Higher order Dirichlet type problems for the two-dimensional Helmholtz operator in complex quaternionic analysis

Abstract

It is well known that the development of methods for solving Dirichlet problems is important and relevant for various areas of mathematical physics related to the Laplace equation, the Helmholtz equation, the Stokes equation, the Maxwell equation, the Dirac equation, and others. In the previous work, the author studied the solvability of Dirichlet problem of the first order and the second order in the quaternion analysis. In the present paper studies the Dirichlet boundary value problem of high order, associated with the two-dimensional Helmholtz equation with complex potential. In this paper, the existence and uniqueness and a representation formula for the solution of Dirichlet boundary value problems in the two-dimensional case are proved. Most of Dirichlet boundary value problems are focused to the 3D case. Note that the case is not a simple consequence of the three-dimensional case. For solving the problem, we use the method of orthogonal decomposition of the quaternion Sobolev space. This orthogonal decomposition of space is also the basis for the study of many elliptic boundary value problems that are in various areas of mathematics and mathematical physics. Orthogonal decompositions of the quaternionic-valued Sobolev space with respect to the Dirac operator of high order as well as the corresponding orthoprojections onto the subspaces of theses decompositions are obtained.