Notions of Dirichlet problem for functions of least gradient in metric measure spaces

We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a $(1,1)$-Poincar\'e inequality. Since one of the two notions is not amenable to the direct method of the calculus of variations, we construct, based on an approach of [23, 29], solutions by considering the Dirichlet problem for $p$-harmonic functions, $p>1$, and letting $p\to 1$. Tools developed and used in this paper include the inner perimeter measure of a domain.

Existence, uniqueness, continuity, and stability of solutions to the Dirichlet problem for p-harmonic functions in metric measure space setting is now reasonably well understood when 1 < p < ∞.The corresponding problem for p = 1, that is, finding a BV function of least gradient in the given domain, with prescribed trace on the boundary, is not well understood.Part of the problem is that without additional curvature restrictions for the boundary of the given domain, solutions to the Dirichlet problem, where the trace of the BV function is prescribed, are known to not always exist.Thus alternate notions of Dirichlet problem for the least gradient functions need to be explored.Based on the notion of Dirichlet problem set forth in [16], in [17] a notion of Dirichlet problem was proposed ( [17] considers the area functional, but the results are easily applicable to the total variation functional).It was shown in [17] that for a wide class of domains in metric measure spaces equipped with a doubling measure supporting a (1, 1)-Poincaré inequality, solutions always exist if the boundary data are themselves given by a BV function.The notion proposed there required extension of the BV solution to the exterior of the domain of the problem.
In this paper we discuss an alternate notion of the Dirichlet problem for least gradient functions that does not require extension of the BV solution to the complement of the domain of interest.The boundary data is given by a fixed Lipschitz function.However, unlike in [17], the direct method of the calculus of variations does not yield existence of solutions for this notion of the Dirichlet problem.Thus an alternate method of verifying existence needs to be adopted.In [23,Theorem 3.1] it was shown, using the tools of viscosity solutions, that the limit of a sequence of p-harmonic functions in a Euclidean domain, as p → 1, must be a function of least gradient.In the recent paper [29] it was shown that such a limit function, again in the Euclidean setting, satisfies the notion of Dirichlet problem considered in this paper.The key tool used in [29] is the divergence theorem.In our setting of metric measure spaces we do not have access to the divergence theorem nor notions of viscosity solutions.We instead employ a careful study of inner trace of BV functions for a class of domains.
We start by showing that if there is a sequence u p k of p k -harmonic functions with (p k ) k a monotone decreasing sequence of real numbers larger than 1 such that lim k p k = 1, and u p k converges to u in L 1 , then the limit function u is a function of least gradient, see Theorem 3.3.In the case of p-energy with p > 1, there is no ambiguity in the sense in which we want to fix the boundary values of the function, if the boundary values are themselves restrictions of Sobolev functions.Note that Lipschitz functions are a priori in the Sobolev class N 1,p for each 1 ≤ p < ∞.However, when p = 1 and the solutions are merely functions of bounded variation, it is not clear what notion of the Dirichlet problem is the correct one.
In this paper, we propose two ways of defining solutions to the Dirichlet problem: the first one, described in Definition 4.1(B), is based on minimizing the BV-energy in the closure of the domain.In the second one, given in Definition 4.1(T), extension of solutions to the complement of the domain is not required, but the energy being minimized includes the integral of the jump in the inner trace of the BV function (in comparison with the boundary data) measured with respect to the interior perimeter of the domain.
The drawback of the first approach is that the structure of the underlying space close to the boundary but outside the domain also affects the minimization problem.This phenomenon occurs already in weighted Euclidean spaces; see the discussion following Definition 4.1.On the other hand, the advantage of the first approach is that the energy being minimized is lower semicontinuous with respect to L 1convergence, and hence existence of solutions can be proven using the direct method of the calculus of variations.In the Euclidean setting, Dirichlet problems related to minimizing convex functionals with linear growth have been studied in [7], and the notion of Dirichlet problem considered there is also equivalent to the notion given by Definition 4.1(B) here.The second approach given in Definition 4.1(T) avoids the impact of the part of the complement of the domain that is near the boundary of the domain, but the drawback is that proving the existence of solutions using the direct method of the calculus of variations is not possible.In the setting of metric measure spaces considered here, we do not even have the tools of divergence or Green's theorem, and hence our proof is more involved.
One benefit of the proof we provide here is that the results hold even in a wider class of Euclidean domains; the standard theory from [29] only consider smooth domains, while [7] considers Euclidean Lipschitz domains.
The structure of this paper is as follows.In Section 2 we explain the notation and definitions of concepts used in this paper.In Section 3 we show that functions that arise as L 1 -limits of p-harmonic functions are functions of least gradient, see Theorem 3.3.The focus of the fourth section is to describe the two notions of solution to the Dirichlet problem, see Definition 4.1, while the fifth section gives a way of finding good Lipschitz approximations of BV functions via discrete convolutions.Such discrete convolutions are used in Section 6 to compare the inner perimeter measure P + (Ω, •) of the bounded domain Ω with its perimeter measure P (Ω, •), see Theorem 6.9.
In Section 7, we show that the least gradient functions, obtained as L 1 -limits of p-harmonic functions that are solutions to the Dirichlet problem with the fixed Lipschitz boundary data, are necessarily solutions to the Dirichlet problem defined in Definition 4.1(T) with the same Lipschitz boundary data.This result is Theorem 7.7.For this result, we need some additional assumptions on Ω.More precisely, we need to assume that Ω is of finite perimeter and that at H-a.e. boundary point of Ω the complement of Ω has positive density.
The focus of Section 8 is to show that in addition to perturbing the BV energy to the L p -energy (via p-harmonic functions), if we also perturb the domain by approximating the domain from outside, then the corresponding p-harmonic solutions have a subsequence that converges to a solution to the Dirichlet problem as given in Definition 4.1(B).While the problem (T) is associated with approximating the domain from inside, the results of Section 8 show that the problem (B) is associated with approximating the domain from outside; see Theorem 8.3.It should be noted that the restrictions placed on the domain in relation to problem (T) as in Section 7 are not needed in Section 8. Finally, in Section 9 we consider alternate notions of functions of least gradient, and show that all these notions coincide.For the convenience of the reader, in the appendix we provide a proof of the fact that the inner perimeter measure P + (Ω, •) as considered in Definition 2.23 is indeed a Radon measure.
Acknowledgements.The research of N.S. is partially supported by the grant # DMS-1500440 of NSF (U.S.A.).P.L. was supported by a grant from the Finnish Cultural Foundation.Part of the research was conducted during the visit of N.S. to Aalto University, and during the visit of X.L. to University of Cincinnati.Some parts of the research was conducted during the time spent by X.L. as a postdoctoral scholar at Aalto University.The authors wish to thank these institutions for their kind hospitality.The authors also thank Juha Kinnunen for making them aware of the reference [29] and for fruitful discussions on the topic.

