Smith Colloquium

October 18, 2018

Ian Tice (Carnegie Mellon University)

Title: Trace operators for homogeneous Sobolev spaces in infinite strip-like domains

Abstract: Sobolev spaces are an indispensable tool in the modern theory of partial differential equations. Trace embeddings show that functions in Sobolev spaces, which are a priori defined as elements of \(L^p\) and hence are only defined almost everywhere, can actually be restricted to sufficiently regular hypersurfaces in a bounded way. Characterizing the resulting trace spaces and constructing bounded right inverses (lifting results) then plays an essential role in using Sobolev spaces to study boundary-value problems in PDE. The use of Sobolev spaces to study equations in unbounded, infinite-measure sets often requires employing homogeneous Sobolev seminorms, in which only the highest-order derivatives are controlled in \(L^p\). In this setting, the classical trace results may fail for certain choices of sets that appear naturally in PDE applications, such as infinite strip-like sets \( \mathbb{R}^{n-1}\times (0,b) \subset \mathbb{R}^n\). In this talk we will survey the classical theory and then turn to recent developments in the homogeneous trace theory. In particular, we will show that in strip-like sets the homogeneous trace spaces are characterized by a new type of fractional homogeneous Sobolev regularity and an interaction between the traces on the different connected components of the boundary.

Smith Colloquium Fall 2018