MIXED BOUNDARY VALUE PROBLEMS FOR THE HELMHOLTZ EQUATION

Abstract

We consider a new approach to investigating the mixed boundary value problem (BVP) for the Helmholtz equation in the case of a three-dimensional unbounded domain ${\mathrm{\Omega }}^{-}\subset {ℝ}^3$ with compact boundary surface $S=\partial {\mathrm{\Omega }}^{-}$, which is divided into two disjoint parts, $S_D$ and $S_N$, where the Dirichlet and Neumann type boundary conditions are prescribed respectively. Our approach is based on the classical potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP to a system of integral equations containing neither extensions of the Dirichlet or Neumann data nor a Steklov–Poincaré type operator involving the inverse of the single layer boundary integral operator, which is not available explicitly for arbitrary boundary surfaces. The right-hand sides of the resulting boundary integral equations system are functions coinciding with the given Dirichlet and Neumann data of the problem under consideration. We show that the corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate $L_2$-based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the class of functions belonging to the Sobolev space $W_{2,\mathrm{\text{ loc}}}^1({\mathrm{\Omega }}^{-})$ and satisfying the Sommerfeld radiation conditions. We also show that the pseudodifferential matrix operator thus obtained is invertible in the $L_p$-based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses the $C^{\alpha }$-Hölder continuity property in the closed domain ${\overline{\mathrm{\Omega }}}^{-}$ with $\alpha =\frac{1}{2}-𝜀$, where $𝜀>0$ is an arbitrarily small number.