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

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.