Mathematical Sciences - Colloquium - "Sharp Sobolev trace inequalities via conformal geometry" by Jeffrey Case (Penn State)

Friday, September 28, 2018
11:00 am to 12:00 pm


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Speaker: Jeffrey Case (Penn State)

Title:  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 principal for the conformal 3/2- power of the Laplacian, will also be discussed.
Friday, September 28, 2018 11:00AM - 12:00PM Stratton Hall 203