On homeomorphisms for an elliptic equation in domains with corners (Q1343073)

The mixed boundary value problem for the Poisson equation \(- \Delta u = f\) in a domain \(\Omega\) with polygonal boundary is considered. On each side of \(\Gamma\) either Dirichlet or Neumann boundary conditions are posed. Generalizing Roitberg's and Seftel's idea, a dual theory of distribution solutions for this mixed problem is constructed. This construction is made for distribution solutions \(u \in L_2 (\Omega)\). The data space associated with the mixed problem, and the ``trace space'' associated with a function \(u \in L_2 (\Omega)\) are defined. The existence of a distribution solution of the mixed boundary value problem is shown. The solution may be not unique.