The Helmholtz equation in heterogeneous media: A priori bounds, wellposedness, and resonances
Abstract
We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation ∇ ⋅ (A∇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^{∞} and satisfy certain monotonicity conditions, and thereby obtain new results both about the wellposedness 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 starshaped; 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.
 Publication:

Journal of Differential Equations
 Pub Date:
 March 2019
 DOI:
 10.1016/j.jde.2018.08.048
 arXiv:
 arXiv:1801.08095
 Bibcode:
 2019JDE...266.2869G
 Keywords:

 35J05;
 35J25;
 35B34;
 35P25;
 78A45;
 Mathematics  Analysis of PDEs;
 35J05;
 35J25;
 35B34;
 35P25;
 78A45