A Wasserstein approach to the one-dimensional sticky particle system

We present a simple approach to study the one-dimensional pressureless Euler system via adhesion dynamics in the Wasserstein space of probability measures with finite quadratic moments. Starting from a discrete system of a finite number of "sticky" particles, we obtain new explicit estimates of the solution in terms of the initial mass and momentum and we are able to construct an evolution semigroup in a measure-theoretic phase space, allowing mass distributions with finite quadratic moment and corresponding L^2-velocity fields. We investigate various interesting properties of this semigroup, in particular its link with the gradient flow of the (opposite) squared Wasserstein distance. Our arguments rely on an equivalent formulation of the evolution as a gradient flow in the convex cone of nondecreasing functions in the Hilbert space L^2(0,1), which corresponds to the Lagrangian system of coordinates given by the canonical monotone rearrangement of the measures.Comment: Added reference

Contraction of general transportation costs along solutions to Fokker-Planck equations with monotone drifts

We shall prove new contraction properties of general transportation costs along nonnegative measure-valued solutions to Fokker-Planck equations in Rd, when the drift is a monotone (or lambda-monotone) operator. A new duality approach to contraction estimates has been developed: it relies on the Kantorovich dual formulation of optimal transportation problems and on a variable-doubling technique. The latter is used to derive a new comparison property of solutions of the backward Kolmogorov (or dual) equation. The advantage of this technique is twofold: it directly applies to distributional solutions without requiring stronger regularity and it extends the Wasserstein theory of Fokker-Planck equations with gradient drift terms started by Jordan-Kinderlehrer-Otto [14] to more general costs and monotone drifts, without requiring the drift to be a gradient and without assuming any growth conditions

Geometric Aspects of Ambrosetti-Prodi operators with Lipschitz nonlinearities

For Dirichlet boundary conditions on a bounded domain, what happens to the critical set of the Ambrosetti-Prodi operator if the nonlinearity is only a Lipschitz map? It turns out that many properties which hold in the smooth case are preserved, despite of the fact that the operator is not even differentiable at some points. In particular, a global Lyapunov-Schmidt decomposition of great convenience for numerical inversion is still available

Generating and Adding Flows on Locally Complete Metric Spaces

As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in \cite{BC}. In that paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem i.e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution curves for this sum.Comment: 29 pages,6 figure

Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains

We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $\phi (\Omega)$ parametrized by Lipschitz homeomorphisms $\phi$ defined on a fixed reference domain $\Omega$. Given two open sets $\phi (\Omega)$, $\tilde \phi (\Omega)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $\|\tilde \phi -\phi \|_{W^{1,p}(\Omega)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.Comment: 34 pages. Minor changes in the introduction and the refercenes. Published in: Around the research of Vladimir Maz'ya II, pp23--60, Int. Math. Ser. (N.Y.), vol. 12, Springer, New York 201

Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction

We study a singular-limit problem arising in the modelling of chemical reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the {\epsilon}-problem converge to a solution of the limiting problem.Comment: Added two sections, corrected minor typos, updated reference

Nonsmooth analysis of doubly nonlinear evolution equations

In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional,for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.Comment: 45 page

Finite Element Convergence for the Joule Heating Problem with Mixed Boundary Conditions

We prove strong convergence of conforming finite element approximations to the stationary Joule heating problem with mixed boundary conditions on Lipschitz domains in three spatial dimensions. We show optimal global regularity estimates on creased domains and prove a priori and a posteriori bounds for shape regular meshes.Comment: Keywords: Joule heating problem, thermistors, a posteriori error analysis, a priori error analysis, finite element metho

The mixed problem in L^p for some two-dimensional Lipschitz domains

We consider the mixed problem for the Laplace operator in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. The boundary of the domain is decomposed into two disjoint sets D and N. We suppose the Dirichlet data, f_D has one derivative in L^p(D) of the boundary and the Neumann data is in L^p(N). We find conditions on the domain and the sets D and N so that there is a p_0>1 so that for p in the interval (1,p_0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L^p

The mixed problem for the Laplacian in Lipschitz domains

We consider the mixed boundary value problem or Zaremba's problem for the Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We assume that the boundary between the sets where we specify Dirichlet and Neumann data is a Lipschitz surface. We require that the Neumann data is in L^p and the Dirichlet data is in the Sobolev space of functions having one derivative in L^p for some p near 1. Under these conditions, there is a unique solution to the mixed problem with the non-tangential maximal function of the gradient of the solution in L^p of the boundary. We also obtain results with data from Hardy spaces when p=1.Comment: Version 5 includes a correction to one step of the main proof. Since the paper appeared long ago, this submission includes the complete paper, followed by a short section that gives the correction to one step in the proo