New problems on minimizing movements tt

In this paper 1 intend to deepen the idea of minimizing movernenl which bas bcen presented in the conference [10]. Such idea seems to be suitable to unify many problems in calcul us of variations, differential equations, geometric measure theory: among others, steepest descent methods, heat equation, mean curvature flow, monotone operators, various evolution problems, eLc. This paper is completely self-contained and may be read independently of the papers quoted in the bibliography; nevertheless, we remark that the main definitions of this paper may be considered as slight generalizations of the definitions given in [10] and that the paper [10] has been inspired lIlainly by the paper [1]. Minimizing movements arc tied in variou~ ways [,0 penalization methods, rconvergence, singular perturbation, geometric measure theory, etc., hence the bibliograpllic indications will be unavoidably partial and far from being complete. In many cases the reader can surely find many other interesting rflfflren ces , as weil as many interesting examples, problems, conjectures suggested by his own experience, which cou Id be more interesting and expressive than those presented in this paper. One could think of finding general hypotheses on F and S such that the set of minimizing movcmcnts 1\4 M (F, S) or the set of generalized minimiziny 1TWVp.1np.nts G MM (F, S) are nonvoid, or finite or such that their elements can be characterized by some differential equation, and/or some other meaningful condition. 1 believe that the idea of uùnimizing movement is the natural meeting point of many problems of analysis, geomet,ry, rnat.hematical physics and numerical analysis, aJld ils rlflvelopment will requiTe the contribution of many researchers with different backgrounds. 1 wish to thank L.Ambrosio, A.Leaci, S.Mortola and D.Pallara for their cooperation in preparing this paper.