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

TL;DR

This paper analyzes the eigenvalues and eigenfunctions related to second order Sobolev embeddings with Navier boundary conditions, providing a complete description and linking higher order cases to first order embeddings.

Contribution

It offers a comprehensive characterization of s-numbers and extremal functions for second order Sobolev embeddings, revealing conditions under which they relate to first order cases.

Findings

Eigenvalues and eigenfunctions are explicitly described.

The relation to first order embeddings holds if and only if 1/p + 1/q = 1.

Provides new insights into the structure of higher order Sobolev spaces.

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.