2024 FFT

The Relevance of the Shapiro-Lopatinsky Condition for Elliptic PDE’s in Nonsmooth Settings

Marius Mitrea (Baylor)

Abstract: The bedrock of the theory of elliptic boundary value problems in smooth domains is the notion of regular elliptic problem, in which the key requirement is that the corresponding differential operator satisfies the Shapiro-Lopatinsky condition relative to the type of boundary condition (Dirichlet, Neumann, etc.) considered. The upshot is that being regular elliptic guarantees, and in fact is equivalent to, the Fredholm solvability of the boundary value problem in question (formulated in Sobolev-Besov spaces) in smooth domains. In this talk, I will discuss recent progress in answering the longstanding open question as to whether the Shapiro-Lopatinsky remains relevant in nonsmooth settings.