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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 $$L^2(D)$$.
##### MSC:
 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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