Preliminaries
Throughout this paper we assume that (X, d, µ) is a complete metric space equipped with a Borel regular outer measure µ that satisfies a doubling property and supports a (1, 1)-Poincaré inequality (see definitions below).We assume that X consists of at least 2 points.The doubling property means that there exists a constant for every ball B(x, r) ⊂ X.Given a ball B = B(x, r) and τ > 0, we denote by τ B the ball B(x, τ r).In a metric space, a ball does not necessarily have a unique center and radius, but whenever we use the above abbreviation we will consider balls whose center and radii have been pre-specified.
In general, C ≥ 1 will denote a generic constant whose particular value is not important for the purposes of this paper, and might differ between each occurrence.When we want to specify that a constant C depends on the parameters a, b, . . ., we write C = C(a, b, . ..).Unless otherwise specified, all constants only depend on the doubling constant C d and the constants C P , λ associated with the Poincaré inequality defined below.
A complete metric space with a doubling measure is proper, that is, closed and bounded sets are compact.Since X is proper, for any open set Ω ⊂ X we define Lip loc (Ω) to be the space of functions that are Lipschitz in every Ω ′ ⋐ Ω.Here Ω ′ ⋐ Ω means that Ω ′ is open and that Ω ′ is a compact subset of Ω.We define other local spaces similarly.
For any set A ⊂ X, and 0 < R < ∞, the restricted spherical Hausdorff content of codimension 1 is defined by The codimension 1 Hausdorff measure of a set A ⊂ X is The codimension 1 Minkowski content of a set A ⊂ X is defined for any positive Radon measure ν by Definition 2.2.The measure theoretic boundary ∂ * E of a set E ⊂ X is the set of all points x ∈ X at which both E and its complement have positive upper density, i.e. lim sup The measure theoretic interior I E is the set of all points x ∈ X for which lim and the measure theoretic exterior O E is the set of all points x ∈ X for which lim See the discussion following (2.13) for more on the relationship between the measure theoretic boundary and the perimeter measure.
A curve is a rectifiable continuous mapping from a compact interval into X.Definition 2.3.A nonnegative Borel function g on X is an upper gradient of an extended real-valued function u on X if for all curves γ on X, we have where x and y are the end points of γ.We interpret |u(x) − u(y)| = ∞ whenever at least one of |u(x)|, |u(y)| is infinite.
By replacing X with a set A ⊂ X and considering curves γ in A, we can talk about a function g being an upper gradient of u in A. Upper gradients were originally introduced in [21].
We define the local Lipschitz constant of a locally Lipschitz function u ∈ Lip loc (X) by Lip u(x) := lim sup Then Lip u is an upper gradient of u, see e.g.[12,Proposition 1.11].
It is easy to check that if u, v ∈ Lip loc (X) and α, β ≥ 0, then we have the subadditivity Let Γ be a family of curves, and let 1 ≤ p < ∞.The p-modulus of Γ is defined by where the infimum is taken over all nonnegative Borel functions ρ such that γ ρ ds ≥ 1 for every γ ∈ Γ.If a property fails only for a curve family with p-modulus zero, we say that it holds for p-almost every (a.e.) curve.Definition 2.7.If g is a nonnegative µ-measurable function on X and (2.4) holds for p-almost every curve, then g is a p-weak upper gradient of u.It is known that if u has an upper gradient g ∈ L p loc (Ω) in Ω, then there exists a minimal p-weak upper gradient of u in Ω, which we always denote by g u , satisfying g u (x) ≤ g(x) for µ-a.e.x ∈ Ω, for any p-weak upper gradient g ∈ L p loc (Ω) of u in Ω, see [8,Theorem 2.25].Remark 2.8.Note that a priori the minimal p-weak upper gradient g u of u may depend on p.However, if u has a minimal q-weak upper gradient g 0 in Ω with 1 ≤ q < p, then g 0 ≤ g u µ-a.e. in Ω because a p-weak upper gradient of u is automatically a q-weak upper gradient of u.Also, a minimal p-weak upper gradient in Ω is also a minimal p-weak upper gradient in any open U ⊂ Ω.
From the results in [12] (see [22] for further exposition on this) it follows that when the measure µ on X is doubling and supports a (1, 1)-Poincaré inequality, the minimal p-weak upper gradient of a locally Lipschitz function u on Ω is Lip u for all 1 < p < ∞.
We consider the following norm with the infimum taken over all upper gradients g of u.
Definition 2.9.The substitute for the Sobolev space W 1,p (R n ) in the metric setting is the following Newton-Sobolev space N 1,p (X) := {u : u N 1,p (X) < ∞}/∼, where the equivalence relation ∼ is given by u ∼ v if and only if Similarly, we can define N 1,p (Ω) for any open set Ω ⊂ X.For more on Newton-Sobolev spaces, we refer to [34,22,8].
The p-capacity of a set A ⊂ X is given by where the infimum is taken over all functions u ∈ N 1,p (X) such that u ≥ 1 in A.
Remark 2.10.When µ is doubling and supports a (1, p)-Poincaré inequality, then Lipschitz functions are dense in N 1,p (X).When X is complete and µ is doubling, even if X does not support a (1, p)-Poincaré inequality Lipschitz functions are still dense in N 1,p (X); this follows from the deep results in [5].
Next we recall the definition and basic properties of functions of bounded variation on metric spaces, see [30].See also e.g.[4,15,16,35] for the classical theory in the Euclidean setting.For u ∈ L 1 loc (X), we define the total variation of u on X to be where each g u i is the minimal 1-weak upper gradient of u i .Note that instead of merely requiring u i → u in L 1 loc (X) we could require u i −u → 0 in L 1 (X).It turns out that even with this stricter definition, the norm Du (X) does not change; see Lemma 5.5.Note also that by [2, Theorem 1.1] and Remark 2.8, we can replace g u i by the minimal pweak upper gradient Lip u i , for p > 1.
We say that a function u ∈ L 1 (X) is of bounded variation, and denote . By replacing X with an open set U ⊂ X in the definition of the total variation, we can define Du (U).The BV norm is given by It was shown in [30,Theorem 3.4] that for u ∈ BV(X), Du is the restriction to the class of open sets of a finite Radon measure defined on the class of all subsets of X.This outer measure is obtained from the map U → Du (U) on open sets U ⊂ X via the standard Carathéodory construction.Thus, for an arbitrary set A ⊂ X, For any Borel sets E 1 , E 2 ⊂ X, we have by [30,Proposition 4.7] The proof works equally well for µ-measurable E 1 , E 2 ⊂ X and with X replaced by any open set, and then by approximating an arbitrary set A ⊂ X from the outside by open sets we obtain We have the following coarea formula from [30, Proposition 4.2]: if F ⊂ X is a Borel set and u ∈ BV(X), then (2.12) In particular, the map t → P ({u > t}, F ) is Lebesgue measurable on R.
