Degeneracy Results for Fully Nonlinear Integral Operators

TL;DR

This paper demonstrates that fully nonlinear integral operators exhibit degeneracy phenomena similar to superposition operators, being Fréchet differentiable only under affine conditions, with implications for complex function spaces and models like coupled oscillators.

Contribution

It extends degeneracy results to a broad class of nonlinear integral operators and function spaces, including vector functions and infinite measure spaces.

Findings

Integral operators are Fréchet differentiable only if affine in the argument.

Degeneracy results apply under local Lipschitz or compactness conditions.

Operators in coupled Kuramoto oscillators are non-differentiable and non-compact.

Abstract

It is shown that integral operators of the fully nonlinear type $K(x)(t)=\int_\Omega k(t,s,x(t),x(s))\,ds$ exhibit similar degeneracy phenomena in a large class of spaces as superposition operators $F(x)(t)=f(t,x(t))$. In particular, $K$ is Fr\'echet differentiable in $L_p$ only if it is affine with respect to the "$x(t)$" argument. Similar degeneracy results hold if $K$ satisfies a local Lipschitz or compactness condition. Also vector functions, infinite measure spaces, and a much richer class of function spaces than only $L_p$ are considered. As a side result, degeneracy assertions for superposition operators are obtained in this more general setting, complementing the known results for scalar functions. As a particular example, it is shown that the operators arising in continuous limits of coupled Kuramoto oscillators fail everywhere to be Fr\'echet differentiability or locally compact.