##### Department of Mathematics,

University of California San Diego

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### Math 248 - Analysis Seminar

## Jeffrey Case

#### Penn State University

## Sharp Sobolev trace inequalities via conformal geometry

##### Abstract:

Escobar proved a sharp Sobolev inequality for the embedding of $W^{1,2}(X^{n+1})$ into $L^{2n/(n-1)}(\partial X)$ by exploiting the conformal properties of the Laplacian in X and the normal derivative along the boundary. More recently, an alternative proof was given by using a Dirichlet-to-Neumann operator along the boundary and its close relationship to the 1/2-power of the Laplacian. In this talk, I describe a new relationship between the conformally covariant fractional powers of the Laplacian due to Graham--Zworski and higher-order Dirichlet-to-Neumann operators in the interior, and use it to prove sharp Sobolev inequalities for embeddings of $W^{k,2}$. Other consequences of this relationship, such as a surprising maximum principle for the conformal 3/2-power of the Laplacian, will also be discussed.

Hosts: Peter Ebenfelt and Sean Curry

### May 9, 2019

### 2:00 PM

### AP&M 7218

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