We assume that X supports a (1, 1)-Poincaré inequality, meaning that there are constants C P > 0 and λ ≥ 1 such that for every ball B(x, r), for every locally integrable function u on X, and for every upper gradient g of u, we have Given a set E ⊂ X of finite perimeter, for H-a.e. x ∈ ∂ * E we have where γ ∈ (0, 1/2] only depends on the doubling constant and the constants in the Poincaré inequality, see [1,Theorem 5.4].We denote the set of all such points by Σ γ E. For any open set Ω ⊂ X, any µ-measurable set E ⊂ X with P (E, Ω) < ∞, and any Borel set A ⊂ Ω, we know that where [1,Theorem 5.3] and [6,Theorem 4.6].
The jump set of u ∈ BV(X) is the set where u ∧ (x) and u ∨ (x) are the lower and upper approximate limits of u defined respectively by and By [6,Theorem 5.3], the variation measure of a BV function can be decomposed into the absolutely continuous and singular part, and the latter into the Cantor and jump part, as follows.Given an open set Ω ⊂ X and u ∈ BV(Ω), we have for any Borel set A ⊂ Ω θ {u>t} (x) dt dH(x), (2.17) where a ∈ L 1 (Ω) is the density of the absolutely continuous part and the functions θ {u>t} are as in (2.14).Definition 2.18.Let Ω ⊂ X be a µ-measurable set and let u be a µ-measurable function on Ω.Let N Ω be the collection of all points x ∈ ∂Ω for which there is some r > 0 with µ(B(x, r) ∩ Ω) = 0.A function T + u : ∂Ω \ N Ω → R is the interior trace of u if for H-a.e.
x ∈ ∂Ω we have lim 19.Given an open set U ⊂ X, the family BV c (U) is the collection of all functions u ∈ BV(X) whose support is a compact subset of U. By BV 0 (U) we mean the collection of all functions u ∈ BV(U) for which T + u exists and T + u = 0 H-a.e. in ∂U.
where each g Ψ i is the minimal 1-weak upper gradient of Ψ i in U, and where the infimum is taken over all sequences (Ψ i ) ⊂ Lip loc (U) such that Ψ i − χ Ω → 0 in L 1 (U) and Ψ i = 0 in U \ Ω for each i ∈ N. Furthermore, for any A ⊂ X we let In the Appendix we show that if P + (Ω, X) < ∞, then P + (Ω, •) is a Radon measure on X, which we call the inner perimeter measure of Ω.
Note that P (Ω, A) ≤ P + (Ω, A) for any A ⊂ X.We will show in Section 6 that the two quantities P (Ω, X) and P + (Ω, X) are in fact comparable when Ω is open and bounded and satisfies the exterior measure density condition lim sup Given f ∈ N 1,p (X), we say that a function u is a p-harmonic solution to the Dirichlet problem in Ω with boundary data The direct method of the calculus of variation yields existence of p-harmonic solutions to the Dirichlet problem (p > 1); see [33,8] for this fact and for more on p-harmonic functions.If f : ∂Ω → R is a Lipschitz function and Ω is bounded, we can extend f to a boundedly supported Lipschitz function on X; such a function is necessarily in N 1,p (X) for all p ≥ 1.Thus we can also talk about solutions to the Dirichlet problem with Lipschitz boundary data f : ∂Ω → R. In this paper we will always assume that the boundary data is a boundedly supported Lipschitz function on X.
We will often assume that Cap 1 (X \ Ω) > 0, because then Cap p (X \ Ω) > 0 for all p > 1.This follows from the fact that if Cap p (X \Ω) = 0, then χ X\Ω N 1,p (X) = 0 by [8, Proposition 1.61], and so χ X\Ω N 1,1 (X) = 0 by Remark 2.8.Definition 2.23.Let Ω ⊂ X be a an open set.We say that a function The principal objects of study in this paper are functions of least gradient as defined above.

Convergence to a function of least gradient
In this section we show that if there is an L 1 -convergent sequence (u p ) of p-harmonic functions with p → 1 + , then the limit is a function of least gradient.In this section, g up always denotes the minimal p-weak upper gradient of u p ∈ N 1,p (X) on X.If g p is the minimal 1-weak upper gradient of u p on X, then for any open set U ⊂ X, by the fact that locally Lipschitz functions are dense in N 1,1 (U) (see [8,Theorem 5.47]) and by Remark 2.8, we have For a Lipschitz function f , g f will denote the minimal p-weak upper gradient of f for any p > 1. Observe from Remark 2.8 that g f is indeed independent of the choice of p.
First we note that while we do not know whether a sequence of p-harmonic functions is L 1 -convergent as p → 1 + , a convergent subsequence always exists.
Proof.By the maximum principle for the Dirichlet problem for pharmonic functions, u p L ∞ (X) ≤ f L ∞ (X) , and so for all p > 1 Let L be the global Lipschitz constant of f .Then On the other hand, see [8,Lemma 2.19].Thus by (3.1), We conclude that the sequence (u p ) p is a bounded sequence in BV(X), and so by the compact embedding given in [30,Theorem 3.7], a subsequence converges in L 1 loc (X) and hence in L 1 (X) to some function u ∈ L 1 (X), and by the lower semicontinuity of the total variation, we have u ∈ BV(X).
Theorem 3.3.Let Ω ⊂ X be a nonempty bounded open set with Cap 1 (X \ Ω) > 0, and let f ∈ Lip(X) be boundedly supported.For each p > 1 let u p ∈ N 1,p (X) be a p-harmonic function in Ω such that u p | X\Ω = f .Suppose that (u p ) p>1 is a sequence of such p-harmonic functions and that Proof.By the proof of Lemma 3.2, we have u ∈ BV(X).Let ψ ∈ BV c (Ω) and K := spt(ψ).Clearly where L is the global Lipschitz constant of f , and therefore (g p up ) 1<p<2 is uniformly bounded in L 1 (Ω).Consequently, there exists a subsequence, still written as (g p up ) p>1 , and a positive Radon measure of finite mass ν on Ω such that g p up dµ → dν weakly* in Ω as p → 1 + .
We now choose K ⋐ Ω such that K ⊂ K and ν(∂ K) = 0.For small enough ε > 0, where g Ψ k is the minimal p-weak upper gradient of Ψ k in K ε , for p > 1, see the discussion on page 7. We set Since u p is p-harmonic, we have Therefore, by lower semicontinuity and (3.1) we get which in turn leads to Letting k → ∞, we get by (3.4) Then letting ε → 0, by the fact that ν(∂ K) = 0 we get The claim follows from this.

