Sobolev embeddings and distance functions

TL;DR

This paper explores the relationship between Sobolev space embeddings and distance function properties on open sets, revealing equivalences and counterexamples across different cases, and applies these findings to analyze solutions of the Lane-Emden equation and eigenfunctions.

Contribution

It establishes the equivalence between Sobolev embeddings and distance function summability in the superconformal case and provides counterexamples in other cases, also studying asymptotic behaviors of solutions.

Findings

Equivalence between Sobolev embedding and distance function summability in superconformal case.

Counterexamples showing non-equivalence in subconformal and conformal cases.

Asymptotic analysis of Lane-Emden solutions and eigenfunctions as p diverges.

Abstract

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space $\mathcal{D}^{1,p}_0$ into $L^q$ and the summability properties of the distance function. We prove that in the superconformal case (i.e. when $p$ is larger than the dimension) these two facts are equivalent, while in the subconformal and conformal cases (i.e. when $p$ is less than or equal to the dimension) we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behaviour of the positive solution of the Lane-Emden equation for the $p-$Laplacian with sub-homogeneous right-hand side, as the exponent $p$ diverges to $\infty$. The case of first eigenfunctions of the $p-$Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies.