Small Perturbation Solutions for Elliptic Equations

In this work we present a general regularity result for small perturbation solutions of elliptic equations. Our approach was motivated by the analysis of flat level sets in Ginzburg-Landau phase transitions models, which were considered in Savin (2003). When dealing with uniformly elliptic equations of the form (1), the classical approach to regularity is to differentiate the equation with respect to a direction e. Then, ue solves the linearized equation which is treated as a linear equation with bounded measurable coefficients. If F is not uniformly elliptic in the whole domain, then, in order to bound the coefficients of the linearized equation, one needs a priori bounds on u, Du and D2u. This is the case in several problems such as the minimal surface or the MongeAmpere equation. In this paper we discuss the regularity of “flat” viscosity solutions of (1). We prove interior C2 estimates for such solutions provided that F is smooth and uniformly elliptic in a neighborhood of the set