

g(x, y)dm_x(y) = 0.

We notice that fory∈E^x₊we have g(x, y) = 1, for y∈E₋^x we haveg(x, y) =−1, so 0 =

Z

X

g(x, y)dmx(y) = Z

E₊^x

g(x, y)dmx(y) + Z

E₋^x

g(x, y)dmx(y)

+ Z

E₀^x

g(x, y)dmx(y) =mx(E^x₊)−mx(E₋^x) + Z

E₀^x

g(x, y)dmx(y).

Askgk_∞≤1, we reorder this equality and estimate mx(E₋^x) =mx(E^x₊) +

Z

E^x₀

g(x, y)dmx(y)≤mx(E₊^x) +mx(E₀^x)

and

m_x(E₊^x) =m_x(E₋^x)− Z

E₀^x

g(x, y)dm_x(y)≤m_x(E₋^x) +m_x(E₀^x).

We put these two estimates together and obtain (2.14), henceusatisfies the nonlocal mean value property.

In the other direction, it is sufficient to check thatg(x, y) defined as

g(x, y) =







1 ifuψ(y)> uψ(x) 0 ifuψ(y) =uψ(x)

−1 ifuψ(y)< uψ(x)