The Helmholtz equation in heterogeneous media: A priori bounds, well-posedness, and resonances
The Helmholtz equation in heterogeneous media: A priori bounds, well-posedness, and resonances
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DOI：
10.1016/j.jde.2018.08.048
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发表时间：
2018-01
影响因子：
2.4
通讯作者：
I. Graham;O. R. Pembery;E. Spence
中科院分区：
文献类型：
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作者：
I. Graham;O. R. Pembery;E. Spence
考虑非齐次Helmholtz方程的Dirichlet外问题，即方程K2（A <$u）+k2 nu =− f，其中A和n都是位置的函数.我们证明了新的先验界的解决方案的条件下，在A，n，和域，确保非捕获的射线;新奇是，这些界限是明确的k，A，n，和几何参数的域。当A和n为L∞且满足一定的单调性条件时，我们证明了这些先验界成立，从而得到了关于这类问题的适定性和声传输问题的共振性的新结果（即A和n不连续），其中传输接口仅假设为C 0和星形;后一结果的新奇在于，直到最近，关于声传输问题的共振的唯一已知结果是对于具有严格正曲率的C∞凸界面。
We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, ie the equation∇⋅(A∇ u)+ k 2 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 (ie 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.