On homeomorphisms for an elliptic equation in domains with corners

Our purpose is to construct a space of distribution solutions in a simple case for which the data is not smooth. We consider the mixed boundary value problem for the equation Δu = f in a domain ω with polygonal boundary Γ. On each side of Γ we impose either Dirichlet or Neumann boundary conditions. This is a "corner problem" whose solution contains corner singularities that are well understood [2, 3]. We are therefore in a position to construct a dual theory of distribution solutions for this mixed problem. In this paper we make this construction for distribution solutions u ∈ L₂(ω); that is, the case when the solution is one step below the energy space in regularity. For this, we must give a careful definition of the data space associated with the mixed problem, and the "trace space" associated with a function u ∈ L₂(ω). We find that there is always a distribution solution to the mixed boundary value problem, but the solution may not be unique; there may be distribution solutions to the homogeneous problem constructed with the help of the corner singular functions.