2008, 2nd ed., Lectures in mathematics ETH Zürich, ISBN 3764387211, vii, 334

This book is devoted to a theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure. It consists...

Measure theory | Metric spaces | Monotone operators | Differential equations, Partial | Evolution equations, Nonlinear | Mathematics | Differential equations, Parabolic | Probability Theory and Stochastic Processes | Differential Geometry | Measure and Integration

Measure theory | Metric spaces | Monotone operators | Differential equations, Partial | Evolution equations, Nonlinear | Mathematics | Differential equations, Parabolic | Probability Theory and Stochastic Processes | Differential Geometry | Measure and Integration

Book

Inventiones mathematicae, ISSN 0020-9910, 2/2014, Volume 195, Issue 2, pp. 289 - 391

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak...

52C23 | 49J52 | 35K90 | Mathematics, general | Mathematics | 58J35 | 49Q20 | 31C25 | EXISTENCE | MATHEMATICS | EQUATIONS | CURVATURE CONDITIONS | GEOMETRY | Mathematical analysis | Dirichlet problem | Calculus | Heat transmission | Entropy | Density | Curvature | Heat transfer

52C23 | 49J52 | 35K90 | Mathematics, general | Mathematics | 58J35 | 49Q20 | 31C25 | EXISTENCE | MATHEMATICS | EQUATIONS | CURVATURE CONDITIONS | GEOMETRY | Mathematical analysis | Dirichlet problem | Calculus | Heat transmission | Entropy | Density | Curvature | Heat transfer

Journal Article

The Annals of Probability, ISSN 0091-1798, 1/2015, Volume 43, Issue 1, pp. 339 - 404

The aim of the present paper is to bridge the gap between the Bakry–Émery and the Lott–Sturm–Villani approaches to provide synthetic and abstract notions of...

Gamma calculus | Ricci curvature | Metric measure space | Dirichlet form | Barky-émery condition | METRIC-MEASURE-SPACES | EXISTENCE | CONVEXITY | metric measure space | INEQUALITY | STATISTICS & PROBABILITY | LIPSCHITZ FUNCTIONS | Barky-Emery condition | EULERIAN CALCULUS | DIRICHLET FORMS | MANIFOLDS | HEAT-FLOW | GEOMETRY | Probability | Mathematics | Functional Analysis | Metric Geometry | Analysis of PDEs | Barky–Émery condition | 30L99 | 49Q20 | 47D07

Gamma calculus | Ricci curvature | Metric measure space | Dirichlet form | Barky-émery condition | METRIC-MEASURE-SPACES | EXISTENCE | CONVEXITY | metric measure space | INEQUALITY | STATISTICS & PROBABILITY | LIPSCHITZ FUNCTIONS | Barky-Emery condition | EULERIAN CALCULUS | DIRICHLET FORMS | MANIFOLDS | HEAT-FLOW | GEOMETRY | Probability | Mathematics | Functional Analysis | Metric Geometry | Analysis of PDEs | Barky–Émery condition | 30L99 | 49Q20 | 47D07

Journal Article

Inventiones mathematicae, ISSN 0020-9910, 3/2018, Volume 211, Issue 3, pp. 969 - 1117

We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces....

Mathematics, general | Mathematics | METRIC-MEASURE-SPACES | MATHEMATICS | INEQUALITY | EQUATIONS | CURVATURE-DIMENSION CONDITION | RICCI CURVATURE | GEOMETRY | Entropy | Functionals | Transport | Radon | Mathematics - Optimization and Control

Mathematics, general | Mathematics | METRIC-MEASURE-SPACES | MATHEMATICS | INEQUALITY | EQUATIONS | CURVATURE-DIMENSION CONDITION | RICCI CURVATURE | GEOMETRY | Entropy | Functionals | Transport | Radon | Mathematics - Optimization and Control

Journal Article

Discrete and Continuous Dynamical Systems- Series A, ISSN 1078-0947, 04/2014, Volume 34, Issue 4, pp. 1641 - 1661

We prove that the linear "heat" flow in a RCD(K, infinity) metric measure space (X, d, m) satisfies a contraction property with respect to every...

