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 $\nabla \cdot (A\nabla u)+k^{2}nu=-f$ 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 $L^{\infty }$ 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 $C^{\infty }$ convex interfaces with strictly positive curvature.