On the Dirichlet Problem Generated by the Maz'ya--Sobolev Inequality

TL;DR

This paper investigates the conditions under which the best constants in Maz'ya--Sobolev inequalities are achieved in various geometric domains, including wedges and bounded regions.

Contribution

It provides new insights into the attainability of sharp constants for Maz'ya--Sobolev inequalities across different domain types.

Findings

Attainability of sharp constants in wedges and perturbed wedges.

Conditions for attainability in bounded domains.

Analysis of the influence of domain geometry on inequality constants.

Abstract

We discuss the attainability of sharp constants for the Maz'ya--Sobolev inequalities in wedges, "perturbed" wedges and bounded domains.