Bi-Laplacians on graphs: self-adjoint extensions and parabolic theory

Abstract: Elastic beams have been studied by means of hyperbolic equations driven by bi-Laplacian operators since the early 18th century: several properties of the corresponding parabolic equation on the whole Euclidean space have been discovered since the 1960s by Krylov, Hochberg, and Davies, among others. On a bounded domain or a metric graph, the bi-Laplacian is generally not anymore acting as a squared operator, though: this strongly affects the features of associated PDEs.

I am going to present a full characterization of self-adjoint extensions of the bi-Laplacian, focusing on a class of realizations that encode dynamic boundary conditions. Maximum principles of parabolic equations will also be discussed: after a transient time, I am going to show that solutions often display Markovian features.

This is joint work with Federica Gregorio.

mathematical physicsanalysis of PDEs

Audience: researchers in the topic

TAMU: Mathematical Physics and Harmonic Analysis Seminar