Dirichlet-Type Problems for the Two-Dimensional Helmholtz Operator in Complex Quaternionic Analysis

Abstract

This study aims to study a class of Dirichlet-type problems associated with the two-dimensional Helmholtz equation with complex potential. Orthogonal decompositions of the complex quaternionic-valued Sobolev space as well as the corresponding orthoprojections onto the subspaces of theses decompositions are obtained. Analytic representation formulas for the underlying solutions in terms of hypercomplex integral operators are established.