Estimativas do gradiente na fronteira para soluções de desigualdades diferenciais totalmente não lineares.

Abstract

In this work we obtain an estimate and a regularity of the gradient for solutions to fully nonlinear differential inequalities with unbounded coefficients and quadratic growth on the gradient. The boundary data is C^(1,Dini) and solutions are understood in the viscosity sense. More specifically, the drift term and the RHS are in L^q with q>n. We prove that u ∈ C^1 on the flat boundary with some modulus of continuity with the estimates. Our results can be seen as extended versions of remarkble estimates obtained by N. Krylov (1983) and O. Ladyzhenskaya and N. Ural’tseva (1989). Finally, we also show that in the case RHS is in L^n the result does not hold and solutions may fail to be even Lipschitz on a neighborhood of the boundary wich means that, in the RHS sense, this theorem is sharp.