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PADS Seminar: Trace Theorems for Non-Differentiable Functions

Author: Lona | Image: Lona

Speaker: Mikil Foss, University of Nebraska-Lincoln

Abstract: Given a uniformly continuous function on an open domain, there is a unique extension to the boundary that preserves the continuity. The trace operator provides a function that captures the boundary values for this extension. Gagliardo’s trace theorem extends this concept to the Sobolev spaces. There have been generalizations of Gagliardo’s theorem in many directions. Typically, trace theorems require some differentiability of the function in its domain and some regularity of the domain’s boundary. These assumptions ensure a there is a well-defined boundary value function. Moreover, this trace will, itself, possess some differentiability and a certain Lebesgue point property. I will present a trace theorem that provides a well-defined boundary-value function that exists in a fractional Sobolev space and has the Lebesgue point property yet requires no differentiability within the domain and allows very irregular boundaries. The result is motivated by boundary-value problems involving nonlocal operators that are defined for integrable but not necessarily differentiable functions.