Bernstein-type Techniques for 2d Free Boundary Graphs

that appears in many applications (see [AC], [F].) In [C1],[C2],[C3], the author introduced the notion of “viscosity” solution to (1.1), and developed the theory of existence and regularity of viscosity free boundaries. In particular, the regularity theory is inspired by the regularity theory for minimal surfaces, precisely by the “oscillation decay” method of De Giorgi, according to which if S is a minimal surface in the unit ball B1, and S is the graph of a Lipschitz function, then S is C (hence smooth) in B1/2. Analogously, if F (u) is a Lipschitz free boundary in B1, then F (u) is C 1,α in B1/2. Higher regularity results of [KN] then yield the local analyticity of F (u) in the interior. Thus, a natural question arises, that is how to obtain the Lipschitz continuity of a viscosity free boundary. In the theory of minimal surfaces, in the special case of minimal graphs, this is achieved via an a-priori gradient bound for solutions to the minimal surface equation, originally proved in [BMG]. Analogously, an a-priori bound for the Lipschitz constant of smooth free boundary graphs is needed, in order to obtain that viscosity free boundary graphs are smooth in the interior. In this note we provide this tool in the 2D and 3D case. Our proof is based on the so-called Bernstein technique, which is been widely used in literature (see for example [GT].) A similar approach for minimal surfaces is used in [WX].