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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:266

Article

Graham, I. G.¹  Pembery, O. R.¹  Spence, E. A.¹ 

[1] Univ Bath, Dept Math Sci, Bath BA2 7AY, Avon, England

关键词: Helmholtz equation;    Heterogeneous;    Variable wave speed;    High frequency;    Transmission problem;    Nontrapping;    Resolvent;    Uniqueness;    Resonance;    Semiclassical;   

DOI  :  10.1016/j.jde.2018.08.048

来源: Elsevier

【 摘 要 】

We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation del. (A del 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-infinity 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-infinity convex interfaces with strictly positive curvature. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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