Boundary Value Problems for the Helmholtz Equation in an Octant

Abstract.

We consider a class of boundary value problems for the three-dimensional Helmholtz equation that appears in diffraction theory. On the three faces of the octant, which are quadrants, we admit first order boundary conditions with constant coefficients, linear combinations of Dirichlet, Neumann, impedance and/or oblique derivative type. A new variety of surface potentials yields 3 × 3 boundary pseudodifferential operators on the quarterplane ${\mathbb{R}}^{2}_{++}$ that are equivalent to the operators associated to the boundary value problems in a Sobolev space setting. These operators are analyzed and inverted in particular cases, which gives us the analytical solution of a number of well-posed problems.