Definitions of the Dirichlet problem for p = 1
The focus of this paper is to show that the limit of p-harmonic functions with Lipschitz boundary data f , as p → 1 + , solves a reasonable notion of a Dirichlet problem with boundary data f .The issue is to give such a notion.In the case of the p-energy, there is no ambiguity in the sense in which we want to fix the boundary values of the function, if the boundary values are themselves restrictions of Newton-Sobolev functions.In the case p = 1, we propose the following two ways of defining solutions to the Dirichlet problem.Definition 4.1.Let Ω ⊂ X be a nonempty bounded open set with Cap 1 (X \ Ω) > 0, and let f ∈ Lip(X) be boundedly supported.We say that a function u is a solution to the Dirichlet problem for functions of least gradient with boundary data f in the sense of (B) (respectively in the sense of (T)) if it is a solution to the following minimization problem: Note that in definition (T), we need to make extra assumptions on Ω to ensure that the boundary integral is well defined.In both definitions, the solution is allowed to have jumps on the boundary of Ω.In definition (B), this is taken into account by including the variation measure from the boundary ∂Ω as well.The advantage of this approach is that its energy is more straightforward to calculate, and we need fewer assumptions on Ω.The drawback is that contrary to the formulation (T), the structure of the underlying space X close to the boundary but outside Ω also affects the minimization problem.For instance, let X be the Euclidean space R n equipped with the Euclidean metric, and let Ω be the unit ball centered at the origin.Let α ∈ (0, 1] and equip X with the measure where L n is the n-dimensional Lebesgue measure.It can be shown that for u ∈ BV(X), Du (Ω) = D Euc u (Ω) + α D Euc u (∂Ω), where D Euc u is the total variation with respect to L n .Similarly, in this setting we have P + (Ω, X) = 2π but P (Ω, X) = 2απ.

Discrete convolutions
A tool that is commonly used in analysis on metric spaces is the discrete convolution.Given any open set U ⊂ X and a scale R > 0, we can choose a Whitney-type covering {B j = B(x j , r j )} ∞ j=1 of U such that (see e.g.[9, Theorem 3.1]) (1) for each j ∈ N, for each k ∈ N, the ball 10λB k intersects at most C 0 = C 0 (C d , λ) balls 10λB j (that is, a bounded overlap property holds), (3) if 10λB j intersects 10λB k , then r j ≤ 2r k .Given such a covering of U, we can take a partition of unity {φ j } ∞ j=1 subordinate to the covering, such that 0 ≤ φ j ≤ 1, each φ j is a C/r j -Lipschitz function, and supp(φ j ) ⊂ 2B j for each j ∈ N (see e.g.[9,Theorem 3.4]).Finally, we can define the discrete convolution v of any u ∈ L 1 loc (U) with respect to the Whitney-type covering by In general, v ∈ Lip loc (U), and hence v ∈ L 1 loc (U).Let v be the discrete convolution of u ∈ L 1 loc (U) with Du (U) < ∞, with respect to a Whitney-type covering Moreover (noting that v depends on the scale R), see the proof of [26, Proposition 4.1]; note that u does not need to be in L 1 (U), only in L 1 loc (U).Now let (v i ) be a sequence of discrete convolutions of u ∈ BV loc (U) with respect to Whitney-type coverings at scales R i ց 0. According to [26,Proposition 4.1], we have for some constant γ ∈ (0, 1/2] for H-a.e. y ∈ U; recall the definitions of the lower and upper approximate limits from (2.15) and (2.16).By applying discrete convolutions, we can show that in the definition of the total variation, we can replace convergence in L 1 loc (Ω) with convergence in L 1 (Ω).Lemma 5.5.Let Ω ⊂ X be an open set and let u ∈ L 1 loc (Ω) with Du (Ω) < ∞.Then there exists a sequence of functions (w i ) ⊂ Lip loc (Ω) with w i − u → 0 in L 1 (Ω) and Ω g w i dµ → Du (Ω), where each g w i is the minimal 1-weak upper gradient of w i .

Note that we cannot write w
Proof.For every δ > 0, let Fix ε > 0 and x ∈ X, and choose δ ∈ (0, 1) such that which is a 4/δ-Lipschitz function.
Let each v i ∈ Lip loc (Ω) be a discrete convolution of u in Ω, at scale 1/i.From the definition of the total variation we get a sequence of functions and (5.3), and by the Leibniz rule of [8, Lemma 2.18],

Now define
Here g w i , g u i , g v i , and g η all denote minimal 1-weak upper gradients.Since By a diagonalization argument, where we also let ε → 0 (and hence δ → 0), we complete the proof.

Comparability of P + and P
Recall the definition of P + (Ω, •) from Definition 2.20.As shown by the example found in the discussion following Definition 4.1, P + (Ω, •) does not necessarily agree with P (Ω, •).In light of this, the current section aims to compare P + (Ω, •) and P (Ω, •).The main result of this section is Theorem 6.9.
An analog of P + (Ω, X) was studied in [32], where it was shown that for certain open sets Ω ⊂ R n , one has , where H n−1 is the (n − 1)-dimensional Hausdorff measure.We obtain in Corollary 6.11 a weak analog of this result.In fact, our corollary is applicable to a wider class of Euclidean domains than the result of [32], since we can permit the part of the boundary in which Ω is "thin" to be very large.
In the following lemma, we essentially follow an argument that can be found e.g. in [31, p. 67].Lemma 6.1.Let K ⊂ X be compact, and let α ∈ [0, 1) and ε > 0. Take a sequence (v i ) ⊂ C(K) with 0 ≤ v i ≤ 1 for every i ∈ N, and for every x ∈ K. Then there exists a convex combination of v i , denoted by v, such that v(x) ≤ α + ε for every x ∈ K.
Proof.We have lim i→∞ max{v i (x), α} = α for every x ∈ K.Note that the functions max{v i (x), α}, and the constant function α, are continuous and take values between 0 and 1.Thus for any signed Radon measure ν on K we have by Lebesgue's dominated convergence theorem that Since K is compact, we have C(K) = C c (K) and then by the Riesz representation theorem we conclude that max{v i (x), α} → α weakly in the space C(K).By Mazur's lemma, see [31,Theorem 3.13], we can find convex combinations of the functions w i := max{v i , α}, denoted by w i , which converge strongly in the space C(K) to α.In other words, w i → α uniformly in K. Thus for a sufficiently large choice of i ∈ N, we have w i (x) ≤ α + ε for all x ∈ K.With w i = N j=1 λ i,j w j for some N ∈ N and the appropriate choice of numbers λ i,j ∈ [0, 1] such that Proposition 6.2.Let Ω, U ⊂ X be open sets with P (Ω, U) < ∞, and suppose that there exists Then there exists a sequence and U is the unit disk, we have P (Ω, U) = 0 and the set A can be taken to be the slit.If we add countably many slits, see Example 6.14 below, we still have P (Ω, U) = 0 but H(A) = ∞ and the conclusion of the proposition becomes meaningless.
