5 O ct 1 99 7 VARIATIONAL EVOLUTION PROBLEMS AND NONLOCAL GEOMETRIC MOTION

We consider two variational evolution problems related to MongeKantorovich mass transfer. These problems provide models for collapsing sandpiles and for compression molding. We prove the following connection between these problems and nonlocal geometric curvature motion: The distance functions to surfaces moving according to certain nonlocal geometric laws are solutions of the variational evolution problems. Thus we do the first step of the proof of heuristics developed in earlier works. The main techniques we use are differential equations methods in the Monge-Kantorovich theory. In this paper we study two models involving limits as p → ∞ of solutions of pLaplacian evolution problems. One is a model of collapsing sandpiles proposed by L. C. Evans, R. F. Gariepy and the author [EFG97]. Another is a model of compression molding proposed by G. Aronsson and L. C. Evans [AE]. In both models the limits were characterized as solutions of variational evolution problems related to MongeKantorovich mass transfer. Solutions that have the form of the distance function to a moving boundary arise naturally in the both models. The equations of the motion of the boundary for both models were derived heuristically in [EFG97] and [AE]. According to the equations the outer normal velocity of the boundary at a point depends on both the local geometry (curvatures) and the nonlocal geometry of the boundary. Thus, the motion of the boundary is a nonlocal geometric motion. In this paper we prove rigorously the connection between geometric and variational evolution problems. Namely, assuming that a moving surface satisfying the geometric equation is given, we prove that the distance function defines a solution of the corresponding variational evolution problem. The main assumption is that the surface remains convex (or, more generally, semiconvex) during the evolution. For the collapsing sandpiles model we also prove the corresponding result for the solutions that have the form of the maximum of several distance functions (such solutions represent several sand cones interacting in the process of collapse). In the proofs we utilize the connection between the models and the Monge-Kantorovich mass transfer. This allows us to use the differential equations methods in the Monge-Kantorovich theory, which have been developed recently by L. C. Evans and W. Gangbo [EGan]. The examples suggest that solutions of the geometric equations derived in [EFG97] can develop singularities, even if the initial data is smooth. Thus we do not assume below that the moving surface is smooth. This however makes the technique more involved. Research is supported in part by NSF grants DMS-9701755 (MSRI) and DMS-9623276. 1