Hardy Inequalities for Nonconvex Domains

Abstract

We obtain a series of Hardy type inequalities for domains involving both distance to the boundary and distance to the origin. In particular, we obtain the Hardy─Sobolev inequality for the class of symmetric functions in a ball and prove that for d ≥ 3 the Hardy inequality involving the distance to the boundary holds with the constant 1/4 in a large family of domains not necessarily convex. We also present an example showing that for any positive fixed constant there is an ellipsoid layer such that the Hardy inequality with the distance to the boundary fails.