Proof of Proposition 6.2.Let each v i , i ∈ N, be the discrete convolution of χ Ω with respect to a Whitney-type covering of U at scale 1/i.We can add to the set A the H-negligible set where (5.4) fails with u = χ Ω .Fix ε > 0. We can pick balls B(x j , s j ) intersecting A with and j∈N µ(B(x j , s j )) Furthermore, we can choose radii r j ∈ [s j , 2s j ] such that for each j ∈ N, see [24, Lemma 6.2].
For brevity, let us write B j := B(x j , r j ), j ∈ N. Then by the subadditivity (2.11) and the lower semicontinuity of perimeter, we have Let each vi , i ∈ N, be the discrete convolution of χ Ω\ j∈N B j with respect to the same Whitney-type covering of U at scale 1/i used also in defining the functions v i .By the properties (5.2) and (5.3) of discrete convolutions, we have vi for each i ∈ N. Note that vi (x) ≤ v i (x) for every x ∈ U. Thus for every x ∈ ∂Ω ∩ U \ A we have by (5.4) note that χ ∧ Ω (x) = 0 for every x ∈ ∂Ω ∩ U \ A by (6.3).Moreover, lim i→∞ vi (x) = 0 for every x ∈ A, since χ Ω\ j∈N B j = 0 in a neighborhood of every x ∈ A. Note that U ′ \ Ω is a compact set.Using Lemma 6.1, we find for every i ∈ N a convex combination of the func- Clearly we still have and by (6.6) and the subadditivity (2.6), Next, let Then by (6.7), wi = 0 in U ′ \ Ω.Again, we still have wi ∈ Lip loc (U) with wi − χ Ω\ j∈N B j → 0 in L 1 (U), and by (6.5) and (6.8), We can do the above for each ε = 1/k, k ∈ N. Denote Ω k := Ω \ j∈N B j , with the balls B j picked corresponding to the choice ε = 1/k.Thus we obtain sequences wk,i with wk,i − χ Ω k → 0 in L 1 (U) as i → ∞.Then for each k ∈ N we can pick a sufficiently large Note that we always have P (Ω, U) ≤ P + (Ω, U), since the definition of the latter involves a more restricted class of approximating functions.Now we can show the following.Theorem 6.9.Let Ω ⊂ X be a bounded open set with P (Ω, X) < ∞, and suppose that lim sup for H-a.e. x ∈ ∂Ω.Then P + (Ω, X) < ∞, and for any open set U ⊂ X, we have P (Ω, U) ≤ P + (Ω, U) ≤ CP (Ω, U).
Proof.Take a bounded open set U ′ ⊂ X with Ω ⋐ U ′ .Let w k ∈ Lip loc (X) be the sequence given by Proposition 6.2 with the choice U = X (note that now H(A) = 0).Then w k = 0 in U ′ \ Ω, and in fact from the proof of Proposition 6.2 it is easy to see that w k = 0 in X \ Ω.
Then by the definition of P + (Ω, •) and the fact that the local Lipschitz constant is an upper gradient, For an open set U ⊂ X, the first inequality of the second claim is clear.To prove the second inequality, fix ε > 0. For some U ′ ⋐ U we have be the sequence given by Proposition 6.2.Then By letting ε → 0, we obtain the result.
To conclude this section, we prove two corollaries of Proposition 6.2 that will not be needed in the sequel, but may be of independent interest.First we need a lemma.Lemma 6.10.For any w ∈ Lip c (X), where C co only depends on the doubling constant of the measure.
Proof.By [27, Proposition 3.5] (which is based on [11]) the following coarea inequality holds: for any w ∈ Lip c (X), Since H(A) ≤ C 3 d µ + (A) for any A ⊂ X (see e.g.[27, Proposition 3.12]), we obtain the result.Corollary 6.11.Let Ω ⊂ X be a bounded open set with P (Ω, X) < ∞, and suppose that there exists A ⊂ ∂Ω with H(A) < ∞ such that lim sup for every x ∈ ∂Ω \ A. Then there exists a sequence of open sets Ω j ⋐ Ω with χ Ω j (x) → 1 for every x ∈ Ω and for each j ∈ N.

Proof.
Choose an open set U ′ with Ω ⋐ U ′ ⋐ X. Apply Proposition 6.2 with U = X to obtain a sequence w k ∈ Lip loc (X) with and w k = 0 in U ′ \ Ω.From the proof of Proposition 6.2 it is easy to see that in fact w k = 0 in X \ Ω, so that w k ∈ Lip c (X) for each k ∈ N.
From the proof of Proposition 6.2 we can also see that w k (x) → 1 for every x ∈ Ω, so that for any t ∈ (0, 1), χ {w k >t} (x) → 1 for every x ∈ Ω.By Lemma 6.10, for all k ∈ N. Thus for any fixed k ∈ N we find a set T k ⊂ (0, 1) with Now, if for every t ∈ (0, 1) there were an index N t ∈ N such that t / ∈ T k for all k ≥ N t , then by the Lebesgue dominated convergence theorem we would have which is a contradiction.Thus there exists t ∈ (0, 1) such that for some subsequence k j , we have t ∈ T k j for all j ∈ N.
Thus we can define Ω j := {w k j > t}.
We know the following fact about the extension of sets of finite perimeter: if Ω ⊂ X is an open set with H(∂Ω) < ∞ and E ⊂ Ω is a µ-measurable set with P (E, Ω) < ∞, then P (E, X) < ∞ and in fact P (E, X) ≤ P (E, Ω) + CH(∂Ω), (6.12) see [24,Proposition 6.3].Now we can show a partially more general result.Corollary 6.13.Let Ω ⊂ X be a bounded open set with P (Ω, X) < ∞, and suppose that there exists A ⊂ ∂Ω with H(A) < ∞ such that lim sup for every x ∈ ∂Ω \ A. Let E ⊂ Ω be a µ-measurable set with P (E, Ω) < ∞.Then P (E, X) < ∞.
Proof.Take the sequence of sets Ω j ⋐ Ω given by Corollary 6.11.By Lebesgue's dominated convergence theorem, we have µ(Ω \ Ω j ) → 0, and so by the lower semicontinuity of perimeter and (6.12), we have Example 6.14.Without the requirement of the measure density condition for X \ Ω given in the hypothesis of the above corollary, the conclusion of the corollary fails.For example, with D ⊂ R 2 the unit disk in X = R 2 = C centered at 0, set θ n = n j=1 π 2 j and let Ω := D \ {z ∈ C : Arg(z) = θ n for some n ∈ N}.
we see that P (E, Ω) = 0, but as H(∂ * E) = ∞ (note that H is now comparable to the one-dimensional Hausdorff measure), it follows that P (E, X) = P (E, R 2 ) = ∞.

