Singular weighted Sobolev spaces and diffusion processes: an example (due to V.V. Zhikov)

TL;DR

This paper explores the properties of weighted Sobolev spaces with singular weights, demonstrating the failure of smooth function density and constructing associated diffusion processes using Dirichlet forms theory.

Contribution

It provides a detailed analysis of a class of two-dimensional weighted Sobolev spaces with singularities and links them to diffusion processes via Dirichlet forms.

Findings

Smooth functions are not dense in certain weighted Sobolev spaces.

A detailed analytical description of these spaces is provided.

Diffusion processes can be associated with degenerate non-regular spaces.

Abstract

We consider the Sobolev space over $\mathbb{R}^d$ of square integrable functions whose gradient is also square integrable with respect to some positive weight. Tt is well known that smooth functions are dense in the weighted Sobolev space when the weight is uniformly bounded from below and above. This may not be the case when the weight is unbounded. In this paper, we focus on a class of two dimensional weights where the density of smooth functions does not hold. This class was originally introduced by V.V. Zhikov; such weights have a unique singularity point of non-zero capacity. Following V.V. Zhikov, we first give a detailed analytical description of the weighted Sobolev space. Then, we explain how to use Dirichlet forms theory to associate a diffusion process to such a degenerate non-regular space.