On solvability of polyharmonic Dirichlet problem in symmetric Sobolev spaces

Abstract

Let Omega subset of R-n be a bounded domain with sufficiently smooth boundary partial derivative Omega and X (Omega) be a symmetric space defined on the measure space (Omega; dx). We consider a Dirichlet problem for 2m-th order polyharmonic equation, and we establish its solvability (in strong sense) in Sobolev space W-X(2m) (Omega) generated by the norm of X (Omega). Such spaces include classical Sobolev spaces, Orlicz-Sobolev spaces, grand Sobolev spaces, and Marcinkiewicz-Sobolev spaces. The obtained results are new for these special cases, too.