Dirichlet problem (T): trace definition
In this section we consider the Dirichlet problem (T) given in Definition 4.1.We show that the limit of p-harmonic functions with boundary data f is a solution to this problem.
In the Euclidean setting, it is known that if a bounded domain Ω has a Lipschitz boundary, the trace operator T + : BV(Ω) → L 1 (∂Ω, H) is continuous under strict convergence, see e.g.[4,Theorem 3.88].In the following proposition we give a generalization of this fact to the metric setting.
Proof.For t > 0, let Note that by lower semicontinuity of the total variation with respect to L 1 -convergence, Since also Du k (Ω) → Du (Ω) by assumption, necessarily We have Since T + is assumed to be linear and bounded, for some constant C Ω > 0 and for any v ∈ BV(Ω) we have By letting ε → 0, we obtain the result.
Lemma 7.3.Let Ω ⊂ X be a bounded open set such that Ω satisfies the exterior measure density condition (2.21), Ω supports a (1, 1)-Poincaré inequality, and there is a constant C ≥ 1 such that whenever x ∈ ∂Ω and 0 < r < diam(Ω), we have Assume also that for all x ∈ ∂Ω and 0 < r < diam(Ω), Let f ∈ Lip(X) be boundedly supported, and let u ∈ BV(Ω).Then there exists a sequence Remark 7.4.Note that some requirement similar to the exterior measure density condition in the above lemma is needed, for without such a requirement we cannot talk about the trace T + u of a function u ∈ BV(Ω).This difficulty is illustrated by the example of the slit disk, see [28,Example 3.2].
Proof.The assumptions on Ω guarantee that the trace operator T + : BV(Ω) → L 1 (∂Ω, H) is linear and bounded, see [28,Theorem 5.5].The assumptions also together imply that Clearly we have in fact η m ∈ Lip(X) for every m ∈ N. It is straightforward to check that then also g ηm dµ → dP + (Ω, •) weakly* in the sense of measures on X.Since Ω supports a (1, 1)-Poincaré inequality, Lipschitz functions are dense in N 1,1 (Ω), see [8,Theorem 5.1].It follows that there exists a sequence (φ k ) ⊂ Lip(Ω) such that φ k → u in L 1 (Ω) and By lower semicontinuity of the total variation with respect to L 1convergence, necessarily also Then ψ k,m ∈ Lip(X) and as m → ∞ and then k → ∞.Furthermore, ψ k,m = f on X \ Ω.By the Leibniz rule of [8,Lemma 2.18], Here g ψ k,m , g φ k , g f , and g ηm all denote minimal 1-weak upper gradients.It follows that As f is a Lipschitz function and η m → 1 in L 1 (Ω), we have Note that the Lipschitz functions φ k have Lipschitz extensions to X, which we still denote by φ k , and that necessarily T + φ k = φ k on ∂Ω.Since g ηm dµ → dP + (Ω, •) weakly* in the sense of measures, lim It follows from Lemma 7.1 that T + φ k → T + u in L 1 (∂Ω, H), and then by (2.14) and Theorem 6.9, also Thus, recalling also that and now we can choose a diagonal sequence {ψ k,m k } k to satisfy the conclusion of the lemma.
In what follows, we denote by T − u the outer trace (if it exists) of a BV function u ∈ BV(X), namely, T − u is the interior trace of u considered with respect to X \ Ω as given in Definition 2.18.We will only need the following proposition for the case where u = f on X \ Ω for some Lipschitz function f ; in this case, we always have T − u = f on ∂Ω \ N X\Ω , in particular, T − u = f on ∂ * Ω. Proposition 7.5.Let Ω ⊂ X be a µ-measurable set with P (Ω, X) < ∞ and let u ∈ BV(X) such that for H-almost every x ∈ ∂ * Ω, T + u(x) and T − u(x) exist.Then Proof.We only need to prove that By [6, Theorem 5.3], we have Du c (∂ * Ω) = 0, and then by the decomposition (2.17), It is fairly easy to check that {u ∧ (x), u ∨ (x)} = {T − u(x), T + u(x)}, whenever both traces exist.This is also proved in [18,Proposition 5.8(v)].Suppose that u ∧ (x) = T − u(x) and u ∨ (x) = T + u(x), the other case being analogous.In the proof of [18,Proposition 5.8(v)] it is also shown that lim for all t ∈ (u ∧ (x), u ∨ (x)).We also have x ∈ ∂ * {u > t} for all t ∈ (u ∧ (x), u ∨ (x)).According to [6, Proposition 6.2], we have θ {u>t} (x) = θ Ω (x) for H-almost every such x.Hence we have For µ-measurable Ω ⊂ X and any κ > 0, define the weighted measure Consider then the space (X, d, µ κ ).It is easy to show that this is still a complete metric space such that µ κ is doubling and supports a (1, 1)-Poincaré inequality.We use the subscript κ to signify that a perimeter or some other quantity is taken with respect to the measure µ κ .
Theorem 7.7.Let Ω ⊂ X be a nonempty bounded open set of finite perimeter such that Cap 1 (X \ Ω) > 0, Ω satisfies the exterior measure density condition (2.21), and Ω supports a (1, 1)-Poincaré inequality.Suppose also that there is a constant C ≥ 1 such that whenever x ∈ ∂Ω and 0 < r < diam(Ω), we have Finally, assume that for all x ∈ ∂Ω and 0 < r < diam(Ω), Let f ∈ Lip(X) be boundedly supported.For each p > 1 let u p be a p-harmonic function in Ω such that u p | X\Ω = f .Suppose that (u p ) p>1 is a sequence of such p-harmonic functions and that u p → u in L 1 (Ω) Observe that each ψ k can act as a test function for testing the pharmonicity of u p .Therefore by (3.1) Letting p → 1 + , we see that lim sup Therefore by now letting k → ∞, we have lim sup Thus we need to prove that lim sup in order to complete the proof.
Recall the definitions of O Ω and I Ω from Definition 2.2.By the exterior measure density condition (2.21), we know that H(∂Ω ∩ I Ω ) = 0. Recall the definition of (X, d, µ κ ) from (7.6).We note that D κ u is absolutely continuous with respect to H, which follows from the BV coarea formula (2.12) and (2.14).Thus D κ u (I Ω \ Ω) = 0. Note also that since u = f on X \ Ω, ∂ * {u − f > t} ∩ O Ω = ∅ for all t ∈ R, and so by the coarea formula and (2.14) Then by the lower semicontinuity of the total variation and Proposition 7.5, we have lim inf Similarly, on the left-hand side we have The inequality (7.10) will follow from the above inequality if we know that lim This is the focus of the rest of this section, and we will complete the proof at the end of the section.
