This paper is devoted to a proof of the convexity of a free boundary for a quasi-linear problem defined on a convex domain in IR 2 and to the obtention of L 1 -bounds on the gradient of a solution. The free boundary can be seen as the boundary of the coincidence set of an obstacle problem. The bound on the gradient explicitely depends on the curvature of the boundary of the domain. The main tool is an estimate of the maximum of the gradient on the level lines, which involves their curvature. The second result is valid in an analytical framework only. We indicate how to extend our results to dimensions higher than 2. Key-words and phrases: Quasi-linear elliptic equations -- Free boundary -- Gradient estimates -- Fr'echet formula -- Curvature of level sets -- Obstacle problem -- Coincidence set 1991 Mathematics Subject Classification: 35J25, 35J67, 35R35. 1 1 Introduction Our first result is concerned with the following free boundary problem. Consider a solution of div ` a(jruj...