Marcelo Montenegro and Enrico Valdinoci Pointwise estimates and monotonicity formulas without maximum principle (247K, pdf) ABSTRACT. We study a second order elliptic partial differential equation for which a maximum principle is not available and whose nonlinearity is not $C^1$. We discuss the role of a pointwise gradient bound, of which we study the optimal constant. As a consequence, we derive a monotonicity estimate near flat points of the free boundary of a minimizer.