Optimal Rellich-Sobolev constants and their extremals

TL;DR

This paper investigates the extremals of Rellich-Sobolev inequalities, proving they have constant sign and showing that the optimal constants are independent of the domain under Navier boundary conditions.

Contribution

It establishes the sign property of extremals and demonstrates the domain-independence of optimal constants in Rellich-Sobolev inequalities.

Findings

Extremals for second order Rellich-Sobolev inequalities have constant sign.

Optimal constants are independent of the domain under Navier boundary conditions.

Abstract

We prove that extremals for second order Rellich-Sobolev inequalities have constant sign. Then we show that the optimal constants in Rellich-Sobolev inequalities on a bounded domain {\Omega} and under Navier boundary conditions do not depend on {\Omega}