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On the completeness of root functions of a holomorphic family of non-coercive mixed problem. (English) Zbl 1344.32011
Summary: We consider a non-coercive mixed boundary value problem in a bounded domain \(D\) of complex space \(\mathbb{C}^n\) for a second order parameter-dependent elliptic differential operator \(A(z,\overline{\partial},\lambda)\) with complex-valued essentially bounded measured coefficients and complex parameter \(\lambda\). The differential operator is assumed to be of divergent form in \(D\), the boundary operator \(B(z,\overline{\partial})\) is of Robin type. The boundary of \(D\) is assumed to be a Lipschitz surface. Under reasonable assumptions the pair (\(A,B\)) induces a family of non-coercive mixed problems and a holomorphic family of Fredholm operators \(L(\lambda):H^+(D)\to H^-(D)\) in suitable Hilbert spaces \(H^+(D)\subset H^{1/2}(D)\), \(H^{-1/2}(D)\subset H^-(D)\) of Sobolev type (here \(H^S(D)\) are the Sobolev-Slobodetskii spaces over \(D\)). If there is a Lipschitz function close enough to the (possibly discontinuous) argument of the complex-valued multiplier of the parameter \(\lambda\) in \(A(z,\overline{\partial},\lambda)\) then we prove that the operators \(L(\lambda)\) are continuously invertible for all \(\lambda\) with sufficiently large modulus \(|\lambda|\) on each angle on the complex plane \(\mathbb{C}\) where the operator \(A(z,\overline{\partial},\lambda)\) is parameter-dependent elliptic. We also describe reasonable conditions for the system of root functions related to the family \(L(\lambda)\) to be (doubly) complete in the spaces \(H^+(D)\), \(H^-(D)\) and the Lebesgue space \(L^2(D)\).
32W50 Other partial differential equations of complex analysis in several variables
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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