We construct various examples of Sobolev-type functions, defined via upper gradients in metric spaces, that fail to be quasicontinuous or weakly quasicontinuous. This is done with quasi-Banach function lattices X as the function spaces defining the smoothness of the Sobolev-type functions. These results are in contrast to the case X = Lp with 1 ≤ p < ∞, where all Sobolev-type functions in N1,p are known to be quasicontinuous, provided that the underlying metric space 𝒫 is locally complete. In most of our examples, 𝒫 is a compact subset of R2 and X = L∞. Four particular examples are the damped topologist’s sine curve, the von Koch snowflake curve, the Cantor ternary set and the Sierpiński carpet. We also discuss several related properties, such as whether the Sobolev capacity is an outer capacity, and how these properties are related. A fundamental role in these considerations is played by the lack of the Vitali–Carathéodory property.

Given a bounded finely open set V and a function f on the fine boundary of V, we introduce four types of upper Perron solutions to the nonlinear Dirichlet problem for p-energy minimizers, 1&lt;p&lt;infinity, with f as boundary data. These solutions are given as pointwise infima of suitable families of fine p-superminimizers in V. We show (under natural assumptions) that the four upper Perron solutions are equal quasieverywhere and that they are fine p-minimizers of the p-energy integral. We moreover show that the upper and the lower Perron solutions coincide quasieverywhere for Sobolev and for uniformly continuous boundary data, i.e. that such boundary data are resolutive. For the uniformly continuous boundary data, the Perron solutions are also shown to be finely continuous and thus finely p-harmonic. We prove our results in a complete metric space X equipped with a doubling measure supporting a p-Poincar & eacute; inequality, but they are new also in unweighted R-n.

Using the Caffarelli–Silvestre extension, we show for a general open set Ω⊂Rn that a boundary point x0 is regular for the fractional Laplace equation (−Δ)s⁢u=0, 0<s<1, if and only if (x0,0) is regular for the extended weighted equation in a subset of Rn+1. As a consequence, we characterize regular boundary points for (−Δ)s⁢u=0 by a Wiener criterion involving a Besov capacity. A decay estimate for the solutions near regular boundary points and the Kellogg property are also obtained.

We initiate the study of fine p-(super)minimizers, associated with p-harmonic functions, on finely open sets in metric spaces, where infinity. After having developed their basic theory, we obtain the p-fine continuity of the solution of the Dirichlet problem on a finely open set with continuous Sobolev boundary values, as a by-product of similar pointwise results. These results are new also on unweighted . We build this theory in a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality.

In a complete metric space equipped with a doubling measure supporting a p-Poincare inequality, we prove sharp growth and integrability results for p-harmonic Green functions and their minimal p-weak upper gradients. We show that these properties are determined by the growth of the underlying measure near the singularity. Corresponding results are obtained also for more general p-harmonic functions with poles, as well as for singular solutions of elliptic differential equations in divergence form on weighted R-n and on manifolds.The proofs are based on a new general capacity estimate for annuli, which implies precise pointwise estimates for p-harmonic Green functions. The capacity estimate is valid under considerably milder assumptions than above. We also use it, under these milder assumptions, to characterize singletons of zero capacity and the p-parabolicity of the space. This generalizes and improves earlier results that have been important especially in the context of Riemannian manifolds.

By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite p-energy p-harmonic and p-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local p-Poincare inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We study the inclusions between these classes of metric measure spaces, and their relationship to the p-hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant p-harmonic functions with finite p-energy as spaces having at least two well-separated p-hyperbolic sequences of sets towards infinity. We also show that every such space X has a function f is an element of/ LP(X) + R with finite p-energy.

We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach is not dependent on function spaces and therefore applies equally well to functions on metric spaces as to operator algebras. In particular, we consider analogues of Dirichlet and obstacle problems, as well as first eigenvalue problems, and formulate conditions for the existence of solutions and their uniqueness. Moreover, we investigate to what extent a lattice structure may be introduced on ( ordered) Banach spaces via a norm-minimizing variational problem. A multitude of examples is provided to illustrate the versatility of our approach. (C) 2015 Elsevier Ltd. All rights reserved.

We study different definitions of Sobolev spaces on quasiopen sets in a complete metric space X equipped with a doubling measure supporting a p-Poincare inequality with 1 amp;lt; p amp;lt; infinity, and connect them to the Sobolev theory in R-n. In particular, we show that for quasiopen subsets of R-n the Newtonian functions, which are naturally defined in any metric space, coincide with the quasicontinuous representatives of the Sobolev functions studied by Kilpelainen and Maly in 1992.