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The Calderón approach to an elliptic boundary problem. (English) Zbl 1216.35028

Summary: We consider a boundary problem for an elliptic system in a bounded region \(\Omega \in \mathbb R^n\) and where the spectral parameter is multiplied by a discontinuous weight function \(\omega (x) =\text{diag}(\omega _1(x),\dots ,\omega_N(x))\). 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 \(\omega _j(x)^{-1}\) are essentially bounded in \(\Omega\). In this paper we suppose that \(\omega (x)\) vanishes identically in a proper subregion \(\Omega _0\) of \(\Omega \) and that the \(\omega _j(x)^{-1}\) are essentially bounded in \(\Omega \backslash \overline {\Omega _{N_0}}\). Then by using methods which are a variant of those used in constructing the Calderón projectors for the boundary \(\Gamma _{N_0}\) of \(\Omega _{N_0}\), 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

MSC:

35J57 Boundary value problems for second-order elliptic systems
35P05 General topics in linear spectral theory for PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
47F05 General theory of partial differential operators
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