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

We consider the Sobolev space over ℝ^d of square integrable functions whose gradient is also square integrable with respect to some positive weight. It 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.