An Analog of the Neumann Problem for the $1$-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability

We study an inhomogeneous Neumann boundary value problem for functions of least gradient on bounded domains in metric spaces that are equipped with a doubling measure and support a Poincar\'e inequality. We show that solutions exist under certain regularity assumptions on the domain, but are generally nonunique. We also show that solutions can be taken to be differences of two characteristic functions, and that they are regular up to the boundary when the boundary is of positive mean curvature. By regular up to the boundary we mean that if the boundary data is $1$ in a neighborhood of a point on the boundary of the domain, then the solution is $-1$ in the intersection of the domain with a possibly smaller neighborhood of that point. Finally, we consider the stability of solutions with respect to boundary data.


Introduction
The goal of the Neumann boundary value problem for ∆ p in a smooth Euclidean domain Ω ⊂ R n is to find a function u ∈ W 1,p (Ω) such that ∆ p u = −div(|∇u| p−2 ∇u) = 0 in Ω, and where ∂ η u is the derivative of u in the direction of outer normal to ∂Ω and f ∈ L ∞ (∂Ω, H n−1 ) such that ∂Ω f dH n−1 = 0. For p = 1 this problem is highly degenerate, see for example [35,36].
In the study of boundary value problems for PDEs, more attention has generally been given to Dirichlet problems than to Neumann problems. This is especially true in the general setting of a metric space equipped with a doubling measure that supports a Poincaré inequality. In this setting, nonlinear potential theory for Dirichlet problems when p > 1 is now well developed, see the monograph [4] as well as e.g. [5,8,9,41]. By contrast, Neumann problems have been studied very little. The paper [10] dealt mostly with homogeneous Neumann boundary value problem, while in the paper [32], a Neumann problem was formulated as the minimization of the functional where g u is an upper gradient of u and p > 1, see Section 2 for notation. In the Euclidean setting, with Ω a smooth domain, a variant of this boundary value problem was studied in [35], and a connection between the problem for p > 1 and the problem for p = 1 was established through a study of the behavior of solutions u p for p > 1 as p → 1 + . For functions f ∈ L ∞ (∂Ω), the following norm was associated in [35,36]: : w ∈ BV(Ω) with w = 0, ∂Ω w dP (Ω, ·) = 0 .
The problem of minimizing I p corresponding to p = 1 was studied in [36] and then in [35] for Euclidean domains with Lipschitz boundary, and f * ≤ 1. The paper [35] also gave an application of this problem to the study of electrical conductivity. We point out here that the condition f * ≤ 1 gives the minimal energy I 1 (u) = 0, and hence constant functions will certainly minimize the energy. Our focus in the present paper is to study the situation corresponding to f * > 1, in which case there are no minimizers for the energy I 1 if one seeks to minimize I 1 (u) within the class of all functions u ∈ BV(Ω), see the discussion in the proof of [35,Proposition 3.1]. Thus we are compelled to add further natural constraints on the competitor functions u, namely that −1 ≤ u ≤ 1. This constraint is not as restrictive as it might seem, and instead for any β > 0 we can also consider constraints of the form that all competitor functions satisfy −β ≤ u ≤ β. Then u β is a minimizer for the constraint that all competitors v should satisfy −β ≤ v ≤ β if and only if β −1 u β is a minimizer for the constraint that all competitors v satisfy −1 ≤ v ≤ 1. Thus the study undertaken here complements the results in [35,36] in the smooth Euclidean domains setting. For instance, suppose that ∂Ω is of positive mean curvature (either in the sense of Riemannian geometry in Euclidean setting, or in the sense of Definition 5.10 in the more general metric setting). For such a domain, whenever the boundary data f is not H-a.e. zero on ∂Ω and takes on only three values, −1, 0, 1, then f * > 1; this interesting setting, excluded in the studies in [35,36], is covered in Section 5 of the present paper. For an alternate (but equivalent) framing of the Neumann boundary value problem for p = 1, see [38]. The paper [38] also gives an application of the problem to the study of conductivity, see [38,Section 1.1]. The problem as framed in [38] is not tractable in the metric setting as it relies heavily on the theory of divergence free L ∞ -vector fields, a tool that is lacking in the non-smooth setting. In this paper, our goal is to study an analogous problem of minimizing I p in the metric setting when p = 1. In this case, instead of the p-energy it is natural to minimize the total variation among functions of bounded variation. See e.g. [34,36,40,43,45] for previous studies of the Dirichlet problem when p = 1 in the Euclidean setting, and [18,25,29] in the metric setting. In this paper, following the formulation given in [32], we consider minimization of the functional Our goal is to study the existence, uniqueness, regularity, and stability properties of solutions. In Section 3 we consider basic properties of solutions and note that they are generally nonunique. However, in Proposition 3.8 we show that if a solution exists, it can be taken to be of the form χ E 1 − χ E 2 for disjoint sets E 1 , E 2 ⊂ Ω. In most of the rest of the paper, we consider only such solutions. It is clear that these solutions cannot exhibit much interior regularity, but in Proposition 3.14 we show that χ E 1 and χ E 2 are functions of least gradient.
In Section 4 we show that under some regularity assumptions on Ω, and additionally that −1 ≤ f ≤ 1, the functional I(·) is lower semicontinuous with respect to convergence in L 1 (Ω), and we use this fact to establish the existence of solutions; this is Theorem 4.15. In Section 5 we study the boundary regularity of solutions when f only taken the values −1, 0, 1. In the Euclidean setting, f can be interpreted as the relative outer normal derivative of the solution, and so one would expect to have T χ E 1 = 1 where f = −1. This is not always the case, but in Theorem 5.13 we show that T χ E 1 = 1 in the interior points of {f = −1} when Ω has boundary of positive mean curvature.
