Bulk-Boundary eigenvalues for Bilaplacian problems

TL;DR

This paper explores a novel bulk-boundary eigenvalue problem for the Bilaplacian with specific boundary conditions, analyzing eigenvalues in various geometries and discovering bifurcation phenomena related to domain size.

Contribution

It introduces a new eigenvalue problem for the Bilaplacian with boundary conditions from dynamical boundary problems, and analyzes eigenvalues in ball and annulus geometries.

Findings

Eigenvalues characterized by special function equations.

Bifurcation from zero eigenvalue depending on domain size.

Continuity properties under parameter variation.

Abstract

We initiate the study of a bulk-boundary eigenvalue problem for the Bilaplacian with a particular third order boundary condition that arises from the study of dynamical boundary conditions for the Cahn-Hilliard equation. First we consider continuity properties under parameter variation (in which the parameter also affects the domain of definition of the operator). Then we look at the ball and the annulus geometries (together with the punctured ball), obtaining the eigenvalues as solutions of a precise equation involving special functions. An interesting outcome of our analysis in the annulus case is the presence of a bifurcation from the zero eigenvalue depending on the size of the annulus.