Integral Equations of Diffraction Problems with Unbounded Smooth Obstacles

Abstract

The paper is devoted to the boundary integral equations method for the diffraction problems on obstacles D in \({\mathbb{R}^{n}}\) with smooth unbounded boundaries for Helmholtz operators with variable coefficients. The diffraction problems are described by the Helmholtz operators $$\mathcal{H}u(x)=\left( \rho (x)\nabla \cdot \rho ^{-1}(x)\nabla+a(x)\right) u(x),\quad x\in \mathbb{R}^{n}$$ where \({\rho, a}\) belong to the space of the infinitely differentiable functions on \({\mathbb{R}^{n}}\) bounded with all derivatives. We introduce the single and double layer potentials associated with the operator \({ \mathcal{H}}\), and reduce by means of these potentials the Dirichlet, Neumann, and Robin problems to pseudodifferential equations on the infinite boundary \({\partial D}\). Applying the limit operators method we study the Fredholm properties and the invertibility of the boundary pseudodifferential operators in the Sobolev spaces \({H^{s}(\partial D),s\in \mathbb{R}}\).