On the completeness of root functions of a holomorphic family of non-coercive mixed problem

Abstract

We consider a non-coercive mixed boundary value problem in a bounded domain D of complex space for a second order parameter-dependent elliptic differential operator with complex-valued essentially bounded measured coefficients and complex parameter . The differential operator is assumed to be of divergent form in D, the boundary operator 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 in suitable Hilbert spaces , of Sobolev type (here 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 in then we prove that the operators are continuously invertible for all with sufficiently large modulus on each angle on the complex plane where the operator is parameter-dependent elliptic. We also describe reasonable conditions for the system of root functions related to the family to be (doubly) complete in the spaces , and the Lebesgue space .