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

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. 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.

AMS SUBJECT CLASSIFICATION:

• 60J60

COMMUNICATED BY:

• Andrey Piatnitski

KEYWORDS:

• Weighted Sobolev spaces
• Dirichlet form theory
• non-regular Dirichlet form
• non-regular weights
• symmetric extensions

Acknowledgments

The authors would like to thank Professor M. Fukushima for his remarks and for showing interest in the paper. The first author would like to thank the Laboratoire d'Excellence LabEx Archimède for supporting him financially in Marseille where this research was carried out. Finally, but most importantly, the authors would like to thank V.V. Zhikov as the present work was directly inspired by [3]. Not long before Zhikov passed away, he was informed about this project and was kind enough to encourage us to complete it. Vasily Vasilyevich's contribution to analysis is so rich that there is no doubt it will continue to be a source of inspiration for us and many generations of mathematicians.

Disclosure statement

No potential conflict of interest was reported by the authors.