A problem with free boundary

One considers the Dirichlet problem for the equation u=(u), where is the Heaviside function. Under special assumptions one constructs the solution of this problem with convex and smooth level surfaces and, in particular, with a regular free surface, which coincides with the set of level zero. One proves the solvability in the small of the problem in the neighborhood of the constructed regular solution under perturbations of the boundary condition and a smooth boundary of the domain .Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 72–83, 1986.