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 $-\Delta u=f$ in a domain $\Omega$ with polygonal boundary $\Gamma$. On each side of $\Gamma$ 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 $u \in L_2(\Omega)$; 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 $u \in L_2(\Omega)$. 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.
"On homeomorphisms for an elliptic equation in domains with corners." Differential Integral Equations 8 (2) 333 - 352, 1995. https://doi.org/10.57262/die/1369083473