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.