Domains In Metric Spaces Research Articles

Non-locality, non-linearity, and existence of solutions to the Dirichlet problem for least gradient functions in metric measure spaces

We study the Dirichlet problem for least gradient functions for domains in metric spaces equipped with a doubling measure and supporting a (1,1)-Poincaré inequality when the boundary of the domain satisfies a positive mean curvature condition. In this setting, it was shown by Malý, Lahti, Shanmugalingam, and Speight (2019) that solutions exist for continuous boundary data. We extend these results, showing existence of solutions for boundary data that are approximable from above and below by continuous functions. We also show that for each f\in L^1(\partial\Omega) , there is a least gradient function in \Omega whose trace agrees with f at all points of continuity of f , and so we obtain existence of solutions for boundary data which are continuous almost everywhere. This is in contrast to a result of Spradlin and Tamasan (2014), who constructed an L^1 -function on the unit circle which has no least gradient solution in the unit disk in \mathbb{R}^2 . Modifying the example of Spradlin and Tamasan, we show that the space of solvable L^1 -functions on the unit circle is non-linear, even though the unit disk satisfies the positive mean curvature condition.

Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space (X,dX,μX) satisfying a 2-Poincaré inequality. Given a bounded domain Ω⊂X with μX(X∖Ω)>0, and a function f in the Besov class B2,2θ(X)∩L2(X), we study the problem of finding a function u∈B2,2θ(X) such that u=f in X∖Ω and Eθ(u,u)≤Eθ(h,h) whenever h∈B2,2θ(X) with h=f in X∖Ω. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on Ω, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extends the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.

To theory of Q-homeomorphisms in metric spaces

We study the problem of continuation on boundary of the so-called Q-homeomorphisms between domains in metric spaces with measures. We obtain conditions for function Q(x) and domain boundary, under which any homeomorphism allows continuous continuation on boundary.

Existence and almost uniqueness for p-harmonic Green functions on bounded domains in metric spaces

We study (p-harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive capacity, and that they satisfy very precise capacitary identities for superlevel sets. Suitably normalized singular functions are called Green functions. Uniqueness of Green functions is largely an open problem beyond unweighted Rn, but we show that all Green functions (in a given domain and with the same singularity) are comparable. As a consequence, for p-harmonic functions with a given pole we obtain a similar comparison result near the pole. Various characterizations of singular functions are also given. Our results hold in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, or under similar local assumptions.

Prime ends in metric spaces and quasiconformal-type mappings

By using the inner diameter distance condition we define and investigate new, in such a generality, class {mathcal {F}} of homeomorphisms between domains in metric spaces and show that, under additional assumptions on domains, {mathcal {F}} contains (quasi)conformal, bi-Lipschitz and quasisymmetric mappings as illustrated by examples. Moreover, we employ a prime ends theory in metric spaces and provide conditions allowing continuous and homeomorphic extensions of mappings in {mathcal {F}} to topological closures of domains, as well as homeomorphic extensions to the prime end boundary. Domains satisfying the bounded turning condition, locally and finitely connected at the boundary and the structure of prime end boundaries for such domains play a crucial role in our investigations. We apply our results to show the Koebe theorem on arcwise limits for mappings in {mathcal {F}}. Furthermore, relations between the Royden boundary and the prime end boundary are presented. Our work generalizes results due to Carathéodory, Näkki, Väisälä and Zorič.

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

Abstract 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 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.

On the Boundary Behavior of One Class of Mappings in Metric Spaces

We study the problem of extension to the boundary of continually ring Q-homeomorphisms relative to a p-module between continual domains in metric spaces with measures and formulate the conditions for the function Q and the boundaries of domains under which every continually ring Q-homeomorphism admits a continuous or homeomorphic extension to the boundary. The accumulated results yield, in particular, important applications to fractals in ℝ n , n ≥ 2.

Prime ends for domains in metric spaces

In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Carathéodory and Näkki. Modulus ends and prime ends, defined by means of the p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.

Boundary behavior of ring Q-homeomorphisms in metric spaces

We investigate the problem of extension of so-called ring Q-homeomorphisms between domains in metric spaces with measures to the boundary. We establish conditions for the function Q(x) and the boundary of the domain under which any ring Q-homeomorphism admits a continuous or a homeomorphic extension to the boundary. The results are applicable, in particular, to Riemannian manifolds, Lowner spaces, and Carnot and Heisenberg groups.