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

Elliptic boundary value problems in convex polygons with low regularity boundary data via the unified method