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arXiv · 2204.04703

The eigenvalues and eigenfunctions of the non-linear equation associated to second order Sobolev embeddings

Lyonell Boulton·Jan Lang
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Abstract

We consider the non-linear eigenvalue equations characterizing $L^p$ into $L^q$ Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the extremal functions in the general case $(p,q)\in(1,\infty)^2$. Among other results, we show that these can be expressed in terms of those of related first order embeddings, if and only if $\frac{1}{p}+\frac{1}{q}=1$. Our findings shed new light on the surprising nature of higher order Sobolev spaces in the Banach space setting.

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Lyonell Boulton, Jan Lang. 2023-05-20. The eigenvalues and eigenfunctions of the non-linear equation associated to second order Sobolev embeddings. https://arxiv.org/abs/2204.04703

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