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The Helmholtz equation in heterogeneous media
Abstract
We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation ∇⋅(A∇u)+k 2nu=−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 C 0 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.