arXiv subjects

Giuseppe Savaré

Publications and source records attributed to Giuseppe Savaré.

At least 37 records · Page 2Linked to original sources

arXiv2022

Visco-energetic solutions for a model of crack growth in brittle materials

Visco-energetic solutions have been recently advanced as a new solution concept for rate-independent systems, alternative to energetic solutions/quasistatic evolutions and balanced viscosity solutions. In the spirit of this novel concept, we revisit the analysis of the variational model proposed by Francfort and Marigo for the quasi-static crack growth in brittle materials, in the case of antiplane shear. In this context, visco-energetic solutions can be constructed by perturbing the time incremental scheme for quasistatic evolutions by means of a viscous correction inspired by the term introduced by Almgren, Taylor, and Wang in the study of mean curvature flows. With our main result we prove the existence of a visco-energetic solution with a given initial crack. We also show that, if the cracks have a finite number of tips evolving smoothly on a given time interval, visco-energetic solutions comply with Griffith's criterion.

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arXiv2021

Dissipative probability vector fields and generation of evolution semigroups in Wasserstein spaces

We introduce and investigate a notion of multivalued $λ$-dissipative probability vector field (MPVF) in the Wasserstein space $\mathcal{P}_2(\mathsf X)$ of Borel probability measures on a Hilbert space $\mathsf X$. Taking inspiration from the theory of dissipative operators in Hilbert spaces and of Wasserstein gradient flows of geodesically convex functionals, we study local and global well posedness of evolution equations driven by dissipative MPVFs. Our approach is based on a measure-theoretic version of the Explicit Euler scheme, for which we prove novel convergence results with optimal error estimates under an abstract CFL stability condition, which do not rely on compactness arguments and also hold when $\mathsf X$ has infinite dimension. We characterize the limit solutions by a suitable Evolution Variational Inequality (EVI), inspired by the Bénilan notion of integral solutions to dissipative evolutions in Banach spaces. Existence, uniqueness and stability of EVI solutions are then obtained under quite general assumptions, leading to the generation of a semigroup of nonlinear contractions.

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arXiv2021

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 $(\mathcal{P}_2(H),W_2)$ 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 characterizing weakly convergent sequences. We apply this notion to the approximation of fixed points for a non-expansive map in a weakly closed subset of $\mathcal{P}_2(H)$ and of minimizers of a lower semicontinuous and geodesically convex functional $ϕ:\mathcal{P}_2(H)\to(-\infty,+\infty]$ attaining its minimum. In particular, we will show that every solution to the Wasserstein gradient flow of $ϕ$ weakly converge to a minimizer of $ϕ$ as the time goes to $+\infty$. Similarly, if $ϕ$ is also convex along generalized geodesics, every sequence generated by the proximal point algorithm converges to a minimizer of $ϕ$ with respect to the weak topology of $\mathcal{P}_2(H)$.

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arXiv2020

Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergence

This paper is devoted to the study of multi-agent deterministic optimal control problems. We initially provide a thorough analysis of the Lagrangian, Eulerian and Kantorovich formulations of the problems, as well as of their relaxations. Then we exhibit some equivalence results among the various representations and compare the respective value functions. To do it, we combine techniques and ideas from optimal transportation, control theory, Young measures and evolution equations in Banach spaces. We further exploit the connections among Lagrangian and Eulerian descriptions to derive consistency results as the number of particles/agents tends to infinity. To that purpose we prove an empirical version of the Superposition Principle and obtain suitable Gamma-convergence results for the controlled systems.

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arXiv2020

Jump processes as Generalized Gradient Flows

We have created a functional framework for a class of non-metric gradient systems. The state space is a space of nonnegative measures, and the class of systems includes the Forward Kolmogorov equations for the laws of Markov jump processes on Polish spaces. This framework comprises a definition of a notion of solutions, a method to prove existence, and an archetype uniqueness result. We do this by using only the structure that is provided directly by the dissipation functional, which need not be homogeneous, and we do not appeal to any metric structure.

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arXiv2019

Sobolev spaces in extended metric-measure spaces

These lecture notes contain an extended version of the material presented in the C.I.M.E. summer course in 2017, aiming to give a detailed introduction to the metric Sobolev theory. The notes are divided in four main parts. The first one is devoted to a preliminary study of the underlying topological, metric, and measure-theoretic aspects of a general extended metric-topological measure space $\mathbb X=(X,τ,\mathsf d,\mathfrak m)$. The second part is devoted to the construction of the Cheeger energy, initially defined on a distinguished unital algebra of Lipschitz functions. The third part deals with the basic tools needed for the dual characterization of the Sobolev spaces: the notion of $p$-Modulus of a collection of (nonparametric) rectifiable arcs and its duality with the class of nonparametric dynamic plans, i.e.~Radon measures on the space of rectifiable arcs with finite $q$-barycentric entropy with respect to $\mathfrak m$. The final part of the notes is devoted to the dual/weak formulation of the Sobolev spaces $W^{1,p}(\mathbb X)$ in terms of nonparametric dynamic plans and to their relations with the Newtonian spaces $N^{1,p}(\mathbb X)$ and with the spaces $H^{1,p}(\mathbb X)$ obtained by the Cheeger construction. In particular, when $(X,\mathsf d)$ is complete, a new proof of the equivalence between these approaches is given by a direct duality argument. A substantial part of these Lecture notes relies on well established theories. New contributions concern the extended metric setting, the role of general compatible algebras of Lipschitz functions and their density w.r.t.~the Sobolev energy, a compactification trick, the study of reflexivity and infinitesimal Hilbertianity inherited from the underlying space, and the use of nonparametric dynamic plans for the definition of weak upper gradients.

