Sobolev Spaces

Speaker: Dr. Behnam Esmayli (Visiting Assistant Professor, University of Cincinnati)

Lecture - 2

Title: Removable Sets for Sobolev Functions on Weighted R^n

Date: 11th June, 2025 (Wednesday)

Time: 06:30 PM - 07:30 PM (IST)

Abstract: Let the Euclidean space R^n be equipped with a doubling weight. Define H as the completion of smooth functions in the weighted Sobolov norm. Suppose E is a set of (weighted) measure zero. If $u$ is a Sobolev function on R^n\E then is $u$ a Sobolev function on R^n? If this is true for every $u$ then we say E is removable. The subtlety is that none of the smooth functions that approximate $u$ on R^n\E might be smooth on R^n. Removability is a difficult question even in the unweighted Sobolev theory. I will discuss a recent work of mine with Riddhi Mishra on when a set E is removable for Sobolev functions on R^n equipped with a weight that is doubling and satisfies a Poincare inequality.

Lecture - 1

Title: Sobolev Functions on Weighted R^n

Date: 4th June, 2025 (Wednesday) 

Time: 06:30 PM - 07:30 PM (IST)

Abstract: Problems in PDE (e.g. degenerate elliptic equations) and in geometry motivate and necessitate the study of R^n equipped with a measure other than the Lebesgue measure. What should we define as the class of Sobolev functions on this space? I discuss the intricacies lying behind this question and explain the two main classical approaches: the space W via the notion of weak differentiability and the space H as the closure of smooth functions. I will aim to convince you that H is the more natural choice, but I will review the literature on when H=W.