We prove Rellich and improved Rellich inequalities that involve the distance function from a hypersurface of codimension $k$, under a certain geometric assumption. In case the distance is taken from the boundary, that assumption is the convexity of t
he domain. We also discuss the best constant of these inequalities.
We present a unified approach to improved $L^p$ Hardy inequalities in $R^N$. We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where distance is taken from a s
urface of codimension $1<k<N$. In our main result we add to the right hand side of the classical Hardy inequality, a weighted $L^p$ norm with optimal weight and best constant. We also prove non-homogeneous improved Hardy inequalities, where the right hand side involves weighted L^q norms, q eq p.
We study the wave equation in the exterior of a bounded domain $K$ with dissipative boundary condition $partial_{ u} u - gamma(x) u = 0$ on the boundary $Gamma$ and $gamma(x) > 0.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}
, : t geq 0.$ The eigenvalues $lambda_k$ of $G$ with ${rm Re}: lambda_k < 0$ yield asymptotically disappearing solutions $u(t, x) = e^{lambda_k t} f(x)$ having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case $min_{xin Gamma} gamma(x) > 1.$ For strictly convex obstacles $K$ this formula concerns all eigenvalues of $G.$
We consider a general class of sharp $L^p$ Hardy inequalities in $R^N$ involving distance from a surface of general codimension $1leq kleq N$. We show that we can succesively improve them by adding to the right hand side a lower order term with optim
al weight and best constant. This leads to an infinite series improvement of $L^p$ Hardy inequalities.
In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals $((p-1)/p)^p$ whenever Dirichlet boundary conditions are
imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.