While solutions are generally nonunique, in Theorem 6.5 we show that so-called minimal solutions are unique. Finally, in Section 7 we study stability properties of solutions with respect to boundary data, and show that a convergent sequence of boundary data yields a sequence of solutions that converges up to a subsequence; this is Theorem 7.4. Finally, in Theorem 7.9 we present one method of explicitly constructing a solution for limit boundary data.
Note that if Ω is a domain such that µ(X \ Ω) = 0, then as BV functions are insensitive to sets of measure zero, we will always have that P (Ω, ·) is the zero measure and, by the Poincaré inequality, the minimizer of the functional I is a (µ-a.e.) constant function. This is not a very interesting situation to consider. The results in this paper will be significant only for domains Ω with µ(X \ Ω) > 0.

Notation and definitions
In this section we introduce the necessary notation and assumptions.
In this paper, (X, d, µ) is a complete metric space equipped with a Borel regular outer measure µ satisfying a doubling property, that is, there is a constant C d ≥ 1 such that 0 < µ(B(x, 2r)) ≤ C d µ(B(x, r)) < ∞ for every ball B = B(x, r) with center x ∈ X and radius r > 0. If a property holds outside a set of µ-measure zero, we say that it holds almost everywhere, or a.e. We assume that X consists of at least two points. When we want to specify that a constant C depends on the parameters a, b, . . . , we write C = C(a, b, . . .).
A complete metric space with a doubling measure is proper, that is, closed and bounded subsets 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 open Ω Ω.
Here Ω Ω means that Ω is a compact subset of Ω. Other local spaces of functions are defined analogously.
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 given by The measure theoretic boundary ∂ * E of a set E ⊂ X is the set of points x ∈ X at which both E and its complement have positive upper density, i.e. The measure theoretic interior and exterior of E are defined respectively by and A curve γ is a nonconstant rectifiable continuous mapping from a compact interval into X. The length of a curve γ is denoted by γ . We will assume every curve to be parametrized by arc-length, which can always be done (see e.g. [15,Theorem 3.2]). 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. Upper gradients were originally introduced in [20]. If g is a nonnegative µ-measurable function on X and (2.3) holds for 1-a.e. curve, we say that g is a 1-weak upper gradient of u. A property holds for 1-a.e. curve if it fails only for a curve family with zero 1-modulus. A family Γ of curves is of zero 1-modulus if there is a nonnegative Borel function ρ ∈ L 1 (X) such that for all curves γ ∈ Γ, the curve integral γ ρ ds is infinite.
Let Ω ⊂ X be open. By only considering curves in Ω, we can say that g is an upper gradient of u in Ω. We let where the infimum is taken over all upper gradients g of u in Ω. The substitute for the Sobolev space W 1,1 (Ω) in the metric setting is the Newton-Sobolev space We understand Newton-Sobolev functions to be defined everywhere (even though · N 1,1 (Ω) is then only a seminorm). For more on Newton-Sobolev spaces, we refer to [42,4,21]. The 1-capacity of a set A ⊂ X is given by where the infimum is taken over all functions u ∈ N 1,1 (X) such that u ≥ 1 in A. We know that when X supports a (1, 1)-Poincaré inequality (see below), Cap 1 is an outer capacity, meaning that for any A ⊂ X, see e.g. [4,Theorem 5.31]. If a property holds outside a set A ⊂ X with Cap 1 (A) = 0, we say that it holds 1-quasieverywhere, or 1-q.e. Next we recall the definition and basic properties of functions of bounded variation on metric spaces, following [37]. See also e.g. [2,11,12,14,44] for the classical theory in the Euclidean setting. For u ∈ L 1 loc (X), we define the total variation of u in X to be where each g u i is an upper gradient of u i . We say that a function u ∈ L 1 (X) is of bounded variation, denoted by u ∈ BV(X), if Du (X) < ∞. By replacing X with an open set Ω ⊂ X in the definition of the total variation, we can define Du (Ω). For an arbitrary set A ⊂ X, we define We have the following coarea formula from [37,Proposition 4.2]: if Ω ⊂ X is an open set and u ∈ BV(Ω), then for any Borel set A ⊂ Ω, (2.5) We will assume throughout that X supports a (1, 1)-Poincaré inequality, meaning that there exist constants C P > 0 and λ ≥ 1 such that for every ball B(x, r), every locally integrable function u on X, and every upper gradient g of u, we have By applying the Poincaré inequality to approximating locally Lipschitz functions in the definition of the total variation, we get the following for µmeasurable sets E ⊂ X: For an open set Ω ⊂ X and a µ-measurable set E ⊂ X with P (E, Ω) < ∞, we know that for any Borel set A ⊂ Ω, where θ E : X → [α, C d ] with α = α(C d , C P , λ) > 0, see [1,Theorem 5.3] and [3,Theorem 4.6]. The lower and upper approximate limits of a function u on X are defined respectively by The jump set of a function u is the set It is straighforward to check that the trace is always a Borel function on the set where it exists. for every ϕ ∈ BV(Ω) with compact support in Ω.

Preliminary results
In this section we define the Neumann problem and consider various basic properties of solutions.
In this section, we always assume that Ω ⊂ X is a nonempty bounded open set with P (Ω, X) < ∞, such that for any u ∈ BV(Ω), the trace T u(x) exists for H-a.e. x ∈ ∂ * Ω and thus also for P (Ω, ·)-a.e. x ∈ ∂ * Ω, by (2.7). See [31,Theorem 3.4] for conditions on Ω that guarantee that this holds.
For some of our results, we will also assume that the following exterior measure density condition holds: Moreover, in this section we always assume that f ∈ L 1 (∂ * Ω, P (Ω, ·)) such that Throughout this paper we will consider the following functional: for u ∈ BV(Ω), let First we note the following basic property of the functional. We denote u + = max{u, 0} and u − = max{−u, 0}. Proof. Note that for any µ-measurable E ⊂ X, we have P (E, Ω) = P (Ω \ E, Ω). Since µ is σ-finite on X, it follows that for L 1 -a.e. t ∈ R we have µ({u = t}) = 0 and thus P ({u < t}, Ω) = P ({u ≤ t}, Ω), where L 1 is the Lebesgue measure. Thus by the BV coarea formula (2.5), we have Note that for u ≡ 0, I(u) = 0. Thus, if I(u) ≥ 0 for all u ∈ BV(Ω), then we find a minimizer simply by taking the zero function. Hence, we are more interested in the case where I(u) < 0 for some u ∈ BV(Ω). But then Thus, we consider the following restricted minimization problem.