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arXiv2019

Contraction and regularizing properties of heat flows in metric measure spaces

We illustrate some novel contraction and regularizing properties of the Heat flow in metric-measure spaces that emphasize an interplay between Hellinger-Kakutani, Kantorovich-Wasserstein and Hellinger-Kantorvich distances. Contraction properties of Hellinger-Kakutani distances and general Csiszár divergences hold in arbitrary metric-measure spaces and do not require assumptions on the linearity of the flow. When weaker transport distances are involved, we will show that contraction and regularizing effects rely on the dual formulations of the distances and are strictly related to lower Ricci curvature bounds in the setting of $RCD(K,\infty)$ metric measure spaces.

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arXiv2018

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 $(X,\mathsf{d})$ that can be characterized by Evolution Variational Inequalities. We present new results concerning the structural properties of solutions to the $\mathrm{EVI}$ formulation, such as contraction, regularity, asymptotic expansion, precise energy identity, stability, asymptotic behaviour and their link with the geodesic convexity of the driving functional. Under the crucial assumption of the existence of an $\mathrm{EVI}$ gradient flow, we will also prove two main results: the equivalence with the De Giorgi variational characterization of curves of maximal slope and the convergence of the Minimizing Movement-JKO scheme to the $\mathrm{EVI}$ gradient flow, with an explicit and uniform error estimate of order $1/2$ with respect to the step size, independent of any geometric hypothesis (such as upper or lower curvature bounds) on $\mathsf{d}$. 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.

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arXiv2018

A variational approach to the mean field planning problem

We investigate a first-order mean field planning problem of the form \begin{equation} \left\lbrace\begin{aligned} -\partial_t u + H(x,Du) &= f(x,m) &&\text{in } (0,T)\times \mathbb{R}^d, \\ \partial_t m - \nabla\cdot (m\,H_p(x,Du)) &= 0 &&\text{in }(0,T)\times \mathbb{R}^d,\\ m(0,\cdot) = m_0, \; m(T,\cdot) &= m_T &&\text{in } \mathbb{R}^d, \end{aligned}\right. \end{equation} associated to a convex Hamiltonian $H$ with quadratic growth and a monotone interaction term $f$ with polynomial growth. We exploit the variational structure of the system, which encodes the first order optimality condition of a convex dynamic optimal entropy-transport problem with respect to the unknown density $m$ and of its dual, involving the maximization of an integral functional among all the subsolutions $u$ of an Hamilton-Jacobi equation. Combining ideas from optimal transport, convex analysis and renormalized solutions to the continuity equation, we will prove existence and (at least partial) uniqueness of a weak solution $(m,u)$. A crucial step of our approach relies on a careful analysis of distributional subsolutions to Hamilton-Jacobi equations of the form $-\partial_t u + H(x,Du) \leq α$, under minimal summability conditions on $α$, and to a measure-theoretic description of the optimality via a suitable contact-defect measure. Finally, using the superposition principle, we are able to describe the solution to the system by means of a measure on the path space encoding the local behavior of the players.

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arXiv2018

Spatially Inhomogeneous Evolutionary Games

We introduce and study a mean-field model for a system of spatially distributed players interacting through an evolutionary game driven by a replicator dynamics. Strategies evolve by a replicator dynamics influenced by the position and the interaction between different players and return a feedback on the velocity field guiding their motion. One of the main novelties of our approach concerns the description of the whole system, which can be represented by an evolving probability measure $Σ$ on an infinite dimensional state space (pairs $(x,σ)$ of position and distribution of strategies). We provide a Lagrangian and a Eulerian description of the evolution, and we prove their equivalence, together with existence, uniqueness, and stability of the solution. As a byproduct of the stability result, we also obtain convergence of the finite agents model to our mean-field formulation, when the number $N$ of the players goes to infinity, and the initial discrete distribution of positions and strategies converge. To this aim we develop some basic functional analytic tools to deal with interaction dynamics and continuity equations in Banach spaces, that could be of independent interest.

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arXiv2018

Mean-field optimal control as Gamma-limit of finite agent controls

This paper focuses on the role of a government of a large population of interacting agents as a mean field optimal control problem derived from deterministic finite agent dynamics. The control problems are constrained by a PDE of continuity-type without diffusion, governing the dynamics of the probability distribution of the agent population. We derive existence of optimal controls in a measure-theoretical setting as natural limits of finite agent optimal controls without any assumption on the regularity of control competitors. In particular, we prove the consistency of mean-field optimal controls with corresponding underlying finite agent ones. The results follow from a $Γ$-convergence argument constructed over the mean-field limit, which stems from leveraging the superposition principle.

