Global extensions of the weak Harnack inequality and some applications

We review some recent up-to-the-boundary extensions of the classical de Giorgi-Moser (for uniformly elliptic PDE in divergence form) and Krylov-Safonov (for uniformly elliptic PDE in nondivergence form) weak Harnack inequality. We obtain global WHI for the quantity u/d in a sufficiently smooth domain, where u is a nonnegative supersolution of a linear equation, and d is the distance function to the boundary of the domain. In some cases our results quantify the optimal global integrability of positive supersolutions and the Zaremba-Hopf-Oleinik boundary point lemma with respect to each other. If time permits, we will show some applications to a priori bounds for positive solutions of the Dirichlet problem for nonlinear equations.