Γ-calculus | Dirichlet forms | Ricci curvature | Metric-measure spaces | Optimal transport | EXISTENCE | MATHEMATICS, APPLIED | STABILITY | INEQUALITY | optimal transport | FOKKER-PLANCK EQUATIONS | MATHEMATICS | metric-measure spaces | MANIFOLDS | Gamma-calculus | GEOMETRY

Γ-calculus | Dirichlet forms | Ricci curvature | Metric-measure spaces | Optimal transport | EXISTENCE | MATHEMATICS, APPLIED | STABILITY | INEQUALITY | optimal transport | FOKKER-PLANCK EQUATIONS | MATHEMATICS | metric-measure spaces | MANIFOLDS | Gamma-calculus | GEOMETRY

Journal Article

2005, Lectures in mathematics ETH Zürich, ISBN 9783764324285, vii, 333

This book is devoted to a theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure. It consists...

Measure theory | Differential equations, Parabolic | Metric spaces | Monotone operators | Evolution equations, Nonlinear | Mathematics | Probability Theory and Stochastic Processes | Differential Geometry | Measure and Integration

Measure theory | Differential equations, Parabolic | Metric spaces | Monotone operators | Evolution equations, Nonlinear | Mathematics | Probability Theory and Stochastic Processes | Differential Geometry | Measure and Integration

Book

Archive for Rational Mechanics and Analysis, ISSN 0003-9527, 10/2009, Volume 194, Issue 1, pp. 133 - 220

We prove the global existence of non-negative variational solutions to the “drift diffusion” evolution equation $${{\partial_t} u+ div...

Fluid- and Aerodynamics | Theoretical, Mathematical and Computational Physics | Complex Systems | Fisher Information | Classical Mechanics | Planck Equation | Relative Entropy | Physics, general | Lower Semicontinuity | Physics | Logarithmic Sobolev Inequality | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | LOGARITHMIC SOBOLEV INEQUALITIES | INTERFACE | 4TH-ORDER PARABOLIC EQUATION | FLUCTUATIONS | OPTIMAL TRANSPORTATION | GEOMETRY | Studies

Fluid- and Aerodynamics | Theoretical, Mathematical and Computational Physics | Complex Systems | Fisher Information | Classical Mechanics | Planck Equation | Relative Entropy | Physics, general | Lower Semicontinuity | Physics | Logarithmic Sobolev Inequality | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | LOGARITHMIC SOBOLEV INEQUALITIES | INTERFACE | 4TH-ORDER PARABOLIC EQUATION | FLUCTUATIONS | OPTIMAL TRANSPORTATION | GEOMETRY | Studies

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 03/2020, Volume 278, Issue 4, p. 108347

This is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space that can be characterized by Evolution...

Evolution variational inequalities | Gradient flows | Metric spaces | Minimizing movements

Evolution variational inequalities | Gradient flows | Metric spaces | Minimizing movements

Journal Article

Duke Mathematical Journal, ISSN 0012-7094, 2014, Volume 163, Issue 7, pp. 1405 - 1490

In this paper, we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X, d, m) which is stable under measured...

MATHEMATICS | SOBOLEV SPACES | TRANSPORT | INEQUALITIES | STABILITY | COMPACT ALEXANDROV SPACES | FINSLER MANIFOLDS | WASSERSTEIN SPACES | HEAT-FLOW | GRADIENT FLOWS | GEOMETRY | 60J65 | 35K05

MATHEMATICS | SOBOLEV SPACES | TRANSPORT | INEQUALITIES | STABILITY | COMPACT ALEXANDROV SPACES | FINSLER MANIFOLDS | WASSERSTEIN SPACES | HEAT-FLOW | GRADIENT FLOWS | GEOMETRY | 60J65 | 35K05

Journal Article

Journal of the European Mathematical Society, ISSN 1435-9855, 2016, Volume 18, Issue 9, pp. 2107 - 2165

J. Europ. Math. Soc. 18 (2016) 2107-2165 Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent flows by...

Energetic solutions | Existence results | Time discretization | Rate-independent systems | BV solutions | Vanishing viscosity | Mathematics - Analysis of PDEs

Energetic solutions | Existence results | Time discretization | Rate-independent systems | BV solutions | Vanishing viscosity | Mathematics - Analysis of PDEs

Journal Article

Proceedings of the London Mathematical Society, ISSN 0024-6115, 11/2015, Volume 111, Issue 5, pp. 1071 - 1129

The aim of this paper is to discuss convergence of pointed metric measure spaces in the absence of any compactness condition. We propose various definitions,...

