Extremal functions for the sharp $L^2-$ Nash inequality

TL;DR

This paper establishes geometric conditions that guarantee the existence of extremal functions for the sharp $L^2$-Nash inequality, contributing to the understanding of functional inequalities in geometric analysis.

Contribution

It identifies specific geometric conditions ensuring extremal functions exist for the sharp $L^2$-Nash inequality, advancing the theoretical framework of functional inequalities.

Findings

Geometric conditions for extremal functions established

Existence of extremal functions proven under these conditions

Enhances understanding of the $L^2$-Nash inequality

Abstract

We give geometrical conditions under which there exist extremal functions for the sharp $L^2$-Nash inequality.