On non-coercive mixed problems for parameter-dependent elliptic operators

Abstract

We consider a non-coercive mixed boundary  value problem  in a bounded domain $D$ of ${\mathbb R}^n$ for a second order parameter-dependent elliptic differential operator $A (x,\partial, \lambda)$ 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 $B (x,\partial)$ is of Robin type with possible pseudo-differential components on $\partial D$. The boundary of $D$ is  assumed to be a Lipschitz surface. Under these assumptions the pair $(A (x,\partial, \lambda),B)$ induces a  holomorphic family of  Fredholm operators $L(\lambda): H^+(D) \to H^- (D)$ in suitable Hilbert spaces $H^+(D)$ ,  $H^- (D)$  of Sobolev type. If the argument of the complex-valued multiplier of the parameter in $A (x,\partial, \lambda)$ is continuous then we prove that the operators $L(\lambda)$ are continuously invertible for all $\lambda$ with sufficiently large modulus $|\lambda|$ on each ray on the complex plane $\mathbb C$ where the operator $A (x,\partial, \lambda)$ 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 $L^2 (D)$.