On symmetry of extremals in several embedding theorems

TL;DR

This paper investigates the symmetry properties of extremal functions in certain embedding theorems for Sobolev spaces, providing explicit calculations of sharp constants for specific cases.

Contribution

It offers new insights into the symmetry of extremals and explicitly computes sharp constants for particular embedding cases in Sobolev spaces.

Findings

Symmetry and asymmetry of extremal functions are characterized.

Explicit sharp constants are calculated for cases r>k with k=4 and k=6.

Results contribute to understanding extremal functions in Sobolev embedding theorems.

Abstract

We study the symmetry/asymmetry of functions providing sharp constants in the embedding theorems ${\stackrel{\circ}{W}}\vphantom{W}_2^r(-1,1)\hookrightarrow{\stackrel{\circ}{W}}\vphantom{W}_\infty^k(-1,1)$ for various $r$ and $k$. The sharp constants for all $r>k$ in the cases $k=4$ and $k=6$ are calculated explicitly as well.