The Helmholtz equation in heterogeneous media: A priori bounds, well-posedness, and resonances

Abstract

We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation ∇⋅(A∇u)+k2nu=−f where both A and n are functions of position. We prove new a priori bounds on the solution under conditions on A, n, and the domain that ensure nontrapping of rays; the novelty is that these bounds are explicit in k, A, n, and geometric parameters of the domain. We then show that these a priori bounds hold when A and n are L∞ and satisfy certain monotonicity conditions, and thereby obtain new results both about the well-posedness of such problems and about the resonances of acoustic transmission problems (i.e. A and n discontinuous) where the transmission interfaces are only assumed to be C0 and star-shaped; the novelty of this latter result is that until recently the only known results about resonances of acoustic transmission problems were for C∞ convex interfaces with strictly positive curvature.