Convexity Estimates for Nonlinear Elliptic Equations and Application to Free Boundary Problems

We prove the convexity of the set which is delimited by the free boundary corresponding to a quasi-linear elliptic equation in a 2-dimensional convex domain. The method relies on the study of the curvature of the level lines at the points which realize the maximum of the normal derivative at a given level, for analytic solutions of fully nonlinear elliptic equations. The method also provides an estimate of the gradient in terms of the minimum of the (signed) curvature of the boundary of the domain, which is not necessarily assumed to be convex. R esum e. Nous d emontrons la convexit e de l'ensemble d elimit e par la fronti ere libre correspondant a une equation quasi-lin eaire elliptique d eenie sur un domaine convexe en dimension 2. La m ethode repose sur l' etude de la courbure des lignes de niveau aux points qui r ealisent le maximum de la d eriv ee normale pour un niveau donn e, pour des solutions analytiques d' equations elliptiques compl etement non lin eaires. La m ethode donne aussi une estimation du gradient en fonction du minimum de la courbure (sign ee) du bord du domaine, qui n'est pas n ecessairement suppos e convexe.