In this paper we will derive an non-local (``integral'') equation which transforms a three-dimensional acoustic transmission problem with \emph{variable} coefficients, non-zero absorption, and mixed boundary conditions to a non-local equation on a ``skeleton'' of the domain $\Omega\subset\mathbb{R}^{3}$, where ``skeleton'' stands for the union of the interfaces and boundaries of a Lipschitz partition of $\Omega$. To that end, we introduce and analyze abstract layer potentials as solutions of auxiliary coercive full space variational problems and derive jump conditions across domain interfaces. This allows us to formulate the non-local skeleton equation as a \emph{direct method} for the unknown Cauchy data of the solution of the original partial differential equation. We establish coercivity and continuity of the variational form of the skeleton equation based on auxiliary full space variational problems. Explicit expressions for Green's functions is not required and all our estimates are \emph{explicit} in the complex wave number.

翻译：本文推导了一种非局部（“积分”）方程，将三维变系数、非零吸收及混合边界条件的声学透射问题转化为定义在区域Ω⊂ℝ³的“骨架”（即Ω的Lipschitz划分中所有交界面与边界的并集）上的非局部方程。为此，我们引入并分析了作为辅助强制全空间变分问题解的抽象层势，并推导出跨区域交界面的跳跃条件。这使得我们能够将非局部骨架方程表述为原偏微分方程未知柯西数据的直接方法。基于辅助全空间变分问题，我们建立了骨架方程变分形式的强制性与连续性。该方法无需显式格林函数表达式，且所有估计均在复波数意义下显式给出。