Definition 3.4. We say that a function u ∈ BV(Ω) solves the restricted Neumann boundary value problem with boundary data f if −1 ≤ u ≤ 1 and The restricted problem does not always have a solution. It may also have only trivial, i.e., constant, solutions even though the boundary data are nontrivial. Moreover, non-trivial solutions need not be unique. In the Euclidean setting these issues were observed in [35].  See also Example 7.5 for an example of nonuniqueness with I(u) < 0. Next we will show that it suffices to consider only a special subclass of BV functions as candidates for a solution to the restricted Neumann problem. First we note that we have the following version of Cavalieri's principle, which can be obtained from the usual Cavalieri's principle by decomposing ν into its positive and negative parts.
Lemma 3.7. Let ν be a signed Radon measure on X. Then, for any nonnegative h ∈ L 1 (X, |ν|), Proposition 3.8. Let u ∈ BV(Ω) with −1 ≤ u ≤ 1. Then, there exist disjoint µ-measurable sets E 1 , E 2 ⊂ Ω such that Furthermore, if u is a solution to the restricted Neumann problem with boundary data f , then for L 1 -a.e. t 1 , t 2 ∈ (0, 1), the sets give a solution χ E 1 − χ E 2 to the same restricted Neumann problem.
x ∈ ∂ * Ω, Thus, Next, let E 1 , E 2 ⊂ Ω be disjoint sets such that χ E 1 − χ E 2 solves the restricted Neumann problem. If I( χ E 1 ) < I(− χ E 2 ), then by the above, we also have I(− χ Ω\E 1 ) < I(− χ E 2 ). Then, by Lemma 3.3, solves the restricted Neumann problem if and only if , then by Lemma 3.3 and Lemma 3.12 for any two disjoint µ-measurable sets F 1 , F 2 ⊂ Ω. In view of Proposition 3.8, Recall that a function u ∈ BV(Ω) is of least gradient in Ω if for every ϕ ∈ BV(Ω) with compact support in Ω.
Proposition 3.14. Let E 1 , E 2 ⊂ Ω be disjoint sets such that χ E 1 − χ E 2 solves the restricted Neumann problem. Then, χ E 1 and χ E 2 are functions of least gradient in Ω.
Let F be such a set. By Lemma 3.13, so that χ E 1 is of least gradient. The proof for E 2 is analogous.
The above is our main result on the interior regularity of solutions; from the proposition it follows that the sets E 1 , E 2 and their complements are porous in Ω, see [22,Theorem 5.2].
Since solutions can be constructed from sets E of finite perimeter in Ω and since Ω is itself of finite perimeter in X, it is useful to know that the sets E are also of finite perimeter in X.
Note that if Ω satisfies the condition listed in (3.1), then H(N ) = 0 above.
Lemma 3.16. Assume that Ω satisfies the exterior measure density condition (3.1). Let E ⊂ Ω be a µ-measurable set with P (E, X) < ∞. Then, for any Borel set A ⊂ ∂Ω, we have By Theorem 3.15, we can equally well only assume that P (E, Ω) < ∞.
Proof. Note that the trace T χ E (x) is defined for H-a.e. x ∈ ∂ * Ω and can only take the values 0 and 1. Also, P (E, ·) is concentrated on ∂ * E, and , the measure theoretic exterior of E as defined by (2.2). Thus, we have so that x ∈ ∂ * E if and only if x ∈ ∂ * Ω, and according to [2,Proposition 6.2], for H-a.e. such point we have θ E (x) = θ Ω (x); recall (2.7). In total, by (3.17) and by applying (2.7) twice, Lemma 3.18. Suppose that Ω satisfies the exterior measure density condi- Proof. By Theorem 3.15, P (E, X) < ∞. By the definition of the functional and the fact that −1 ≤ f ≤ 1, whereas by Lemma 3.16, P (E, ∂Ω) ≤ P (Ω, X). Thus we get

Existence of solutions
In this section, we prove that under fairly mild assumptions on Ω, solutions to the restricted Neumann problem given on page 10 exist. This is Theorem 4.15. We say that a set A ⊂ X is 1-quasiopen if for every ε > 0 there is an open set G ⊂ X with Cap 1 (G) < ε such that A ∪ G is open. Note that 1-quasiopen sets do not in general form a topology: as is noted in [6], all singletons in unweighted R n , n ≥ 2, are 1-quasiopen, but not all sets are 1-quasiopen. Nonetheless, countable unions as well as finite intersections of 1-quasiopen sets are 1-quasiopen by [13,Lemma 2.3].
The following lemma is well known in the Euclidean setting, and has been proved in the metric setting in [28,Lemma 3.8]. From this lemma it easily follows that 1-quasiopen sets are always Dumeasurable, and we will use this fact without further notice.
The total variation is easily seen to be lower semicontinuous with respect to L 1 -convergence in any open set. We will need the following more general semicontinuity result that follows from [27,Theorem 4.5].
