Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations

The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an interface (Imbert, 2008) on the other hand. In order to do so, we start by constructing such a game for eikonal equations whose speed has a non-constant sign. This provides a (discrete) deterministic control interpretation of these evolution equations. In all our games, two players choose positions successively, and their final payoff is determined by their positions and additional parameters of choice. Because of the non-locality of the problems approximated, by contrast with local problems, their choices have to "collect" information far from their current position. For integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces. For parabolic integro-differential equations, players choose smooth functions on the whole space

A structure-preserving parametric finite element method for area-conserved generalized mean curvature flow

We propose and analyze a structure-preserving parametric finite element method (SP-PFEM) to simulate the motion of closed curves governed by area-conserved generalized mean curvature flow in two dimensions (2D). We first present a variational formulation and rigorously prove that it preserves two fundamental geometric structures of the flows, i.e., (a) the conservation of the area enclosed by the closed curve; (b) the decrease of the perimeter of the curve. Then the variational formulation is approximated by using piecewise linear parametric finite elements in space to develop the semi-discrete scheme. With the help of the discrete Cauchy's inequality and discrete power mean inequality, the area conservation and perimeter decrease properties of the semi-discrete scheme are shown. On this basis, by combining the backward Euler method in time and a proper approximation of the unit normal vector, a structure-preserving fully discrete scheme is constructed successfully, which can preserve the two essential geometric structures simultaneously at the discrete level. Finally, numerical experiments test the convergence rate, area conservation, perimeter decrease and mesh quality, and depict the evolution of curves. Numerical results indicate that the proposed SP-PFEM provides a reliable and powerful tool for the simulation of area-conserved generalized mean curvature flow in 2D.Comment: 24 pages, 8 figure

Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t, the evolving hypersurface M_t meets such tgh ortogonally, we prove that: a) the flow exists while M_t does not touch the axis of rotation; b) throughout the time interval of existence, b1) the generating curve of M_t remains a graph, and b2) the averaged mean curvature is double side bounded by positive constants; c) the singularity set (if non-empty) is finite and discrete along the axis; d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature.Comment: 24 pages. We have added some lines at the beginning explaining the notation, and clarified a little bit more the proofs of Proposition 1 and Theorems 5 and 10, the statements of Proposition 2 and Corollary 3 and an argument in Remark 1. We have also completed reference 18. A version of this paper will appear in Mathematische Zeitschrif

Gradient flows and a Trotter--Kato formula of semi-convex functions on CAT(1)-spaces

We generalize the theory of gradient flows of semi-convex functions on CAT(0)-spaces, developed by Mayer and Ambrosio--Gigli--Savar\'e, to CAT(1)-spaces. The key tool is the so-called "commutativity" representing a Riemannian nature of the space, and all results hold true also for metric spaces satisfying the commutativity with semi-convex squared distance functions. Our approach combining the semi-convexity of the squared distance function with a Riemannian property of the space seems to be of independent interest, and can be compared with Savar\'e's work on the local angle condition under lower curvature bounds. Applications include the convergence of the discrete variational scheme to a unique gradient curve, the contraction property and the evolution variational inequality of the gradient flow, and a Trotter--Kato product formula for pairs of semi-convex functions.Comment: 27 pages; minor revisions; to appear in Amer. J. Mat