Explicit optimal constants of two critical Rellich inequalities for radially symmetric functions

TL;DR

This paper determines explicit optimal constants for two critical Rellich inequalities involving radially symmetric functions in higher order Sobolev spaces, and discusses whether these constants are attainable.

Contribution

It provides explicit values of the optimal constants for these inequalities and analyzes their attainability in the context of radial Sobolev spaces.

Findings

Explicit optimal constants for two critical Rellich inequalities are derived.

The attainability of these optimal constants is analyzed.

Results contribute to the understanding of inequalities with singularities in higher order Sobolev spaces.

Abstract

We consider two critical Rellich inequalities with singularities at both the origin and the boundary in the higher order critical radial Sobolev spaces $W_{0, {\rm rad}}^{k, p}$, where $1< p = \frac{N}{k}$. We give the explicit values of the optimal constants of two critical Rellich inequalities for radially symmetric functions in $W_{0, {\rm rad}}^{k, \frac{N}{k}}$. Furthermore the (non-)attainability of the optimal constants are also discussed.