Some Applications of Bv Functions in Optimal Control and Calculus of Variations 1 a Problem of Calculus of Variations

In this paper we discuss the use of bounded variation functions in the study of some optimal control problems as well as in the calculus of variations. The bounded variation functions are well adapted to the study of parameter identiication problems, such as the coeecients of an elliptic or parabolic operator. These functions are also convenient for the image recovery problems. These problems are well formulated in the space BV (() in the sense that they have a solution under reasonable assumptions. The numerical approximation of these problems is interesting because of the non separability of the space BV ((). A very surprising fact is the in-uence of the chosen norm, among all the possible equivalent norms in BV ((), on the convergence of the numerical approximations. Indeed the approximation by piecewise constant functions fails if the norm is not properly chosen. The image enhancement or image recovery problems, which have recently received a considerable amount of attention, are an example of problems of calculus of variations that can be studied in the space of the functions of bounded variation. To brieey describe this problem let z denote the grey values of an image which extends in a two dimensional domain. The values of z correspond