Sobolev spaces with zero boundary values on metric spaces
Motivated by the interest in developing the calculus of variations in the general setup and, on the other hand, by the interest of presenting a general theory which covers applications to manifolds, groups, vector fields, graphs and fractal sets in the Euclidean space, the aim of the paper is the generalization of the first order Sobolev space with zero boundary values on any metric space equipped with a Borel regular measure. There are two natural ways available to define Sobolev spaces with zero boundary values. In the classical Euclidean case (with Lebesgue measure) the space is defined as the completion of compactly supported smooth functions on the Sobolev norm. The first candidate is based on the completion of Lipschitz continuous functions which belong to the global Sobolev space and vanish in the complement of a given set; the other possibility is to require that the function can be extended to the global Sobolev space and that the trace of the extension vanishes in the complement. In the classical Euclidean case of an open set both of these definitions are equivalent, but the approximation by Lipschitz functions works basically on open sets only. For an arbitrary set in a metric space the latter definition is more general and it is taken by the authors as their starting point in introducing first order Sobolev spaces with zero boundary values on any metric space equipped with a Borel regular measure. In order to define the trace of a Sobolev function they use the notion of capacity in the metric setup. The central part of the paper is devoted to the problem of approximation on open sets; the authors give sufficient conditions which guarantee that a Sobolev function can be approximated by Lipschitz continuous functions vanishing outside an open set. These conditions are based on Hardy type inequalities. Moreover, it is shown that many classical results (completeness, lattice properties and removable sets) can be extended to the metric setting.
scientific article; zbMATH DE number 1455890