Explicit H2-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems

Abstract

It is well known that the H 2-norm and the C 0-norm of a function u ∈ H 2(Ω) (where Ω ⊂ R n is a bounded domain, n ⩽ 3) can be estimated in terms of a given uniformly elliptic second-order differential operator L and some boundary operator B applied to u, if certain regularity assumptions are satisfied. If these bounds shall be used for numerical purposes, the constants occurring in the estimates must be known explicitly. The main goal of the present article is the computation of such explicit constants. For simplicity of presentation, we restrict ourselves to the case where L[ u] = − Δu + c( x) u. As an application, we prove an existence and inclusion result for nonlinear boundary value problems.