BV solutions and viscosity approximations of rate-independent systems

In the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of 'BV solutions' involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting BV solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems

Measure valued solutions of sub-linear diffusion equations with a drift term

In this paper we study nonnegative, measure valued solutions of the initial value problem for one-dimensional drift-diffusion equations when the nonlinear diffusion is governed by an increasing $C^1$ function $\beta$ with $\lim_{r\to +\infty} \beta(r)<+\infty$. By using tools of optimal transport, we will show that this kind of problems is well posed in the class of nonnegative Borel measures with finite mass $m$ and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the so called $L^2$-Wasserstein distance. Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass ${m}_{\rm c}$, which can be explicitely characterized in terms of $\beta$ and of the drift term. If the initial mass is less then ${m}_{\rm c}$, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass $m$ of the solutions is greater than the critical one, the steady state has a singular part in which the exceeding mass ${m} - {m}_{\rm c}$ is accumulated.Comment: 30 page

LECTURE NOTES ON GRADIENT FLOWS AND OPTIMAL TRANSPORT

We present a short overview on the strongest variational formulation for gradi- ent flows of geodesically λ-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School “Optimal transportation: Theory and applications” in Grenoble during the week of June 22-26, 2009

Weak topology and Opial property in Wasserstein spaces, with applications to gradient flows and proximal point algorithms of geodesically convex functionals

In this paper we discuss how to define an appropriate notion of weak topology in the Wasserstein space ðP2ðHÞ; W2Þ of Borel probability measures with finite quadratic moment on a separable Hilbert space H. We will show that such a topology inherits many features of the usual weak topology in Hilbert spaces, in particular the weak closedness of geodesically convex closed sets and the Opial property characterising weakly convergent sequences. We apply this notion to the approximation of fixed points for a non-expansive map in a weakly closed subset of P2ðHÞ and of minimizers of a lower semicontinuous and geodesically convex functional f: P2ðHÞ ! ð-l; þl] attaining its minimum. In particular, we will show that every solution to the Wasserstein gradient flow of f weakly converge to a minimizer of f as the time goes to þl. Similarly, if f is also convex along generalized geodesics, every sequence generated by the proximal point algorithm converges to a minimizer of f with respect to the weak topology of P2ðHÞ

Duality properties of metric Sobolev spaces and capacity

We study the properties of the dual Sobolev space H-1;q(X) = - H1;p(X) '0 on a complete extended metric-topological measure space X = (X; ⊤; d;m) for p 2 (1;1). We will show that a crucial role is played by the strong closure H-1;q pd (X) of Lq(X;m) in the dual H-1;q(X), which can be identified with the predual of H1;p(X). We will show that positive functionals in H-1;q(X) can be represented as a positive Radon measure and we will charaterize their dual norm in terms of a suitable energy functional on nonparametric dynamic plans. As a byproduct, we will show that for every Radon measure μ with finite dual Sobolev energy, Capp-negligible sets are also μ-negligible and good representatives of Sobolev functions belong to L1(X; μ). We eventually show that the Newtonian-Sobolev capacity Capp admits a natural dual representation in terms of such a class of Radon measures

Gradient flows and Evolution Variational Inequalities in metric spaces. I: structural properties

This is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space (X,d) that can be characterized by Evolution Variational Inequalities (EVI). We present new results concerning the structural properties of solutions to the EVI formulation, such as contraction, regularity, asymptotic expansion, precise energy identity, stability, asymptotic behavior and their link with the geodesic convexity of the driving functional. Under the crucial assumption of the existence of an EVI gradient flow, we will also prove two main results: – the equivalence with the De Giorgi variational characterization of curves of maximal slope; – the convergence of the Minimizing Movement-JKO scheme to the EVI gradient flow, with an explicit and uniform error estimate of order 1/2 with respect to the step size, independent of any geometric hypothesis (as upper or lower curvature bounds) on d. In order to avoid any compactness assumption, we will also introduce a suitable relaxation of the Minimizing Movement algorithm obtained by the Ekeland variational principle, and we will prove its uniform convergence as well