Teoria de regularidade para equações elípticas totalmente não lineares com potenciais singulares e problemas de fronteira livre assintóticos

Abstract

In this work we develop a fully nonlinear theory for singularly perturbed elliptic equations problems with high energy activation. We esta-blish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non-degenerated. For problems governed by concave equations, we establish uniform weak geometric properties of approximating level surfaces. We also provide a thorough analysis of the free boundary problem obtained as a limit as the parameter term goes to zero. We find the precise jumping condition of limiting solutions through the phase transi-tion, which involves a subtle homogenization process of the governing fully nonlinear operator. In particular, for rotational invariant operators, $F(D^2u)$, we show the normal derivative of limiting function is constant along the interface. Smoothness properties of the free boundary are also addressed.