To deal with boundary values given by a function f defined only on ∂Ω, we first need to extend f to the whole space in a suitable way. We will consider open sets Ω whose boundary is codimension 1 Ahlfors regular in the following sense: for every x ∈ ∂Ω, every 0 < r ≤ diam(Ω), and some constant C A ≥ 1, Let Ω ⊂ X be a bounded open set whose boundary is codimension 1 Ahlfors regular as given in Proof. This follows from [33] and the proofs therein. Note that the argument of the extension theorem for Besov boundary data [33, Theorem 1.1] needs to be slightly modified to produce a Newtonian extension not only inside Ω but in the whole set X \ ∂Ω. Namely, when constructing a Whitney-type decomposition W X\∂Ω , we consider only balls whose distance from ∂Ω is at most 2 diam(Ω). Such a collection of balls covers Ω as well as the 2 diam(Ω)neighborhood of ∂Ω, leaving out ∂Ω. Then, we relax the requirements on the partition of unity {φ j,i } j,i subordinate to W X\∂Ω by demanding that Using such a "partition of unity" gives us an extension of f in the class The extension theorem for Besov boundary data modified as described above can then be used directly in [33,Theorem 1.2] Note that for any A ⊂ X, by [16,  In the following, given a ball B = B(x, r) we sometimes abbreviate 2B := B(x, 2r).
Proof. Fix i ∈ N. By the compactness of ∂Ω, we find a covering Consider the function is an upper gradient of v i . We show that By the Leibniz rule, see e.g. [4,Theorem 2.15], g i is a 1-weak upper gradient of f i in X \ ∂Ω. Take a curve γ such that the upper gradient inequality is satisfied by f i and g i on all subcurves of γ in X \ ∂Ω; this is true for 1-a.e. γ, by [4,Lemma 1.34].
Note that Thus, γ can be split into a finite number of subcurves each of which lies either entirely in M j=1 B j , or entirely in X \ ∂Ω. If γ 1 is a subcurve lying so the upper gradient inequality is satisfied. If γ 2 is a subcurve lying entirely in X \ ∂Ω, then by our choice of γ. Summing over the subcurves, we obtain Thus, g i is a 1-weak upper gradient of f i in X. By (4.8) we have and thus f i ∈ N 1,1 (X). Since Lipschitz functions are dense in N 1,1 (X), see e.g. [4, Theorem 5.1], we have also f i ∈ BV(X), with Clearly f i → f in L 1 (X) as i → ∞. By lower semicontinuity, Thus, f ∈ BV(X). Recall the decomposition of the variation measure from (2.8). Since f ∈ N 1,1 (X \ ∂Ω), Df s (X \ ∂Ω) = 0. Since H(∂Ω) < ∞, also Df c (∂Ω) = 0 by [3,Theorem 5.3]. Finally, by (2.8),  Proof. Combine Theorem 4.4 and Proposition 4.6.
In Example 3.5, it is crucial that a > 1. If f is restricted in the same way as u, solutions exist at least if Ω is sufficiently regular. The proof relies on the following lower semicontinuity result, which will also be used later in other contexts. Such a restriction is necessary even in Euclidean setting with smooth domains, see [36] and Example 3.5 (which, while is not a smooth domain, can be modified to be one).

Lemma 4.10.
Let Ω ⊂ X be a nonempty bounded open set satisfying the exterior measure density condition (3.1), and such that for any u ∈ BV(Ω), the trace T u(x) exists for H-a.e. x ∈ ∂ * Ω. Assume also that ∂Ω is codimension 1 Ahlfors regular as given in (4.3).
Let E ⊂ Ω such that P (E, Ω) < ∞. By Cavalieri's principle, (4.11) Fix t ∈ (0, 1). Suppose E i ⊂ Ω, i ∈ N, are such that P (E i , Ω) < ∞ and χ E i → χ E in L 1 (Ω) (and thus in fact in L 1 (X)). By Theorem 3.15, also P (E i , X) < ∞. By lower semicontinuity and Lemma 3.18, we have where we can assume the limit on the right-hand side to be finite. Thus P (E, X) < ∞. By Proposition 4.2, we now have Thus, by Lemma 3.16, Since also χ Ω\E i → χ Ω\E in L 1 (X), by the lower semicontinuity of perimeter we also get Note that T χ Ω\E (x) = 1 if and only if T χ E (x) = 0. Thus by Lemma 3.16, By subtracting P (Ω, {f < −t}) from both sides and noting that P (F, A) = P (Ω \ F, A) for any µ-measurable F ⊂ X and any set A ⊂ Ω, we obtain For such t, by (4.12), and (4.13) and using lower semicontinuity once more, in the 1-quasiopen set {−t < f < t}, (4.14) By combining (4.11) and (4.14) and using Fatou's lemma, we obtain Denoting I(·) = I f (·) to make the dependence on f explicit, we have also and thus the claim is proved.

Theorem 4.15.
Let Ω and f be as in Lemma 4.10. Then the restricted Neumann problem given on page 10 has a solution.
Proof. Take a sequence (u i ) ⊂ BV(Ω) with −1 ≤ u i ≤ 1 and By Proposition 3.8 we can assume that By Lemma 3.18 and Lemma 3.3, and similarly for the sets E i 2 . We conclude that the sequences P (E i 1 , X) and P (E i 2 , X) are bounded, and so by [37,Theorem 3.7] there are sets E 1 , E 2 ⊂ Ω such that χ E i 1 → χ E 1 in L 1 (X) and χ E i 2 → χ E 2 in L 1 (X), passing to a subsequence if needed (without relabeling the sequences). Then, clearly also µ(E 1 ∩E 2 ) = 0. By lower semicontinuity, P (E 1 , X) < ∞ and P (E 2 , X) < ∞. Thus by Lemma 3.3 and Lemma 4.10, Thus, χ E 1 − χ E 2 is a solution.
