Morse Index and Symmetry for Elliptic Problems with Nonlinear Mixed Boundary Conditions

TL;DR

This paper establishes symmetry results for solutions to nonlinear elliptic problems with mixed boundary conditions by relating the Morse index to eigenvalues of a linear problem, advancing understanding of solution properties.

Contribution

It introduces a novel connection between the Morse index and eigenvalues in mixed boundary elliptic problems, providing new tools for analyzing solution symmetry.

Findings

Symmetry results for solutions under Morse index conditions

Construction of eigenvalue sequences for mixed boundary problems

Variational characterization of eigenvalues

Abstract

Under a Morse index condition we prove symmetry results for solutions of a nonlinear mixed boundary condition elliptic problem. As an intermediate step we relate the Morse index of a solution to a mixed boundary condition linear eigenvalue problem for which we construct sequences of eigenvalues and provide variational characterization of them.