Abstract
The essential criteria was discovered in the thesis to define free boundary over the nonlinear partial differential equations.The task is to find co-ordinates of each point of the border so that to minimize functional.Such task can be treated as an optimum control problem.It is a problem with restriction.In accordance with a method of the interfaced functions we will pass to a problem of optimization without restrictions, having entered new functional.The purpose of the given work is calculation of the first variation of functional.It also defines a necessary condition of optimality for definition of position of border of an oil layer.The considered method for determining free boundaries is based on solving problems of determining the boundary of a system that describes a nonlinear partial differential equation.