In light of the result from the previous sections that χ E 1 − χ E 2 is a solution for some choice of E 1 , E 2 ⊂ Ω, we see that the "relative outer normal derivative" of the solution (in relation to the total variation of the function) is directed either entirely outward (i.e., ∂ η u/ Du = ±1 in the Euclidean setting) or has vanishing derivative. Thus, in the Euclidean setting, if one is to make sense of f as the relative outer normal derivative of the solution, then the only permissible values one has for f are 0, 1, and −1. This section is dedicated to the study of boundary behavior of solutions χ E 1 − χ E 2 for such f . Suppose E 1 , E 2 ⊂ Ω are disjoint sets such that χ E 1 − χ E 2 solves the restricted Neumann problem. Note that From Lemma 3.12, we can conclude that , Ω) = 0, then by the facts that X supports the relative isoperimetric inequality (2.6) and Ω is connected, we must have either that µ(Ω \ E 1 ) = 0 or µ(E 1 ) = 0, from either of which we would have that I( χ E 1 ) = 0. Thus we must have P (E 1 , Ω) > 0. However, from this and (5.2) we can only infer that On the set ∂ E 1 Ω one should understand that the relative outer normal derivative of χ E 1 − χ E 2 must be −1; thus on the set ∂ E 1 Ω ∩ {f = 1} the relative outer normal derivative of χ E 1 − χ E 2 does not agree with the boundary data f = 1. The above inequality therefore implies that the relative outer normal derivative of χ E 1 − χ E 2 agrees more often than not with the boundary data f where f = 0. We would prefer to obtain a better quantitative version of this statement.
Proposition 5.4. Suppose that Ω, as a metric measure space equipped with the measure µ Ω , supports a (1, 1)-Poincaré inequality and a measure density condition: there is some C ≥ 1 and r 0 > 0 such that for every x ∈ ∂Ω and 0 < r < r 0 . Suppose also that ∂Ω is codimension 1 Ahlfors regular as defined in (4.3). Assume that ∅ = E 1 Ω is such that χ E 1 − χ Ω\E 1 solves the restricted Neumann problem with boundary data f : where the constant C Ω > 1 is independent of f and E 1 . Otherwise, It is straightforward to check that (5.5) can equivalently be required for every x ∈ Ω, possibly with different constants C, r 0 . Moreover, we will see that one can express C Ω = T 1 + 2C P Ω diam(Ω) , where C P Ω > 0 is the constant associated with the Poincaré inequality on Ω and T is the norm of the trace operator T : BV(Ω) → L 1 (∂ * Ω, P (Ω, ·)).
The following example shows that it is in general impossible to obtain an estimate better than (5.3) in case we wish the constants to be independent of E 1 . On the other hand, the situation is different if ∂Ω is of positive mean curvature in the sense of [29], see Definition 5.10 below.
Neumann boundary data: Using the above solution to the given Neumann problem, let us now show that it is in general impossible to obtain an estimate of the form (5.6) The example above heavily relies on the fact that the boundary data are non-zero on flat parts of ∂Ω. In the remaining part of this section, we will discuss the case when ∂Ω is of positive mean curvature in the sense of [29]; see also [43].
Definition 5.10. Let h ∈ BV loc (X). We say that u ∈ BV loc (X) is a weak solution to the Dirichlet problem for least gradients in Ω with boundary data h if u = h on X \ Ω and A weak solution exists whenever h ∈ BV loc (X) with Dh (X) < ∞, see [29, Lemma 3.1].
Definition 5.11. We say that the boundary ∂Ω has positive mean curvature if there exists a non-decreasing function ϕ : (0, ∞) → (0, ∞) and a constant r 0 > 0 such that for all z ∈ ∂Ω and all 0 < r < r 0 with P (B(z, r), X) < ∞, we have that u ∨ ≥ 1 everywhere on B(z, ϕ(r)) for any weak solution u to the the Dirichlet problem for least gradients in Ω with boundary data χ B(z,r) .
Recall that the perimeter measure P (E, ·) relates to H ∂ * E via the function θ E : X → [α, C d ] as stated in (2.7).
Definition 5.12 ([3, Definition 6.1]). We say that X is a local space if, given any two sets of locally finite perimeter The assumption E 1 ⊂ E 2 can in fact be dropped as shown in the discussion after [17,Definition 5.9]. See [3] and [26] for some examples of local spaces. See also [30,Example 5.2] for an example of a space that fails to be local, despite being equipped with a doubling measure that supports a Poincaré inequality.
Theorem 5.13. Suppose X is a local space. Assume that Ω satisfies the exterior measure density condition (3.1), that H(∂Ω) < ∞, and that ∂Ω has positive mean curvature. Suppose that χ E 1 − χ E 2 solves the restricted Neumann problem with boundary data f : Moreover, if u ∈ BV(Ω) is any solution to the restricted Neumann problem with boundary data f and f = −1 on B(z, r) ∩ ∂Ω for some r > 0, then u = 1 on B(z, ϕ(r)) ∩ Ω and hence T u(z) = 1.
In the above, r → ϕ(r) is the function associated with positive mean curvature of ∂Ω as in Definition 5.11.
Proof. If z ∈ ∂Ω such that f = −1 in a neighborhood of z, we find r > 0 such that f = −1 on B(z, r) ∩ ∂ * Ω, and P (B(z, r), X) < ∞ and H(∂B(z, r) ∩ ∂Ω) = 0; the latter two facts hold for L 1 -a.e. r > 0 by the BV coarea formula (2.5) and the fact that H(∂Ω) < ∞. Take K ⊂ X such that χ K is a weak solution to the Dirichlet problem for least gradients in Ω with boundary data χ B(z,r) ; in particular, χ K = χ B(z,r) on X \Ω. We let E = E 1 ∪(B(z, r)\Ω) and claim that K ∩ E is another weak solution to the Dirichlet problem. Suppose it is not. Then P (K, Ω) < P (K ∩ E, Ω).
