Hardy–Rellich type inequalities in domains of the Euclidean space

For smooth functions supported in a domain of the Euclidean space we investigate two Hardy–Rellich type inequalities with weights which are powers of the distance to the boundary of the domain. Similar inequalities in convex domains were studied by P. L. Owen, M. G. Barbatis, A. Tertikas, W. D. Evans, R. T. Lewis and other mathematicians.
We prove that for an arbitrary plane domain there exists a positive constant in each of these inequalities if and only if the boundary of the domain is a uniformly perfect set. Moreover, we obtain explicit estimates of constants in function of geometric domain characteristics. Also, we find sharp constants in these Hardy–Rellich type inequalities for all convex domains and for non-convex domains of dimension $d\geq 2$ provided that the domains satisfy the exterior sphere condition with certain restriction on the radius of spheres.