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arXiv2018

Weighted Energy-Dissipation principle for gradient flows in metric spaces

This paper develops the so-called Weighted Energy-Dissipation (WED) variational approach for the analysis of gradient flows in metric spaces. This focuses on the minimization of the parameter-dependent global-in-time functional of trajectories \[ \mathcal{I}_\varepsilon[u] = \int_0^{\infty} e^{-t/\varepsilon}\left( \frac12 |u'|^2(t) + \frac1{\varepsilon}ϕ(u(t)) \right) \dd t, \] featuring the weighted sum of energetic and dissipative terms. As the parameter $\varepsilon$ is sent to~$0$, the minimizers $u_\varepsilon$ of such functionals converge, up to subsequences, to curves of maximal slope driven by the functional $ϕ$. This delivers a new and general variational approximation procedure, hence a new existence proof, for metric gradient flows. In addition, it provides a novel perspective towards relaxation.

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arXiv2017

Reverse approximation of gradient flows as Minimizing Movements: a conjecture by De Giorgi

We consider the Cauchy problem for the gradient flow \begin{equation} \label{eq:81} \tag{$\star$} u'(t)=-\nablaϕ(u(t)),\quad t\ge 0;\quad u(0)=u_0, \end{equation} generated by a continuously differentiable function $ϕ:\mathbb H \to \mathbb R$ in a Hilbert space $\mathbb H$ and study the reverse approximation of solutions to ($\star$) by the De Giorgi Minimizing Movement approach. We prove that if $\mathbb H$ has finite dimension and $ϕ$ is quadratically bounded from below (in particular if $ϕ$ is Lipschitz) then for every solution $u$ to ($\star$) (which may have an infinite number of solutions) there exist perturbations $ϕ_τ:\mathbb H \to \mathbb R \ (τ>0)$ converging to $ϕ$ in the Lipschitz norm such that $u$ can be approximated by the Minimizing Movement scheme generated by the recursive minimization of $Φ(τ,U,V):=\frac 1{2τ}|V-U|^2+ ϕ_τ(V)$: \begin{equation} \label{eq:abstract} \tag{$\star\star$} U_τ^n\in \operatorname{argmin}_{V\in \mathbb H} Φ(τ,U_τ^{n-1},V)\quad n\in\mathbb N, \quad U_τ^0:=u_0. \end{equation} We show that the piecewise constant interpolations with time step $τ> 0$ of all possible selections of solutions $(U_τ^n)_{n\in\mathbb N}$ to ($\star\star$) will converge to $u$ as $τ\downarrow 0$. This result solves a question raised by Ennio De Giorgi. We also show that even if $\mathbb H$ has infinite dimension the above approximation holds for the distinguished class of minimal solutions to ($\star$), that generate all the other solutions to ($\star$) by time reparametrization.

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arXiv2017

Viscous corrections of the Time Incremental Minimization Scheme and Visco-Energetic Solutions to Rate-Independent Evolution Problems

We propose the new notion of Visco-Energetic solutions to rate-independent systems $(X,\mathcal E,\mathsf d)$ driven by a time dependent energy $\mathcal E$ and a dissipation quasi-distance $\mathsf d$ in a general metric-topological space $X$. As for the classic Energetic approach, solutions can be obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation (quasi-)distance $\mathsf d$ is incremented by a viscous correction $δ$ (e.g.~proportional to the square of the distance $\mathsf d$), which penalizes far distance jumps by inducing a localized version of the stability condition. We prove a general convergence result and a typical characterization by Stability and Energy Balance in a setting comparable to the standard energetic one, thus capable to cover a wide range of applications. The new refined Energy Balance condition compensates the localized stability and provides a careful description of the jump behavior: at every jump the solution follows an optimal transition, which resembles in a suitable variational sense the discrete scheme that has been implemented for the whole construction.

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arXiv2017

Nonlinear diffusion equations and curvature conditions in metric measure spaces

Aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, our new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, our new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD*(K,N) condition of Bacher-Sturm.

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arXiv2016

Global existence results for viscoplasticity at finite strain

We study a model for rate-dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of global-in-time solutions to the related PDE system. We reveal its underlying structure as a generalized gradient system, where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite-strain elasticity as well as the multiplicative decomposition of finite-strain plasticity. Moreover, the dissipation potential depends on the left-invariant plastic rate and thus, depends on the plastic state variable. The existence theory is developed for a class of abstract, nonsmooth, and nonconvex gradient systems, for which we introduce suitable notions of solutions, namely energy-dissipation-balance (EDB) and energy-dissipation-inequality (EDI) solutions. Hence, we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory.

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arXiv2016

Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures

We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, that quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.

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arXiv2015

Optimal transport in competition with reaction: the Hellinger-Kantorovich distance and geodesic curves

We discuss a new notion of distance on the space of finite and nonnegative measures which can be seen as a generalization of the well-known Kantorovich-Wasserstein distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and its properties. In fact the distance can be equivalently described by an optimal transport problem on the cone space over the underlying metric space. We give a construction of geodesic curves and discuss their properties.

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