Boundary value problems on singular domains

We give a geometric approach for boundary value problems of the Laplacian with Dirichlet (or mixed) boundary conditions on domains with singularities. In two dimensions these singularities also include cusps. Our approach is by blowing up the singularities via a conformal change to translate the boundary problem to one on a noncompact manifold with boundary that is of bounded geometry and of finite width. This gives a natural geometric interpretation for the weights that appear and for the additional conditions needed to obtain well-posedness results.

This is joint work with Bernd Ammann and Victor Nistor.