The Calderón approach to an elliptic boundary problem

We consider a boundary problem for an elliptic system in a bounded region Ω ⊂ ℝ n and where the spectral parameter is multiplied by a discontinuous weight function ω ( x ) = diag( ω 1 ( x ), …, ω N ( x )). The problem is considered under limited smoothness assumptions and under an ellipticity with parameter condition. Recently, this problem was studied under the assumption that the ω j ( x ) –1 are essentially bounded in Ω. In this paper we suppose that ω ( x ) vanishes identically in a proper subregion Ω   N   0of Ω and that the ω j ( x ) –1 are essentially bounded in . Then by using methods which are a variant of those used in constructing the Calderón projectors for the boundary Γ   N   0of Ω   N   0, we shall derive results here which will enable us in a subsequent work to apply the ideas of Calderón to develop the spectral theory associated with the problem under consideration here (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)