In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study.We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow.We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the timestep and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as "freezing" or "pinning") and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum

Abstract. In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study.We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow.We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as "freezing" or "pinning") and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation.Mathematics Subject Classification (2010). 34B45, 35R02, 53C44, 53A10, 49K15, 49Q05, 35K05.

Motivated by the question of existence of global solutions, we obtain pointwise upper bounds for radially symmetric and monotone solutions to the homogeneous Landau equation with Coulomb potential. The estimates say that blow up in the L ∞ (R 3 )-norm at a finite time T can occur only if the L 3/2 (R 3 )-norm of the solution concentrates for times close to T . The bounds are obtained using the comparison principle for the Landau equation and for the associated mass function. This method provides long-time existence results for the isotropic version of the Landau equation with Coulomb potential, recently introduced by Krieger and Strain.

In this work we provide an Aleksandrov-Bakelman-Pucci type estimate for a certain class of fully nonlinear elliptic integro-differential equations, the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal, fractional-order setting and on the use of Riesz potentials to interpret second derivatives as fractional order operators. This result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations. Furthermore, this result also gives a new comparison theorem for viscosity solutions of such equations which only depends on the L ∞ and L n norms of the right hand side, in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions. These results appear to be new even for the linear case of the relevant equations.with the additional assumption that 0 ≤ f k ≤ 1. Then we pose the following question:Question 1.1. Under what conditions will it be true that |{x :(We will assume L is 1-homogeneous, and hence the constant 0 function is a supersolution, and so always u k ≤ 0.) In the case that L is a second order, uniformly elliptic operator, L(u, x) = a ij (x)u x i x j (x),

Abstract. The study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge-Ampère type known as generated Jacobian equations. This class of equations, whose general existence theory has been recently developed by Trudinger, goes beyond the framework of optimal transport. We obtain pointwise estimates for weak solutions of such equations under minimal structural and regularity assumptions, covering situations analogous to that of costs satisfying the A3-weak condition introduced by Ma, Trudinger and Wang in optimal transport. These estimates are used to develop a C 1,α regularity theory for weak solutions of Aleksandrov type. The results are new even for all known near-field reflector/refractor models, including the point source and parallel beam reflectors and are applicable to problems in other areas of geometry, such as the generalized Minkowski problem.

We identify a condition for regularity of optimal transport maps that requires only three derivatives of the cost function, for measures given by densities that are only bounded above and below. This new condition is equivalent to the weak Ma-Trudinger-Wang condition when the cost is C 4 . Moreover, we only require (non-strict) c-convexity of the support of the target measure, removing the hypothesis of strong c-convexity in a previous result of Figalli, Kim, and McCann, but at the added cost of assuming compact containment of the supports of both the source and target measures.

Abstract. In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for Monge-Ampère type equations. An application to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically.

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