Solvability of some overdetermined elliptic system in a domain with corners π/n

摘要

AbstractIn the paper we prove the existence and uniqueness of solutions of the overdetermined elliptic systemdocumentclass{article}pagestyle{empty}begin{document}$$ begin{array}{*{20}c} {left( {rm A} right)} & {{rm rot }upsilon {rm = }omega } & {{rm div }upsilon {rm = 0}} & {upsilon cdot {rm }bar n|_{partial Omega } = b} end{array} $$end{document}whereb, ω are given functions, in a domain ΩCR3with corners π/n, n= 2, 3, … The proof is divided on two steps, we construct a solution for the Laplace equation in a dihedral angle π/n, using the method of reflection and we get an estimate in the norms of the Sobolev spaces in some neighbourhood of the edge. In the dihedral angle system (A) reduces to the Dirichlet and Neumann problems for the Laplace equation. In the next step we prove the existence of solutions in the Sobolev spacesWpl(Ω) using the existence of generalized solutions