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 Davila and Davila and Dupaigne for the classical case s = 1 in [23, 24] respectively.