Domains in Metric Measure Spaces with Boundary of Positive Mean Curvature, and the Dirichlet Problem for Functions of Least Gradient

Abstract

We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality. We propose a notion of domain with boundary of positive mean curvature and prove that, for such domains, there is always a solution to the Dirichlet problem for least gradients with continuous boundary data. Here least gradient is defined as minimizing total variation (in the sense of BV functions) and boundary conditions are satisfied in the sense that the boundary trace of the solution exists and agrees with the given boundary data. This extends the result of Sternberg, Williams and Ziemer [27] to the non-smooth setting. Via counterexamples we also show that uniqueness of solutions and existence of continuous solutions can fail, even in the weighted Euclidean setting with Lipschitz weights. * The authors thank Estibalitz Durand-Cartagena, Marie Snipes and Manuel Ritoré for fruitful discussions about the subject of the paper.

Continuity of minimizers to weighted least gradient problems

“…Since F 1 is open, the latter observation yields C 2 ∩ F 1 = ∅, but this contradicts the fact that C 2 ⊂ B r (x) \ F 1 . Thus we have established that (22) A j = (∂ F j ) B r (x) for j = 1, 2, with F 1 ⊂ F 2 .…”

“…Thus A j for each j ∈ {1, 2} is a mass-minimizing current in B r (x) satisfying (22) with spt A j =C j connected, for which spt A 1 ∩ spt A 2 ∩ B r (x) = ∅ due to (21). In light of these observations, we can apply Corollary 3.3 to conclude thatC 1 =C 2 .…”

“…In addition, the time dependent notion of total variation flow has proved to be useful in image processing including denoising and restoration, see for example [2,3,6,8,25]. Further generalizations of least gradient problems in the metric space setting have been explored quite recently in [16,20,22].…”

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

“…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).…”

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

“…However, if α is small enough so that α ∂Ω |f | dH n−1 < Du (Ω), then the solution to the problem (B) is not u. On the other hand, if ∂Ω has positive mean curvature in the sense of [29], then the class of all solutions to the problem (B) and the class of all solutions to the problem (T) is the same, see Proposition 4.15 of [29]. Observe that when α < 1 the boundary of the ball, ∂Ω, no longer satisfies the condition of positive mean curvature in the sense of [29].…”