Extremal maps in best constants vector theory - Part I: Duality and Compactness

TL;DR

This paper investigates sharp potential type Riemannian Sobolev inequalities of order 2, focusing on duality, compactness, and the existence of extremal maps under weak regularity conditions.

Contribution

It introduces a comprehensive framework for analyzing extremal maps in potential type Sobolev inequalities, emphasizing duality and compactness with minimal regularity assumptions.

Findings

Established continuous dependence of optimal constants.

Proved existence of extremal maps.

Analyzed compactness properties under weak regularity.

Abstract

We develop a comprehensive study on sharp potential type Riemannian Sobolev inequalities of order 2 by means of a local geometric Sobolev inequality of same kind and suitable De Giorgi-Nash-Moser estimates. In particular we discuss questions like continuous dependence of optimal constants and existence and compactness of extremal maps. The main obstacle arising in the present setting lies at fairly weak conditions of regularity assumed on potential functions.