2018/01/24 by Graham, Ivan G., Pembery, Owen R., Spence, Euan A. · 3 citations
#35B34 #35J05 #35J25 #35P25 #78A45 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1801.08095
We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation ∇⋅(A ∇ u ) + k2 n u =-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.