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In this paper we consider the following Dirichlet problem with non-homogeneous boundary conditions in a multianisotropic Sobolev space W2M(R2×R+){P(Dx,Dx3)u=f(x,x3),x3>0,x∈R2,Dx3su|x3=0=φs(x),s=0,…,m−1.It is assumed that P(Dx,Dx3) is a multianisotopic regular operator of a special form with a characteristic polyhedron M. We prove unique solvability of the problem in the space W2M(R2×R+), assuming additionally, that f(x,x3) belongs to L2(R2×R+) and has a compact support, boundary functions φs belong to special Sobolev spaces of fractional order and have compact supports.