Weighted Hardy-Rellich type inequalities: improved best constants and symmetry breaking

TL;DR

This paper determines the sharp constants for weighted Hardy-Rellich inequalities with radial derivatives, showing improvements over previous bounds and identifying cases of symmetry breaking in extremal functions.

Contribution

It provides the exact sharp constants for these inequalities and demonstrates that extremal functions can lack radial symmetry in certain cases.

Findings

New sharp constants for weighted Hardy-Rellich inequalities.

Identification of cases with symmetry breaking in extremal functions.

Improved bounds for the inequalities with weighted radial derivatives.

Abstract

When studying the weighted Hardy-Rellich inequality in $L^2$ with the full gradient replaced by the radial derivative the best constant becomes trivially larger or equal than in the first situation. Our contribution is to determine the new sharp constant and to show that for some part of the weights is strictly larger than before. In some cases we emphasize that the extremals functions of the sharp constant are not radially symmetric.