  We use novel integral representations developed by the second author to prove
certain rigorous results concerning elliptic boundary value problems in convex
polygons. Central to this approach is the so-called global relation, which is a
non-local equation in the Fourier space that relates the known boundary data to
the unknown boundary values. Assuming that the global relation is satisfied in
the weakest possible sense, i.e. in a distributional sense, we prove there
exist solutions to Dirichlet, Neumann and Robin boundary value problems with
distributional boundary data. We also show that the analysis of the global
relation characterises in a straightforward manner the possible existence of
both integrable and non-integrable corner-singularities.
