Davies E.B., Hinz A.M.
Explicit constants for Rellich inequalities in $L^p(\Omega)$
(33K, Latex)
ABSTRACT. We obtain explicit and sometimes sharp constants for weighted Rellich
and Hardy inequalities in $L_p(\Omega)$ where $\Omega$ stands
for Euclidean space or a bounded region in a complete Riemannian manifold. The
application of Rellich inequalities to the study of relative bounds of a
potential with
respect to $(-\Delta)^m$ in $L_p(\Omega)$ is facilitated by showing the purely
local character of the determination of these bounds. The proof for this fact
is based on a special partition of unity. The final section is then devoted
to a detailed analysis of potentials with local singularities, and best
constants for Rellich and Hardy inequalities are obtained.