On the best constant in a Hardy–Sobolev inequality

Abstract

In this article the question of finding the best constant in a Hardy–Sobolev inequality is addressed. The inequality under investigation gives the possibility to bound a weighted norm of a function u, with a weight that depends only on a group of variables, in terms of the L^p -norm of its gradient. A complete answer is obtained in a special case.

Keywords:

• Hardy–Sobolev inequalities
• Steiner symmetrization
• isoperimetric inequalities
• AMS Classifications: 26D10
• 47J30
• 51M16