Abstract. Let λ ∗> 0 denote the largest possible value of λ such that 8 < ∆ 2 u = λe u

Dedicated to M. I. Vishik on occasion of his 80th birthday Abstract. We obtain several essential self-adjointness conditions for the Schrödingertype operator HV = D ∗ D + V, where D is a first order elliptic differential operator acting on the space of sections of a hermitian vector bundle E overa manifold M with positive smooth measure dµ, and V is a Hermitian bundle endomorphism. These conditions are expressed in terms of completeness of certain metrics on M naturally associated with HV. These results generalize the

The Hardy-Rellich inequality given here generalizes a Hardy inequality of Davies [2], from the case of the Dirichlet Laplacian of a region\Omega ` R N to that of the higher order polyharmonic operators with Dirichlet boundary conditions. The inequality yields some immediate spectral information for the polyharmonic operators and also bounds on the trace of the associated semigroups and resolvents.

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces W 1,p 0 and in higher-order Sobolev spaces on a bounded domain Ω ⊂ Rn can be refined by adding remainder terms which involve Lp norms. In the higher-order case further Lp norms with lower-order singular weights arise. The case 1 < p < 2 being more involved requires a different technique and is developed only in the space W 1,p 0. 1

We prove that the best constant for the critical embedding of higher order Sobolev spaces does not depend on all the traces. The proof uses a comparison principle due to Talenti [19] and an extension argument which enables us to extend radial functions from the ball to the whole space with no increase of the Dirichlet norm. Similar arguments may also be used to prove the very same result for Hardy-Rellich inequalities. AMS Classification: primary 46E35, secondary 26D10, 35J55 Keywords: optimal constant, Sobolev embedding, Hardy-Rellich inequality