Sharp and improved regularity estimates to fully nonlinear equations and free boundary problems.

Abstract

The thesis consists of the following three papers on regularity estimates for fully non-linear parabolic equations and one-phase singularly perturbed elliptic problems. Sharp regularity estimates for second order fully nonlinear parabolic equations - Joint work with Eduardo V. Teixeira. The purpose of the fi rst chapter is prove sharp regularity estimates for viscosity solutions to fully non-linear parabolic equations of the form @u@t F(D2u; Du; x; t) = f(x; t) in Q 1 = B 1 ( 1; 0]; (Eq1) where F is a uniformly elliptic operator and f 2 L p;q (Q 1 ). The quantity (n; p; q) :=np+2q determines which regularity regime a solution to (Eq1) belongs to. We prove that when 1 < (n; p; q) < 2 ϵ F , solutions are parabolic-Hölder continuous for a sharp, quantitative exponent 0 < (n; p; q) < 1. The case (n; p; q) = 1 is a critical borderline situation as it divides the regularity theory. In this scenario, we obtain a sharp universal Log-Lipschitz regularity estimate. When 0 < (n; p; q) < 1, solutions are locally of class C 1+ ;1+ 2 and in the limiting case (n; p; q) = 0, we show C 1;Log-Lip regularity estimates provided F is convex in the Hessian argument for example. Schauder Type Estimates for “Flat” Viscosity Solutions to Non-convex Fully Nonlinear Parabolic Equations and Applications - Joint work with Disson S. dos Prazeres. In a second moment we establish Schauder type estimates for fl at solutions to non-convex fully non-linear parabolic equations of the following form @u@t F(x; t; D2u) = f(x; t) in Q 1 (Eq2) provided the coeffi cientsof F and the source f are Dini continuous. Furthermore, we prove a partial regularity result, as well as a theorem of Evans-Krylov type. Finally, for problems with merely continuous data we prove that fl at solutions to( Eq2) are parabolic C 1;Log-Lip smooth. Regularity up to the boundary for fully nonlinear singularly perturbed elliptic equations - Joint work with Gleydson C. Ricarte. Posteriorly, we are interested in studying regularity up to the boundary for one-phasesingularly perturbed fully non-linear elliptic problems F(x; Du"; D2u") = ϵ (uϵ) in Ω R n (Eq3) where " behaves asymptotically as the Dirac measure 0 as " goes to zero. We shall establish global gradient bounds independent of the parameter " to viscosity solutions to (Eq3), which allow us to pass the limit and obtain optimal regularity for free boundary problem.