The purpose of this paper is to study a mathematical model of lubricating flow between elastic surfaces obeying the linear Hertzian theory when cavitation takes place. Cavitation is a free boundary phenomenon that is described in this paper by the New Elrod–Adams model. This model introduces the concentration of fluid as well as the pressure as unknown functions and is suggested in preference to the classical variational inequality due to its ability to describe inflow and outflow. This leads to a nonlinear variational and nonlocal equation. Herein, an existence theorem is proved by means of two different techniques.

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Abstract.We consider an optimal control problem for a class of non-linear elliptic equations. A result of existence and uniqueness of the state equation is proven under weaker hypotheses than in the literature. We also prove the existence of an optimal control. Applications to some lubrication problems and numerical results are given.Mathematics Subject Classification. 35J25, 49J20, 49J50.

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