An analog of the Neumann problem for 1-Laplace equation in the metric setting: existence, boundary regularity, and stability 00footnotetext: 2010 Mathematics Subject Classification: 30l99, 26b30, 43a85. Keywords : bounded variation, metric measure space, Neumann problem, positive mean curvature, 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 Poincaré 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 in a neighborhood of a point on the boundary of the domain, then the solution is 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.
The goal of the Neumann boundary value problem for in a smooth Euclidean domain is to find a function such that
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 is now well developed, see the monograph  as well as e.g. [5, 8, 9, 41]. By contrast, Neumann problems have been studied very little. The paper  dealt mostly with homogeneous Neumann boundary value problem, while in the paper , a Neumann problem was formulated as the minimization of the functional
where is an upper gradient of and , see Section 2 for notation. In the Euclidean setting, with a smooth domain, a variant of this boundary value problem was studied in , and a connection between the problem for and the problem for was established through a study of the behavior of solutions for as . For functions , the following norm was associated in [35, 36]:
The problem of minimizing corresponding to was studied in  and then in  for Euclidean domains with Lipschitz boundary, and . The paper  also gave an application of this problem to the study of electrical conductivity. We point out here that the condition gives the minimal energy , and hence constant functions will certainly minimize the energy. Our focus in the present paper is to study the situation corresponding to , in which case there are no minimizers for the energy if one seeks to minimize within the class of all functions , see the discussion in the proof of [35, Proposition 3.1]. Thus we are compelled to add further natural constraints on the competitor functions , namely that . This constraint is not as restrictive as it might seem, and instead for any we can also consider constraints of the form that all competitor functions satisfy . Then is a minimizer for the constraint that all competitors should satisfy if and only if is a minimizer for the constraint that all competitors satisfy . Thus the study undertaken here complements the results in [35, 36] in the smooth Euclidean domains setting. For instance, suppose that 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 is not -a.e. zero on and takes on only three values, , then ; 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 , see . The paper  also gives an application of the problem to the study of conductivity, see [38, Section 1.1]. The problem as framed in  is not tractable in the metric setting as it relies heavily on the theory of divergence free -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 in the metric setting when . In this case, instead of the -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 in the Euclidean setting, and [18, 25, 29] in the metric setting. In this paper, following the formulation given in , 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 for disjoint sets . 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 and are functions of least gradient.
In Section 4 we show that under some regularity assumptions on , and additionally that , the functional is lower semicontinuous with respect to convergence in , 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 only taken the values . In the Euclidean setting, can be interpreted as the relative outer normal derivative of the solution, and so one would expect to have where . This is not always the case, but in Theorem 5.13 we show that in the interior points of when 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 is a domain such that , then as BV functions are insensitive to sets of measure zero, we will always have that is the zero measure and, by the Poincaré inequality, the minimizer of the functional is a (-a.e.) constant function. This is not a very interesting situation to consider. The results in this paper will be significant only for domains with .
2 Notation and definitions
In this section we introduce the necessary notation and assumptions.
In this paper, is a complete metric space equipped with a Borel regular outer measure satisfying a doubling property, that is, there is a constant such that
for every ball with center and radius . If a property holds outside a set of -measure zero, we say that it holds almost everywhere, or a.e. We assume that consists of at least two points. When we want to specify that a constant depends on the parameters we write .
A complete metric space with a doubling measure is proper, that is, closed and bounded subsets are compact. Since is proper, for any open set we define to be the space of functions that are Lipschitz in every open . Here means that is a compact subset of . Other local spaces of functions are defined analogously.
For any set and , the restricted spherical Hausdorff content of codimension is defined by
The codimension Hausdorff measure of a set is given by
The measure theoretic boundary of a set is the set of points at which both and its complement have positive upper density, i.e.
The measure theoretic interior and exterior of are defined respectively by
A curve is a nonconstant rectifiable continuous mapping from a compact interval into . The length of a curve is denoted by . 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 on is an upper gradient of an extended real-valued function on if for all curves on , we have
where and are the end points of . We interpret whenever at least one of , is infinite. Upper gradients were originally introduced in .
If is a nonnegative -measurable function on and (2.3) holds for -a.e. curve, we say that is a -weak upper gradient of . A property holds for -a.e. curve if it fails only for a curve family with zero -modulus. A family of curves is of zero -modulus if there is a nonnegative Borel function such that for all curves , the curve integral is infinite.
Let be open. By only considering curves in , we can say that is an upper gradient of in . We let
where the infimum is taken over all upper gradients of in . The substitute for the Sobolev space in the metric setting is the Newton-Sobolev space
The -capacity of a set is given by
where the infimum is taken over all functions such that in . We know that when supports a -Poincaré inequality (see below), is an outer capacity, meaning that
for any , see e.g. [4, Theorem 5.31]. If a property holds outside a set with , we say that it holds -quasieverywhere, or -q.e.
