Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions

We present some comparison results for solutions to certain non local elliptic and parabolic problems that involve the fractional Laplacian operator and mixed boundary conditions, given by a zero Dirichlet datum on part of the complementary of the domain and zero Neumann data on the rest. These results represent a non local generalization of a Hopf’s lemma for elliptic and parabolic problems with mixed conditions. In particular we prove the non local version of the results obtained by J. D'avila and J. D'avila-L. Dupaigne for the classical case respectively.