On Dirichlet data for the spectral Laplacian

We present a construction for nontrivial harmonic functions associated to the spectral fractional Laplacian operator, that is a fractional power of the Dirichlet Laplacian giving rise to a nonlocal operator of fractional order. These harmonic functions present a divergent profile at the boundary of the prescribed domain, and they can be classified in terms of a singular boundary trace. We introduce a notion of L1-weak solution, in the spirit of Stampacchia, and we produce solutions of linear and nonlinear problems (possibly with measure data) where one prescribes such a singular boundary trace, therefore providing with a nonhomogeneous boundary value problem for this operator. We also present some results entailing the existence of large solutions in this context.