Sharp Nash Inequalities on the unit sphere. The influence of symmetries

TL;DR

This paper determines the optimal constants for Nash inequalities on the unit sphere, explores the existence of extremal functions, and analyzes the impact of symmetries under specific group actions.

Contribution

It establishes the best constants for Nash inequalities on the sphere and examines how symmetries influence these constants and extremal functions.

Findings

Best constants for Nash inequalities on the sphere are identified.

Existence of extremal functions is addressed.

Symmetry groups affect the best constants and extremal functions.

Abstract

In this paper both we establish the best constants for the Nash inequalities on the standard unit sphere $\mathbb{S}^n$ of $\mathbb{R}^{n+1}$ and we give answers on the existence of extremal functions on the corresponding problems. Also we study the problem of the best constants in the case, where the data are invariant under the action of the group $G=O(k)\times O(m)$, and we find the best constants.