By [29,Corollary 4.6], we have T χ K (x) = χ B(z,r) (x) for H-a.e. x ∈ ∂Ω, and thus H(∂ * K ∩ ∂Ω) = 0, whence P (K, ∂Ω) = 0 by (2.7). Thus P (K, Ω) = P (K, Ω). Now we also have T χ K∩E ≤ χ B(z,r) H-a.e. on ∂Ω, and so H(∂ * (K ∩ E) ∩ ∂Ω \ B(z, r)) = 0. Note that P (E 1 , X) < ∞ by Theorem 3.15, and then P (K ∩E, X) < ∞ by [37,Proposition 4.7]. Thus by the fact that P (K ∩E, ·) is a Borel outer measure and (2.7), It is straightforward to verify that where I K and O E stand for the measure theoretic interior and exterior, respectively, as defined by (2.1) and (2.2). By (2.7) and by X being local, we obtain that Combining this with (5.14), we get On the other hand, comparing E against E ∪ K in the Neumann problem (note that also P (E ∪ K, X) < ∞ by [37,Proposition 4.7]), by Lemma 3.13 we obtain x ∈ ∂Ω. Similarly as before, it is straightforward to verify that By (2.7) and the fact that X is local, we now see that Combining (5.17) with (5.16) yields that It follows that Since (5.15) is in contradiction with (5.18), we have established the claim that K ∩ E is a weak solution to the Dirichlet problem for least gradients in Ω with boundary data χ B(z,r) . Therefore, by the definition of positive mean curvature, B(z, ϕ(r)) ⊂ K ∩ E ⊂ E (up to a µ-negligible set) and in particular, T χ E 1 (z) = T χ E (z) = 1.
We complete the proof of this theorem by considering a solution u for boundary data f with f = −1 on B(z, r) ∩ ∂ * Ω. By the last part of Proposition 3.8, we can find two sequences t 1,k , t 2,k ∈ (0, 1) with lim k→∞ t 1,k = 1 and lim k→∞ t 2,k = 1 such that each χ {u>t 1,k } − χ {u<−t 2,k } is a solution to the same Neumann problem. Thus, by the above argument, we have that u ≥ t 1,k in B(z, ϕ(r)) ∩ Ω for each k ∈ N, and thus the desired conclusion follows by letting k → ∞.
In particular, it follows from the above result that every z in the interior of the set {x ∈ ∂Ω : f (x) = −1} satisfies z ∈ ∂ E 1 Ω. Conversely, z ∈ ∂ E 1 Ω whenever z lies in the interior of the set {x ∈ ∂Ω : f (x) = 1}. Compare this to the situation regarding the Dirichlet problem on domains whose boundary has positive mean curvature. It is known that if the Dirichlet boundary data are continuous, then the solution to the least gradient problem on the domain has trace on the boundary that agrees with the boundary data, see [29]. However, if the boundary data are not continuous, no such control over the trace of the solution is known except in special circumstances such as characteristic functions of relatively open subsets F ⊂ ∂Ω for which H(∂Ω ∩ ∂F ) = 0. Indeed, in the Euclidean setting, with a Euclidean ball playing the role of the domain, there are known to be boundary data, taken from the class L 1 of the boundary sphere, for which solutions to the Dirichlet problem fail to have the correct trace, see [34].
A natural question is whether we have any control over the solution near the part of the boundary where f = 0.
Example 5.20. Consider the simple example of Ω = B(0, 1) ⊂ R 2 (unweighted) with the boundary data We can easily see that it is impossible to determine what value a solution u will have near the boundary points where f = 0. Indeed, the problem is solved by each of the following three functions: Then, T u 1 ≡ 1, T u 2 ≡ −1, and T u 3 ≡ 0 on the set {f = 0}.
One might therefore wonder whether the zero Neumann data in a neighborhood of a boundary point guarantee that the solution is constant in a neighborhood of this point. In the following example, where a disk in the unweighted plane is discussed, we will see that such a conclusion indeed holds true. However, the subsequent two examples will prove the unweighted planar domain to be highly misleading.
Suppose for the sake of contradiction that u is not constant on B(z 0 , r) for any r > 0. Fix R > 0 such that f (z) = 0 for all z ∈ B(z 0 , R) ∩ ∂Ω. Since χ E 1 is a function of least gradient by Proposition 3.14, we can assume that ∂E 1 ∩Ω consists of straight line segments that connect points in ∂Ω and do not intersect each other. Consider the two closed half-disks whose union is Ω and whose intersection contains z 0 . Take all the line segments of ∂E 1 that reach B(z 0 , R) ∩ ∂Ω and lie within one of these half-disks. Then, move their endpoints that lie within B(z 0 , R) ∩ ∂Ω to ∂B(z 0 , R) ∩ ∂Ω within the respective half-disk. Such a modification of E 1 will decrease the perimeter of E 1 inside Ω but the boundary integral will remain unchanged (since f = 0 at all points where the trace of χ E 1 changed). In other words, such a modification will decrease the value of the functional I(·) and hence u could not have been a solution. Figure 1: The perimeter of E 1 inside Ω is decreased by moving the endpoints of ∂E 1 from ∂Ω ∩ B(z 0 , R) to ∂Ω ∩ ∂B(z 0 , R).
Let us now consider a domain in 3-dimensional Euclidean space, where the situation turns out to be very different from the plane. otherwise.
x f = 1 Based on Theorem 5.13, the trace of a solution to the restricted minimization problem u = χ E 1 − χ Ω\E 1 necessarily attains the values of −f in the region where f = 0. Therefore, the set E 1 has to cover the surface of a unit half-ball with x < 0, perhaps apart from the thin slit |y| < 1 100 . However, if E 1 consisted of at least two connected components, one for each component of the set {f = −1}, then the perimeter of E 1 inside Ω would be greater than the perimeter of the halfball B(0, 1) ∩ {x < 0}, which equals the area of a unit disk {(0, y, z) ∈ Ω}. Hence, E 1 consists of a single connected component.
Then, ∂E 1 connects the two half-circles on ∂Ω with x < 0 and y = ±1 100 . If the set E 1 := {(x, y, z) ∈ ∂E 1 : x < 0, |y| < 1 100 } lies entirely inside Ω, then the perimeter of this portion of ∂E 1 can be bounded below by a half of the surface area of a cylinder of height 2 100 and radius 1 − ( 1 100 ) 2 1/2 . Thus, the perimeter of E 1 inside Ω will decrease if a sufficiently large part of E 1 lies on ∂Ω. Therefore, the jump set of the trace of the solution u has a nonempty intersection with the interior of the set {f = 0} and so the solution is nonconstant near the said intersection.
