Skeleton Integral Equations for Acoustic Transmission Problems with Varying Coefficients

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.