Eigenvalues for radially symmetric non-variational fully nonlinear operators

TL;DR

This paper develops a simplified theory for eigenvalues and eigenfunctions of radially symmetric, fully nonlinear, 1-homogeneous operators, characterizing them by the number of zeroes, contrasting with more complex viscosity solution frameworks.

Contribution

It introduces a straightforward approach to determine all eigenvalues and eigenfunctions for radially symmetric 1-homogeneous fully nonlinear operators, based on zeroes count.

Findings

Complete set of eigenvalues characterized by zeroes

Simplified theory for radially symmetric operators

Eigenfunctions explicitly described by zeroes

Abstract

In this paper we present an elementary theory about the existence of eigenvalues for fully nonlinear radially symmetric 1-homogeneous operators. A general theory for first eigenvalues and eigenfunctions of 1-homogeneous fully nonlinear operators exists in the framework of viscosity solutions. Here we want to show that for the radially symmetric operators (and one dimensional) a much simpler theory can be established, and that the complete set of eigenvalues and eigenfuctions characterized by the number of zeroes can be obtained.