Resonances and Eigenvalues for the Constant Mean Curvature Equation

TL;DR

This paper investigates the spectral properties of the linearized constant mean curvature equation around bubble solutions, identifying resonances and eigenvalues, with implications for dispersive estimates in related wave equations.

Contribution

It provides the first analysis of eigenvalues for higher degree bubbles in the constant mean curvature equation, extending previous work on resonances only.

Findings

Resonances are found for degree one bubbles.

Eigenvalues are proven to occur for higher degree bubbles.

Results set the stage for dispersive estimates in wave equations related to H-systems.

Abstract

In this paper we study resonances and eigenvalues for the nonlinear constant mean curvature eqn. linearized around the bubbles found by Brezis-Coron. This nonlinear eqn. is also called a H-system eqn. For degree one bubbles(the degree relates to a certain winding number) we only find resonances. For higher degree we prove eigenvalues do occur. Our goal is to eventually obtain dispersive estimates for the wave eqn. associated to the H-systems eqn. in its linear and non-linear form, a study of which was initiated by Chanillo-Yung.