Let us now consider only candidates for solutions that are of least gradient in Ω and of the form w = χ E 1 − χ E 2 such that the jump set of w does not reach to the interior of the set {f = 0}. It is easy to verify for all α, β ∈ [− π 4 , π 4 ] (and similarly for all α, β ∈ [ 3π 4 , 5π 4 ]) that the path of least weighted length that connects the boundary point (cos α, sin α) with (cos β, sin β) is a straight line segment. Thus, letting w 0 (x, y) = χ (−1,−1/ In particular, I(w) > I(v). Thus, the jump set of a solution u = χ E 1 − χ E 2 does reach to the interior of the set {f = 0}, i.e., there is z 0 ∈ ∂Ω and r 0 > 0 such that f ≡ 0 in ∂Ω ∩ B(z 0 , r 0 ), but u is not constant in B(z 0 , r) for any r < r 0 . It can be verified that y) ∈ Ω : x > max{0.1, |y|/9}.

Minimal solutions and their uniqueness
In this section, we assume that Ω ⊂ X is a nonempty bounded open set with P (Ω, X) < ∞, such that for any u ∈ BV(Ω), the trace T u(x) exists for Ha.e. x ∈ ∂ * Ω. We also assume that the boundary data f ∈ L 1 (∂ * Ω, P (Ω, ·)) satisfies (3.2).
We saw in Example 3.6 that solutions to the restricted Neumann problem need not be unique. However, we will see in this section that minimal solutions exist and are unique. Proof. We have P (E ∩ K, Ω) + P (E ∪ K, Ω) ≤ P (E, Ω) + P (K, Ω) by [37,Proposition 4.7]. Then by linearity of traces, H-a.e. on ∂Ω we have The claim follows. Definition 6.2. A solution u = χ E 1 − χ E 2 to the restricted Neumann problem is said to be minimal if whenever E 1 , E 2 ⊂ Ω are disjoint sets such that v = χ E 1 − χ E 2 is a solution, it follows that µ(E 1 \ E 1 ) = 0 and µ(E 2 \ E 2 ) = 0. By Lemma 3.12, it is enough to compare with solutions of the form χ E − χ Ω\ E . Lemma 6.3. Suppose that u a = χ Ea − χ Ω\Ea and u b = χ E b − χ Ω\E b are both solutions to the restricted Neumann problem. Then, so are Proof. By Lemma 6.1 we know that By Lemma 3.12 we obtain that I(u) = 2I( χ Ea∩E b ), I(v) = 2I( χ Ea∪E b ), and analogously for I(u a ) and I(u b ) as well. Then, Theorem 6.5. Assume that Ω satisfies the exterior measure density condition (3.1), that ∂Ω is codimension 1 Ahlfors regular as given in (4.3), and that −1 ≤ f ≤ 1. Then there exists a unique (up to sets of µ-measure zero) minimal solution to the restricted Neumann problem.
Proof. By Theorem 4.15 we know that a solution exists. Let β = inf E µ(E), where the infimum is taken over all sets E such that u = χ E − χ Ω\E is a solution. By Proposition 3.8 and the fact that Ω is bounded, β < ∞.
are also solutions by Lemma 6.3.
Now if E ⊂ Ω is such that χ E − χ Ω\E is a solution, by Lemma 6.3 we know that χ Ea∩E − χ Ω\(Ea∩E) is also a solution, and so µ(E a ∩ E) ≥ β. Since µ(E a ) = β, necessarily µ(E a \ E) = 0. By the same argument, we obtain that E a is the unique set with these properties, up to sets of µ-measure zero.
By an entirely analogous argument, we find a unique (up to sets of µmeasure zero) set E b ⊂ Ω such that χ Ω\E b − χ E b is a solution, and whenever χ Ω\E − χ E is another solution, then µ(E b \E) = 0. By Lemma 3.13, χ Ea − χ E b is the desired unique minimal solution.

Stability
In this section, we always assume that Ω ⊂ X is a nonempty bounded open set with P (Ω, X) < ∞, such that for any u ∈ BV(Ω), the trace T u(x) exists for H-a.e. x ∈ ∂ * Ω.
The goal here is to investigate stability of solutions to the restricted Neumann problem. By stability we mean that if a sequence of Neumann boundary data converges in L 1 (∂ * Ω) to a function, then the corresponding sequence of solutions converges (perhaps up to a subsequence) to a solution to the Neumann problem with the limit boundary data. Stability properties give us a method by which we can, by hand, construct a solution to the Neumann problem for complicated boundary data by using solutions to simpler boundary data. otherwise.
It is easy to see that there are two types of minimal solutions based on the value of θ k . If θ k ∈ [ π 3 , π 2 ], then a solution can be expressed as u(x, y) = − sgn(y), which is also minimal in case θ k > π 3 . However, if θ k ∈ (0, π 3 ], then the minimal solution u k = χ E k 1 − χ E k 2 is determined by four disk segments whose arcs cover the connected components of {f k = 0}, i.e., The minimal solution u k if θ k ≤ π 3 Thus, u 2k = u for all k = 1, 2, . . ., and trivially u 2k → u as k → ∞. On the other hand are the sets as in (7.2) for θ ∞ = π 3 . Consequently, the sequence of solutions {u k } ∞ k=1 does not have any limit even though the sequence of boundary data functions converges in L 1 (∂ * Ω, P (Ω, ·)).
Note however that both functions u and u ∞ are solutions to the restricted Neumann problem with boundary data given by f = lim k f k . This observation suggests that a weaker notion of stability might apply here. Indeed, Theorem 7.4 below will show that stability can be recovered if we allow for passing to a subsequence of the sequence of solutions.