We will need the following approximation of a set of finite perimeter by "regular" sets.This is inspired by a similar result in [3], but note that we use a somewhat different, "two-sided" definition of the Minkowski content, as given in (2.1).First recall that by [27, Proposition 3.5] (which is based on [11]) the following coarea inequality holds: for any w ∈ Lip c (X), where ν is any positive Radon measure.From this it follows in a straightforward manner that for any w ∈ Lip loc (X), Lemma 7.14.Let E ⊂ X be a set of finite perimeter.Fix 0 < δ < 1.
Then there exists a sequence of open sets of finite perimeter Proof.By Lemma 5.5, we can pick a sequence (v i ) ⊂ Lip loc (X) with v i − χ E → 0 in L 1 (X) and X g v i dµ → P (E, X), where each g v i is the minimal 1-weak upper gradient of v i .We may also choose the functions so that v i ≥ 0. Furthermore, Lip v i ≤ Cg v i µ-almost everywhere, see [12,Proposition 4.26] or [22,Proposition 13.5.2].According to the coarea formula for BV functions, see (2.12), for every i ∈ N we have Now by Chebyshev's inequality, note that this holds also if X g v i dµ = 0, as then P ({v i > t}, X) = 0 for a.e.t ∈ [0, 1].Therefore there is a measurable set for all t ∈ A i .Moreover, since the sets ∂{v i > t} ⊂ {v i = t} are disjoint for distinct values of t, we have µ(∂{v i > t}) = 0 for a.e.t ∈ [0, 1].By the version of the coarea formula found in (7.13), we have Thus for each i ∈ N, there exists t i ∈ A i with and Now we need to show that χ {v i >t i } − χ E → 0 in L 1 (X).Note that for any t ∈ [δ/4, 1 − δ/4], for any x such that we have and it follows that The reason for utilizing the Minkowski content is that it scales nicely according to the parameter κ in µ κ , in the following sense.Lemma 7.15.Let Ω ⊂ X be µ-measurable, let A ⊂ X \ Ω, let β > 0, and suppose that there is some R > 0 for which for every x ∈ A. Then we have Proof.For any x ∈ A and radii r ∈ (0, R), Fix 0 < r < R/5 and consider the collection of balls {B(x, r)} x∈A .By the 5-covering theorem we can pick a countable collection of disjoint balls B(x j , r) such that the balls B(x j , 5r) cover x∈A B(x, r).We have By taking the limit infimum as r → 0 on both sides, we obtain Moreover, we have the following simple estimate for the Minkowski content and Hausdorff measure that we will need in the proof of Proposition 7.19.The estimate can be proved by a simple covering argument, see [27,Proposition 3.12].Lemma 7.17.For any A ⊂ X, we have H(A) ≤ C 3 d µ + (A).It is less clear that this estimate would hold if we used a "one-sided" definition of the Minkowski content, as for example in [3].
Proof.By [20, Lemma 2.6], for each i ∈ N we can find a function The conclusion follows by the definition of P + (Ω, •).Proposition 7.19.Let Ω ⊂ X be a bounded open set with P (Ω, X) < ∞, and assume that for some constant β > 0, we have for H-a.e. x ∈ ∂Ω.Then we have Proof.By Theorem 6.9 we have P + (Ω, X) < ∞.Note that for any κ > 0 we have P + (Ω, X) ≥ P κ (Ω, X), so only the other inequality needs to be proved.Fix 0 < δ < 1, and fix κ > 0. By Lemma 7.14 we can find a sequence of open sets Ω i ⊂ X of finite perimeter such that and lim sup In the following, we will repeatedly use the measure property and the subadditivity property (2.11) of sets of finite perimeter.Since By the lower semicontinuity of perimeter and the fact that the perimeter of a set is concentrated on its measure theoretic boundary, we estimate It follows that P (Ω, I Ω i ) ≤ P (Ω i , X \ Ω) + ε i , (7.23) where ε i → 0 as i → ∞.For any sets A, B ⊂ X, we have Thus we have By (2.11), P (Ω i \ Ω, X) < ∞, and then by using (2.14), we obtain Combining this with (7.23), we obtain that for all i ∈ N Note that H| ∂Ω i is a Borel measure of finite mass, since by Lemma 7.17, Note that for any fixed r > 0, the map is a Borel map as the ratio of two lower semicontinuous functions.Hence for each τ > 0 the function f τ given by is also Borel measurable, and so is So for each i ∈ N, by Egorov's theorem we can choose a set A i ⊂ ∂Ω i with H(∂Ω i \ A i ) < ε i and such that f τ → f ∞ uniformly in A i .Thus (7.16) is satisfied for A = A i \Ω and some R > 0. By Lemma 7.17 and Lemma 7.15, we get Then by (2.14) so by combining with (7.24), we have Recall that µ(Ω i \ Ω i ) = µ(∂Ω i ) = 0 for all i ∈ N. Thus by (7.22), For A ⊂ X, we set Let us denote by D ⊂ ∂Ω the H-negligible set where (7.20) fails.Note that µ(X) > 0, and so we can assume that µ(Ω i \ Ω) < µ(X)/2 for all i ∈ N. Now by the boxing inequality, see [25,Remark 3.3(1)], we can find a collection of balls {B(x i j , r i j )} j∈N covering (Ω i \ Ω) β such that the balls B(x i j , r i j /5) are disjoint, and Note that in [25] it is assumed that µ(X) = ∞, but the condition µ(Ω i \ Ω) < µ(X)/2 is sufficient for the proof to work.Then by (7.26), But by (7.25), Thus we can pick another collection {B(y i k , s i k )} k∈N of balls covering and so by (7.22), lim sup Note that the collections {B(x i j , r i j )} j∈N and {B(y i k , s i k )} k∈N together cover all of Ω i \ Ω.By [24, Lemma 6.2] we can pick radii r i j ∈ [r i j , 2r i j ] such that and similarly we find radii Note that these are closed sets contained in Ω, and thus compact.By (7.28) and (7.29), we have lim sup i→∞ P (K i , X) ≤ lim sup i→∞ P (Ω i , X) + j∈N P (B(x i j , r i j ), X) where we used (7.21) in the last step (and absorbed a factor 2 into the constant C).By (7.27) we have since the balls B(x i j , r i j /5) are disjoint.Now by the fact that χ Ω i → χ Ω in L 1 (X) and (7.29), we obtain χ K i → χ Ω in L 1 (X).Thus by Lemma 7.18 Letting κ → ∞ and then δ → 0, we obtain the result.End of the proof of Theorem 7.7.Note that by (7.8), Ω satisfies the assumptions of Proposition 7.19 (and thus Corollary 7.30) with β = κ.By Cavalieri's principle, Corollary 7.30, and Lebesgue's monotone convergence theorem, we have This proves (7.12), thus completing the proof of Theorem 7.7.
