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.

We study local connectedness, local accessibility and finite connectedness at the boundary, in relation to the compactness of the Mazurkiewicz completion of a bounded domain in a metric space. For countably connected planar domains we obtain a complete characterization. It is also shown exactly which parts of this characterization fail in higher dimensions and in metric spaces.

We characterise regular boundary points of the parabolic p-Laplacian in terms of a family of barriers, both when p greater than 2 and 1 less than p less than 2. By constructing suitable families of such barriers, we give some simple geometric conditions that ensure the regularity of boundary points.

We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain the Adams criterion for the solubility of the single obstacle problem and establish connections with fine potential theory. We also study when the minimal p-weak upper gradient of a function remains minimal when restricted to a nonopen subset. Many of the results are new even for open E (apart from those which are trivial in this case) and also on R-n.

In this paper we study the Perron method for solving the -harmonic Dirichlet problem on the topologists comb. For functions which are bounded and continuous at the accessible points, we obtain invariance of the Perron solutions under arbitrary perturbations on the set of inaccessible points. We also obtain some results allowing for jumps and perturbations at a countable set of points.

In this paper we develop the Perron method for solving the Dirichlet problem for the analog of the p-Laplacian, i.e. for p-harmonic functions, with Mazurkiewicz boundary values. The setting considered here is that of metric spaces, where the boundary of the domain in question is replaced with the Mazurkiewicz boundary. Resolutivity for Sobolev and continuous functions, as well as invariance results for perturbations on small sets, are obtained. We use these results to improve the known resolutivity and invariance results for functions on the standard (metric) boundary. We also illustrate the results of this paper by discussing several examples. (C) 2015 Elsevier Inc. All rights reserved.