(Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as t →∞. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts. 1. Introduction. Let Ω be a bounded set in R N with Lipschitz-continuous boundary ∂Ω. We are interested in the problem ∂u Du = div(

We consider the forward-backward splitting method for finding a zero of the sum of two maximal monotone mappings. This method is known to converge when the inverse of the forward mapping is strongly monotone. We propose a modification to this method, in the spirit of the extragradient method for monotone variational inequalities, under which the method converges assuming only the forward mapping is monotone and (Lipschitz) continuous on some closed convex subset of its domain. The modification entails an additional forward step and a projection step at each iteration. Applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed.

We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.

We prove the existence of a nite extinction time for the solutions of the Dirichlet problem for the total variational ow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in nite time. The asymptotic prole of the solutions of the Dirichlet problem is also studied. It is shown that the proles are non zero solutions of an eigenvalue type problem which seems to be unexplored in the previous literature. The propagation of the support is analyzed in the radial case showing a behaviour enterely dierent to the case of the problem associated to the p-Laplacian operator. Finally, the study of the radially symmetric case allows us to point out other qualitative properties which are peculiar of this special class of quasilinear equations. Key words: Total variation ow, nonlinear parabolic equations, asymptotic behaviour, eigenvalue type problem, propagation of the support. AMS (MOS) subject classication: 35K65, 35K55. 1 Introduction Let be a ...

. We regard the limit as p ! 1 of the flow governed by the p-Laplacian as providing a simplistic model for the "collapse of an initially unstable sandpile." Upon rescaling to stretch out the initial layer we obtain some simple dynamics and provide fairly explicit solutions in certain cases. In particular we note that such models entail "instantaneous " mass transfer governed by Monge-Kantorovich theory. 1. Introduction We study in this paper a "fast/slow diffusion" PDE problem, which can be very loosely interpreted as modelling the collapse of a sandpile from an initially unstable configuration. The mathematical issue is to understand the behavior of the solution u p of the initial value problem ae u p;t \Gamma \Delta p u p = 0 in R n \Theta (0; 1) u p = g on R n \Theta ft = 0g (1.1) in the "infinitely fast/infinitely slow diffusion" limit p ! 1. Here 1 p ! 1, \Delta p u p = div(jDu p j p\Gamma2 Du p ) is the p-Laplacian, and Du p denotes the gradient of u p with respect...

We study the asymptotic behavior at infinity of solutions of a second order evolution equation with linear damping and convex potential. The differential system is defined in a real Hilbert space. It is proved that if the potential is bounded from below then the solution trajectories are minimizing for it, and converge weakly towards a minimizer of Phi if there exists one; this convergence is strong when Phi is even or when the optimal set has a nonempty interior. We introduce a second order proximal-like iterative algorithm for the minimization of a convex function. It is defined by an implicit discretization of the continuous evolution problem and is valid for any closed proper convex function. We find conditions on some parameters of the algorithm in order to have a convergence result similar to the continuous case.

We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f(x; ) = p 1 + kk 2 , which corresponds with the time-dependent minimal surface equation. We also study the asymptotic behavoiur of the solutions.

We study Markov processes associated with stochastic differential equations, whose nonlinearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a Fokker-Planck equation as the steepest descent flow of the relative Entropy functional in the space of probability measures, endowed with the Wasserstein distance. Applications include stochastic partial differential equations and convergence of equilibrium fluctuations for a class of random interfaces.

We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in R N. This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, satisfying a comparison principle, and defining a continuous semigroup in time. We prove that, when the initial set is convex, our evolution coincides with the flat φ-curvature flow in the sense of Almgren-Taylor-Wang. As a by-product, it turns out that the flat φ-curvature flow starting from a compact convex set is unique.

We study the space Mm of all m-accretive operators on a Banach space X endowed with an appropriate complete metrizable uniformity and the space M ∗ m which is the closure in Mm of all those operators which have a zero. We show that for a generic operator in Mm all infinite products of its resolvents become eventually close to each other and that a generic operator in M ∗ m has a unique zero and all the infinite products of its resolvents converge uniformly on bounded subsets of X to this zero.