How a minimal surface leaves an obstacle

This paper is an investigation of the curve of separation determined by the solution to a variational inequality for minimal surfaces.A strictly convex domain ~ in the z =x 1 +ix~ plane is given together with a smooth function ~p which assumes a positive maximum in ~ and is negative on ~, the boundary of ~.Let u denote the Lipschitz function which minimizes area among all Lipschitz functions in ~ constrained to lie above ~p in and to vanish on a~.For such u there is a coincidence set I c ~ consisting of those points z where u(z)=yJ(z).Let us call F={ (xl, x~, xa): xa=u(z)=y~(z), zE~I} the "curve" of separation.The object of this paper is to show that F is analytic, as a function of its arc length parameter, provided that y; is strictly concave and analytic.