In this section, we use the abbreviation L 1 (∂ * Ω) := L 1 (∂ * Ω, P (Ω, ·)). Lemma 7.3. If u is a solution to the restricted Neumann problem with L 1 (∂ * Ω)-boundary data f and v is a solution with L 1 (∂ * Ω)-boundary data h, then Proof. Note that −1 ≤ v ≤ 1 and −1 ≤ u ≤ 1. Therefore, and It follows that and In the above, we used the facts that v is a solution for I h and that u is a solution for I f . The desired conclusion now follows.
Theorem 7.4. Assume that Ω satisfies the exterior measure density condition (3.1) and that ∂Ω is codimension 1 Ahlfors regular as given in (4.3).
are solutions to the restricted Neumann problem with boundary data f k , for disjoint sets E k 1 , E k 2 ⊂ Ω. Then, there is a subsequence {u k j } ∞ j=1 and a function u = χ E 1 − χ E 2 such that u k j → u in L 1 (Ω) and u is a solution to the restricted Neumann problem with boundary data f .
Proof. Clearly f also satisfies (3.2). By Theorem 4.15, we know that there exists a solution v ∈ BV(Ω) for boundary data f . By Lemma 7.3, |I f (v) − for boundary data given by the limit function f ∞ is determined by the sets E 1 = {z ∈ Ω : z > 1 2 } and E 2 = {z ∈ Ω : z < −1 2 }. In particular, The minimal solution for f k , k ∈ N.
The minimal solution for f ∞ .
In light of the above example, we give one explicit construction of a solution (but not necessarily a minimal one) for limit boundary data. We first need the following more general lemma.
In what follows, for E ⊂ Ω of finite perimeter in Ω, we denote Lemma 7.6. For each k ∈ N, assume that f k ∈ L 1 (∂ * Ω) satisfies (3.2) and suppose that E k 1 , E k 2 ⊂ Ω are disjoint sets such that χ E k 2 − χ E k 1 is a solution to the restricted Neumann problem with boundary data f k . Denote E k := E k 1 . Then for each n ∈ N and for each choice of k 1 , · · · , k n ∈ N with k 1 < · · · < k n , we have Proof. The first inequality follows from Lemma 3.13. To prove the second, note first that for K ⊂ Ω of finite perimeter in Ω and for k ∈ N, we have that I f k (E k ) ≤ I f k (E k ∩ K). Moreover, by Lemma 6.1 we know that and so I f k (E k ∪ K) ≤ I f k (K). Now by an iterated ((n − 1)-times) application of (7.7) followed by (7.8), and finally by Lemma 7.3, we obtain I f kn (E k 1 ∪ · · · ∪ E kn ) ≤ I f kn (E k 1 ∪ · · · ∪ E k n−1 ) ≤ I f k n−1 (E k 1 ∪ · · · ∪ E k n−1 ) + f k n−1 − f kn L 1 (∂ * Ω) ≤ . . .
Thus we obtain the desired inequality.
Theorem 7.9. Suppose that Ω satisfies the exterior measure density condition (3.1), and that ∂Ω is codimension 1 Ahlfors regular as given in (4.3). For each k ∈ N, suppose that f k : ∂ * Ω → [−1, 1] satisfies (3.2), that f k − f k+1 L 1 (∂ * Ω) ≤ 2 −k , and that E k 1 , E k 2 ⊂ Ω are disjoint sets such that χ E k 1 − χ E k 2 is a solution for boundary data f k . Set E k = E k 1 . Then the limit supremum gives a solution χ E + − χ Ω\E + for the boundary data f := lim k f k .
Proof. By Theorem 4.15 we know that there exists a solution v ∈ BV(Ω) for boundary data f . From Lemma 7.3 we see that By letting n → ∞ and recalling (7.10), we obtain by Lemma 4.10, since χ Kn → χ E + in L 1 (Ω). Thus χ E + is also a solution for boundary data f .
It can be seen from Example 7.5 that the set E + constructed in Theorem 7.9 need not yield a minimal solution to the limit Neumann problem.
So far we have looked at the most general situation where it is possible to have I f (u) = Du (Ω) + ∂ * Ω T u f dP (Ω, ·) < 0 for some u ∈ BV(Ω). To overcome the fact that should I f (u) < 0 for some u then the minimal value of I f is −∞, we considered minimization only over u ∈ BV(Ω) for which −1 ≤ u ≤ 1. In the special case where inf u∈BV(Ω) I f (u) ≥ 0, the minimal energy must necessarily be 0; hence constant functions (and in particular, the zero function) would be a solution to the given Neumann boundary value problem with boundary data f . In this case we do not here need to restrict our attention to −1 ≤ u ≤ 1 alone, but to all functions in the class BV(Ω). In this case it would be interesting to see under what conditions we would have nonconstant minimizers of I f exist. If there is one, then there are infinitely many distinct (in the sense that they do not differ only by a constant) minimizers, as seen by multiplying by a scalar. In this study we take inspiration from [38]. We do not have a criterion that guarantees existence of a nonconstant minimizer. In the Euclidean setting, the PDE approach helps in forming such a guarantee, but we do not have such an approach in the metric setting. However, we do obtain a criterion under which there is no nonconstant minimizer, see Proposition 8.1 below. As a consequence of Proposition 8. 3 we also obtain that if there are no minimizers for the unrestricted problem for the boundary data f , then there is a positive number λ(−f ) such that the boundary data λ(−f )f does have a minimizer.
Note that if λ(g) < 1, then there is some u ∈ BV(Ω) such that I −g (u) < 0, and hence the unrestricted minimization problem for f = −g has no solution.
Proposition 8.1. If λ(g) ≥ 1, then there is a solution to the unrestricted minimization problem for the energy I −g on Ω. Furthermore, if λ(g) > 1 then the only minimizers are constant functions.
Proof. We will prove the claim of the proposition by showing that for each w ∈ BV(Ω), we have I −g (w) ≥ 0.