Boundary Triples in Banach Spaces and Applications to Boundary Value Problems

The choice of boundary conditions plays a decisive role in the analysis of elliptic operators on (smooth) domains. For elliptic boundary value problems in Lp-spaces, classical conditions such as the Lopatinskii–Shapiro condition provide powerful criteria ensuring sectoriality of the associated realizations.

In this talk, we focus on situations in which such classical elliptic regularity is not available. Motivated by these cases, we introduce a notion of boundary triples in Banach spaces that is inspired by the theory of boundary triples for dual pairs in Hilbert spaces. After establishing basic structural properties of this framework, we prove a Krein-type resolvent formula. This formula allows one to characterize sectoriality of abstract boundary value problems explicitly in terms of parameter-dependent estimates for the boundary operators.

We then apply the abstract results to elliptic boundary value problems that fail to satisfy the classical Lopatinskii–Shapiro condition.

This is joint work with Robert Denk and Amru Hussein.