Corollary 7.31.Let Ω ⊂ X satisfy the assumptions of Theorem 7.7, and let f ∈ Lip(X) be boundedly supported.Then the minimization problem (T ) of Definition 4.1 has a solution.
Proof.For every p > 1, there exists a p-harmonic function u p in Ω such that u p | X\Ω = f .Then the result follows by combining Lemma 3.2 and Theorem 7.7.

Dirichlet Problems (T), (B) and perturbation of the domain
From the definition of P + (Ω, •) it is clear that Problem (T) of Definition 4.1 is associated with approximating the bounded open set Ω from inside.Moreover, if for each k ∈ N we have Ω k ⋐ Ω such that Ω = k∈N Ω k , and v p k is the p k -harmonic solution to the Dirichlet problem on Ω k with boundary data f , then under reasonable hypotheses on Ω we have v p k → u with u a solution to Problem (T) in Ω with boundary data f .Indeed, suppose that for each p > 1 there are constants C p ≥ 1 and β p > 0 such that Ω satisfies the condition that whenever u p is a p-harmonic solution to the Dirichlet problem on Ω with boundary data f we have osc Ω∩B(x,ρ) for all x ∈ ∂Ω and 0 < ρ < r/2.Let p k be any sequence as obtained in Lemma 3.2.Take a sequence ε k → 0 as k → ∞, and let L > 0 such that f is L-Lipschitz continuous.For each k ∈ N we can fix 0 < r k < diam(Ω)/2 such that 4Lr k < ε k .Then by (8.1), whenever x ∈ ∂Ω and 0 < ρ < r k /2 and y ∈ B(x, ρ) ∩ Ω, We can then choose 0 < ρ k < r k /2 such that have by the comparison principle for p-harmonic functions (see [8]) as k → ∞, where we know from the previous section that u satisfies the Dirichlet problem (T) on Ω with boundary data f .Examples of domains where (8.1) hold include the domains whose complements are uniformly 1-fat, see [10]; in particular, domains whose complements are porous satisfy this requirement.
In contrast to problem (T), the Dirichlet problem (B) of Definition 4.1 is associated with approximation of Ω from outside, as we will see next.
Let Ω ⊂ X be a nonempty bounded open set with for all sufficiently large k, and so we may as well assume that this is true for all k ∈ N.For each k ∈ N, fix a decreasing sequence (p k,m ) m such that p k,m > 1 and lim m→∞ p k,m = 1.
Let f ∈ Lip(X) be boundedly supported, and let u k,m ∈ N 1,p k,m (X) be the p k,m -harmonic function solving the Dirichlet problem on Ω k with boundary data f .According to Lemma 3.2, by passing to a subsequence of (p k,m ) m (not relabeled), we find a function u k ∈ BV(X) As in Section 3, g u k,m always denotes the minimal p-weak upper gradient of u k,m , and for a Lipschitz function f , g f denotes the minimal p-weak upper gradient of f for any p > 1.By (3.1), By the lower semicontinuity of the total variation, We have |f | ≤ M on X for some M ≥ 0. By the comparison principle, we also have |u k,m | ≤ M, and then Then by the compact embedding given in [30,Theorem 3.7], by passing to a subsequence of k (not relabeled), we obtain u k → u in L 1 (X) as k → ∞, for u ∈ BV(X).By passing to a further subsequence (not relabeled), we can assume that Note that in this section we do not need Ω to satisfy the extra conditions imposed in Section 7.
Proof.Take a test function ψ ∈ BV(X) such that ψ = 0 on X \ Ω.We can choose a sequence (Ψ j ) ⊂ Lip loc (X) such that Ψ where g Ψ j is the minimal p-weak upper gradient of Ψ j in X, for any p > 1, see the discussion on page 7. Then also g Ψ j dµ → d D(u + ψ) weakly* in the sense of measures (see e.g.[18,Proposition 3.8]), so that for each k ∈ N, By the Leibniz rule of [8, Lemma 2.18], Note that this function agrees with f in X \ Ω k .As noted above, |u k | ≤ M for all k ∈ N, and so |u| ≤ M. As truncation decreases total variation, we can also assume that |u + ψ| ≤ M and that the approximating functions Ψ j also satisfy |Ψ j | ≤ M. Then we have (assuming p k,m < 2 and M ≥ 1) For each k ∈ N, by (8.4) we can choose j k ∈ N large enough so that and then it follows that for all m and It then follows by (8.6) that Combining (8.8) with (8.2), we have By lower semicontinuity of the total variation, Hölder's inequality, and the fact that ψ j k ,k can be used to test the p k,m -harmonicity of u k,m k , we get In the above, we used (8.5) to arrive at the fourth inequality, and (8.9), (8.7) in obtaining the penultimate inequality.

Alternate definitions of functions of least gradient
In this section we consider possible definitions of what it means for a function u ∈ BV(Ω) to be of least gradient in an open set Ω ⊂ X.This is not to be conflated with the notions of solutions to the Dirichlet problems studied in the previous sections, as such solutions must in addition satisfy a boundary condition.
Proof.We set U 1 = W δ 1 and U 2 = W \ W δ 2 .We can find sequences (u i ) ⊂ Lip loc (U 1 ) and (v i ) ⊂ Lip loc (U 2 ) such that u i vanishes in Applying Lemma A.3 with u = u i and v = v i , we obtain functions and Furthermore, by the construction of w i , we see that w i vanishes in W \Ω. From the first of the above two inequalities we see that w i − χ Ω → 0 in L 1 (W ), and hence by the second of the above two inequalities, g w i dµ ≤ P + (Ω, U 1 ) + P + (Ω, U 2 ) + lim sup Note that as i → ∞.The desired conclusion now follows.
Proof.Let (δ k ) be a strictly decreasing sequence of positive numbers such that lim k→∞ δ k = 0.For integers k ≥ 2, let V k := W δ 2k \ W δ 2k−3 .Note then that {V 2k } k∈N is a collection of pairwise disjoint open sets,

Definition 2 . 20 .
Given an open set Ω ⊂ X and an open set U ⊂ X, we define
Then u is a solution to the minimization problem (T ) of Definition 4.1.Note also again that the assumptions on Ω guarantee that the trace operator T + : BV(Ω) → L 1 (∂Ω, H) is linear and bounded, see[28,Theorem 5.5].Let v ∈ BV(Ω).By combining (2.14) and Theorem 6.9, we know that P + (Ω, •) is concentrated on ∂ * Ω.Thus we need to show that