The Method of Potential Operators for Anisotropic Helmholtz Operators on Domains with Smooth Unbounded Boundaries

Abstract

The paper is devoted to the method of potential operators for boundary and transmission problems in domains in $\mathbb{R}^n$ with smooth unbounded boundaries for the anisotropic Helmholtz operators

$$\mathcal{H}u(x)=\bigtriangledown \cdot a(x)\bigtriangledown u(x)+b(x)u(x),\quad x\in \mathbb{R}^N$$

with variable coefficients, where $a\;=\;(a^{k,l})^n_{k,l=1}$ are real-valued symmetric matrices on$\mathbb{R}^n$ with entries $a^{k,l}\in C^\infty_{b}(\mathbb{R}^n),k,l=1,\ldots,n$. We assume that the operator $\mathcal{H}$ is strongly elliptic, $b(x)=\omega^2b_0(x), \omega>0$ s the frequency of the harmonic vibrations, $b_0(x)$ is the refractive index which satisfies the following conditions: $b_0\in C^\infty_{b}(\mathbb{R}^n)$,

$$\mathfrak{R}b_0(x)>0,\quad \mathcal{J}b_0(x)\geq0,\quad x\in \mathbb{R}^n,\quad \liminf_{\mathbb{R}^n\ni x\rightarrow\infty}\mathcal{J}b_0(x)>0. $$

We introduce single and double layer potentials associated with the operator $\mathcal{H}$, and reduce by means of these potentials the Dirichlet, Neumann, Robin, and transmission problems for a domain with unbounded smooth boundary $\partial D$ to pseudodifferential equations on $\partial D$. Applying the method of limit operators, we study Fredholm properties and the invertibility of the boundary pseudodifferential operators in the Sobolev spaces $H^s(\partial D),\;s \in \mathbb{R}$.

Dedicated to my friend and colleague Prof. Roland Duduchava on the occasion of his 70th birthday

Mathematics Subject Classification (2010). Primary 35J25; Secondary 35Q60.