The Helmholtz Equation with $L_2$ Boundary Values

Abstract

This paper deals with the Helmholtz equation in exterior domains with Robin or impedance boundary conditions on a smooth $(C_{1,1} )$ boundary with data in $L_2$ on the boundary. Existence and uniqueness of solutions are established and a constructive method for finding the Green’s function is presented. This involves modifying the fundamental solution by adding suitable combinations of radiating solutions of the Helmholtz equation. The coefficients of these added terms for which the modified Green’s function best approximates the actual Green’s function for the problem are shown to be the elements of the T-matrix which arises in the null field or Waterman method for treating such problems. Use is made of complete families in $L_2$ on the boundary and existing results are extended' to the general case considered here.