Convexity estimates for nonlinear elliptic equations and application to free boundary problems | Estimations de convexité pour des équations elliptiques non linéaires et application à des problèmes de frontière libre

We prove the convexity of the set which is delimited by the free bound- ary corresponding to a quasi-linear elliptic equation in a 2-dimensional convex do- main. 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.