A haracterization of closure of the set of compactly supported functions in Dirichlet generalized integral metric and its applications

Abstract

We obtain conditions under which a function $u(x)$ with finite Dirichlet ge e alized integral over a domain $G$ ($u(x)\in H(G))$ belongs to the closure of the set $C_0^\infty(G)$ in the metrics of this Dirichlet integral (i.e., to the space $H_0(G)$). In the case where $G=R^n \;(n \geq 2)$ using these conditions we construct examples of Dirichlet integrals such that $H(R^n) \neq H_0(R^n)$. For $n=2$ these examples show that in the known Mazia theorem uniform positivity of the Dirichlet integral matrix cannot be replaced with its pointwise positivity. The characterization of the space $H_0(G)$ is also applied to the problem of relative equivalence of the spaces $H(G)$ and $H_0(G)$ concerning the part of the boundary $\Gamma (\Gamma\subseteq \partial G)$. This problem in fact coincides with the problem of possibility to set boundary conditions of corresponding boundary-value problems.