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arXiv · 1908.09934

Degeneracy Results for Fully Nonlinear Integral Operators

Martin Väth
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Abstract

It is shown that integral operators of the fully nonlinear type $K(x)(t)=\int_Ωk(t,s,x(t),x(s))\,ds$ exhibit similar degeneracy phenomena in a large class of spaces as superposition operators $F(x)(t)=f(t,x(t))$. In particular, $K$ is Fréchet 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 Fréchet differentiability or locally compact.

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Martin Väth. 2019-08-26. Degeneracy Results for Fully Nonlinear Integral Operators. https://arxiv.org/abs/1908.09934

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