Hardy–Rellich inequalities in domains of the Euclidean space

Abstract

For test functions supported in a domain of the Euclidean space we consider the Hardy–Rellich inequality: ∫|Δf|2dx≥C2∫|f|2δ−4(x)dx, where C2=const≥0 and δ(x) is the distance from x to the boundary of the domain. M.P. Owen proved that this inequality is valid in any convex domain with C2=9/16 (M.P. Owen (1999) [21]). We examine the inequality in non-convex domains. It is proved that a positive constant C2 for a plane domain exists if and only if its boundary is a uniformly perfect set. For a domain of dimension d≥2 we prove that the inequality holds with the sharp constant C2=9/16, if the domain satisfies an exterior sphere condition with certain restriction on the radius of the sphere. In addition, we obtain similar results for the inequality ∫δ2(x)|Δf|2dx≥C2⁎∫|f|2δ−2(x)dx.