Abstract

The Signorini problem can be equivalently formulated as a thin obstacle problem for an elastic membrane. The resulting free boundary problem turns out to be equivalent to the obstacle problem for the fractional Laplacian on the whole space. We will show how to understand this problem under the light of the language of semigroups that I developed in my PhD thesis (2010). In particular, we are able to consider different kinds of Signorini problems that are equivalent to obstacle problems for fractional powers of operators different than the Laplacian on the whole space. Boundary conditions of different kinds (Dirichlet, Neumann, periodic) and radial solutions can also be treated with this unified language. Another advantage is that this language avoids the use of the Fourier transform. The basic regularity results (Harnack inequalities, Schauder estimates) for these fractional nonlocal operators can be studied by means of the generalization of the Caffarelli—Silvestre extensio n problem that I proved in my PhD thesis. It turns out that the solution for the extension problem can be written in terms of the heat semigroup.