Next we recall the definition and basic properties of functions of bounded variation on metric spaces, following . See also e.g. [2, 11, 12, 14, 44] for the classical theory in the Euclidean setting. For , we define the total variation of in to be
where each is an upper gradient of . We say that a function is of bounded variation, denoted by , if . By replacing with an open set in the definition of the total variation, we can define . For an arbitrary set , we define
If , is a finite Radon measure on by [37, Theorem 3.4]. A -measurable set is said to be of finite perimeter in if , where is the characteristic function of . The perimeter of in is also denoted by
We have the following coarea formula from [37, Proposition 4.2]: if is an open set and , then for any Borel set ,
We will assume throughout that supports a -Poincaré inequality, meaning that there exist constants and such that for every ball , every locally integrable function on , and every upper gradient of , we have
By applying the Poincaré inequality to approximating locally Lipschitz functions in the definition of the total variation, we get the following for -measurable sets :
For an open set and a -measurable set with , we know that for any Borel set ,
The lower and upper approximate limits of a function on are defined respectively by
The jump set of a function is the set
By [3, Theorem 5.3], the variation measure of a 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 and , we have for any Borel set
where is the density of the absolutely continuous part and the functions are as in (2.7).
Let be an open set and let be a -measurable function on . For , the number is the trace of if
It is straighforward to check that the trace is always a Borel function on the set where it exists.
Let be an open set. A function is said to be of least gradient in if
for every with compact support in .
3 Preliminary results
In this section we define the Neumann problem and consider various basic properties of solutions.
In this section, we always assume that is a nonempty bounded open set with , such that for any , the trace exists for -a.e. and thus also for -a.e. , by (2.7). See [31, Theorem 3.4] for conditions on 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 such that
Throughout this paper we will consider the following functional: for , let
First we note the following basic property of the functional. We denote and .
For any , we have .
Note that for any -measurable , we have . Since is -finite on , it follows that for -a.e. we have and thus , where is the Lebesgue measure. Thus by the coarea formula (2.5), we have
Note that for , . Thus, if for all , then we find a minimizer simply by taking the zero function. Hence, we are more interested in the case where for some . But then
Thus, we consider the following restricted minimization problem.
We say that a function solves the restricted Neumann boundary value problem with boundary data if and for all with .
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 non-trivial. Moreover, non-trivial solutions need not be unique. In the Euclidean setting these issues were observed in .
In the unweighted plane (endowed with the Euclidean distance), consider the unit square, i.e., . Fix a constant and let on the middle third portion of the bottom side, on the middle third portion of the top side, and elsewhere on the boundary. If , then
but no admissible function gives this infimum.
Consider again the unit square in the Euclidean plane. Fix a constant and let on the bottom side, on the top side, and on the vertical sides.
If , then , which is attained by any constant function with as well as by any function , , where is an arbitrary decreasing function with , .
See also Example 7.5 for an example of nonuniqueness with .
Next we will show that it suffices to consider only a special subclass of 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 into its positive and negative parts.
Let be a signed Radon measure on . Then, for any nonnegative ,
Let with . Then, there exist disjoint -measurable sets such that
Furthermore, if is a solution to the restricted Neumann problem with boundary data , then for -a.e. , the sets
give a solution to the same restricted Neumann problem.
If and for some , then
Thus, yields that . Conversely, we see that if , then . In conclusion,
However, for -a.e. , since is a finite measure. Thus (3.9) becomes
Thus there is such that , which is the same as .
Denoting to make the dependence on explicit, with the substitutions of by and by , inequality (3.10) becomes
Therefore, there is such that , i.e., . Letting and , we now have by Lemma 3.3
proving the first claim.
If is of finite perimeter in , then . Thus, if are disjoint sets such that solves the restricted Neumann problem, then necessarily , and is also a solution.
If , note that , and that for -a.e. ,
Next, let be disjoint sets such that solves the restricted Neumann problem. If , then by the above, we also have . Then, by Lemma 3.3,
a contradiction. Similarly, is impossible. Moreover, now
so that is also a solution. ∎
Let be disjoint -measurable sets. Then, solves the restricted Neumann problem if and only if
for all -measurable sets .
a contradiction. Similarly, is impossible.
If are such that and for all -measurable sets , then
for any two disjoint -measurable sets . In view of Proposition 3.8, must be a solution. ∎
Recall that a function is of least gradient in if
for every with compact support in .
Let be disjoint sets such that solves the restricted Neumann problem. Then, and are functions of least gradient in .
The above is our main result on the interior regularity of solutions; from the proposition it follows that the sets and their complements are porous in , see [22, Theorem 5.2].
Since solutions can be constructed from sets of finite perimeter in and since is itself of finite perimeter in , it is useful to know that the sets are also of finite perimeter in .
Theorem 3.15 ([25, Corollary 6.13]).
Assume that is a bounded open set with , and suppose that there exists with such that
for every . Let such that . Then is of finite perimeter in .
Note that if satisfies the condition listed in (3.1), then above.
Assume that satisfies the exterior measure density condition (3.1). Let be a -measurable set with