Lema de Hopf-Oleinik não homogêneo para equações lineares da forma divergente via barreiras e aplicações a problemas de fronteira livre

Abstract

In this work we construct new barriers for inhomogeneous linear equations in divergence form with unbounded terms. Such construction is fragmented in some propositions and starts with homogeneous equations in sufficiently small rings, until the general case, which are inhomogeneous equations in unit rings. For this we use some essential ingredients: Maximum Principle, Comparison Principle, Gradient Estimation (Morrey), Harnack Inequalities and Solution Existence Theorems. Some new facts are also observed in some of these auxiliary results, namely, the independence of the norm of the lowest order coefficient of the equation on the Maximum Principle, the inclusion of all the terms of the equation in divergence form for the Principle of Comparison, the proof of an interior estimate of the gradient for non-negative solutions and the proof of the existence and uniqueness of solutions for the linear equations of inhomogeneous divergence form with unbounded terms. Subsequently, we use the constructed barrier to prove a quantitative version of the Hopf-Oleinik lemma and an estimate of the interior gradient for a free boundary problem, showing that solutions of this problem are Lipschitz in the interior. Finally, we use the gradient estimate of the free boundary problem to also prove an interior gradient estimate for a flame propagation problem and in this case we obtain a Lipschitz equicontinuity for its solutions.