arXiv 1801.080952018-01math.AP

非均匀介质中的Helmholtz方程：先验界、适定性和共振

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

Ivan G. Graham, Owen R. Pembery, Euan A. Spence

摘要：

本文考虑非齐次Helmholtz方程的Dirichlet外问题，即方程$\nabla\cdot(A \nabla u ) + k^2 n u =-f$，其中$A$和$n$都是位置函数.我们证明了新的先验界的解决方案的条件下，$A$，$n$，和域，确保非捕获的射线;新颖的是，这些界限是明确的$k$，$A$，$n$，和几何参数的域。当$A$和$n$是$L^\infty$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），其中传输接口仅被假设为$C^0$和星形;后一结果的新颖性在于，直到最近，关于声传输问题的共振的唯一已知结果是针对具有严格正曲率的$C^\infty$凸界面的。

英文摘要：

We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation $\nabla\cdot(A \nabla u ) + k^2 n u =-f$ 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 $L^\infty$ 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.

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摘要：

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