Solution of the Dirichlet Problem for the Poisson's Equation in a Multidimensional Infinite Layer

The paper considers the multidimensional Poisson equation in the domain bounded by two parallel hyperplanes (in the multidimensional infinite layer). For an n-dimensional half-space method of solving boundary value problems for linear partial differential equations with constant coefficients is a Fourier transform to the variables in the boundary hyperplane. The same method can be used for an infinite layer, as is done in this paper in the case of the Dirichlet problem for the Poisson equation. For strip and infinite layer in three-dimensional space the solutions of this problem are known. And in the three-dimensional case Green's function is written as an infinite series. In this paper, the solution is obtained in the integral form and kernels of integrals are expressed in a finite form in terms of elementary functions and Bessel functions. A recurrence relation between the kernels of integrals for n-dimensional and (n + 2) -dimensional layers was obtained. In particular, is built the Green's function of the Laplace operator for the Dirichlet problem, through which the solution of the problem is recorded. Even in three-dimensional case we obtained new formula compared to the known. It is shown that the kernel of the integral representation of the solution of the Dirichlet problem for a homogeneous Poisson equation (Laplace equation) is an approximate identity (δ-shaped system of functions). Therefore, if the boundary values are generalized functions of slow growth, the solution of the Dirichlet problem for the homogeneous equation (Laplace) is written as a convolution of kernels with these functions.DOI: 10.7463/mathm.0415.081294