ON CORRECT SOLVABILITY OF DIRICHLET PROBLEM IN A HALF-SPACE FOR REGULAR EQUATIONS WITH NON-HOMOGENEOUS BOUNDARY CONDITIONS

Abstract

In this paper we consider the following Dirichlet problem with non-homogeneous boundary conditions in a multianisotropic Sobolev space $W_2^{\mathfrak{M}}(R^2 \times R_+)$ $$\begin{cases} P(D_x, D_{x_3}) u = f(x, x_3), \quad x_3 > 0, \quad x \in R^2, \\ D_{x_3}^s u \big\rvert_{x_3 = 0} = \varphi_s(x),\quad s = 0, \dots, m-1. \end{cases} $$ It is assumed that $P(D_x, D_{x_3})$ is a multianisotopic regular operator of a special form with a characteristic polyhedron $\mathfrak{M}$. We prove unique solvability of the problem in the space $W_2^{\mathfrak{M}}(R^2 \times R_+)$, assuming additionally, that $f(x, x_3)$ belongs to $L_2(R^2 \times R^+)$ and has a compact support, boundary functions $\varphi_s$ belong to special Sobolev spaces of fractional order and have compact supports.