EXISTENCE | MATHEMATICS | TANGENT-CONES | RIEMANNIAN-MANIFOLDS | UNIQUENESS | GEOMETRY | Equivalence | Mathematical analysis | Heat transmission | Spectra | Formulas (mathematics) | Curvature | Heat transfer | Convergence

EXISTENCE | MATHEMATICS | TANGENT-CONES | RIEMANNIAN-MANIFOLDS | UNIQUENESS | GEOMETRY | Equivalence | Mathematical analysis | Heat transmission | Spectra | Formulas (mathematics) | Curvature | Heat transfer | Convergence

Journal Article

ESAIM - Control, Optimisation and Calculus of Variations, ISSN 1292-8119, 01/2012, Volume 18, Issue 1, pp. 36 - 80

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...

Differential inclusions | Vanishing-viscosity contact potential | Vanishing-viscosity limit | Doubly nonlinear | Viscous regularization | Generalized gradient flows | Parameterized solutions | viscous regularization | EXISTENCE | MATHEMATICS, APPLIED | BEHAVIOR | LIMIT | MODEL | generalized gradient flows | FORMULATION | parameterized solutions | BRITTLE FRACTURES | QUASI-STATIC EVOLUTION | CRACK-GROWTH | vanishing-viscosity contact potential | differential inclusions | vanishing-viscosity limit | AUTOMATION & CONTROL SYSTEMS | PLASTICITY | Studies | Viscosity | Mathematical models | Mathematics | Calculus of variations | Mathematics - Analysis of PDEs

Differential inclusions | Vanishing-viscosity contact potential | Vanishing-viscosity limit | Doubly nonlinear | Viscous regularization | Generalized gradient flows | Parameterized solutions | viscous regularization | EXISTENCE | MATHEMATICS, APPLIED | BEHAVIOR | LIMIT | MODEL | generalized gradient flows | FORMULATION | parameterized solutions | BRITTLE FRACTURES | QUASI-STATIC EVOLUTION | CRACK-GROWTH | vanishing-viscosity contact potential | differential inclusions | vanishing-viscosity limit | AUTOMATION & CONTROL SYSTEMS | PLASTICITY | Studies | Viscosity | Mathematical models | Mathematics | Calculus of variations | Mathematics - Analysis of PDEs

Journal Article

Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 1/2013, Volume 46, Issue 1, pp. 253 - 310

In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We...

35K85 | 35K50 | 35A15 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 58E99 | Mathematics | 49Q20 | ELASTICITY | MATHEMATICS | MATHEMATICS, APPLIED | SPACES | LOWER SEMICONTINUITY | PRINCIPLE | ELLIPTIC-EQUATIONS | GRADIENT FLOWS | FORMULATION | FUNCTIONALS | TRANSFORMATIONS | RATE-INDEPENDENT SYSTEMS | Partial differential equations | Mathematical analysis | Dissipation | Nonlinear evolution equations | Mathematical models | Banach space | Calculus of variations | Cauchy problem | Mathematics - Analysis of PDEs

35K85 | 35K50 | 35A15 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 58E99 | Mathematics | 49Q20 | ELASTICITY | MATHEMATICS | MATHEMATICS, APPLIED | SPACES | LOWER SEMICONTINUITY | PRINCIPLE | ELLIPTIC-EQUATIONS | GRADIENT FLOWS | FORMULATION | FUNCTIONALS | TRANSFORMATIONS | RATE-INDEPENDENT SYSTEMS | Partial differential equations | Mathematical analysis | Dissipation | Nonlinear evolution equations | Mathematical models | Banach space | Calculus of variations | Cauchy problem | Mathematics - Analysis of PDEs

Journal Article

IEEE Transactions on Information Theory, ISSN 0018-9448, 05/2014, Volume 60, Issue 5, pp. 2687 - 2693

We associate to the pth Rényi entropy a definition of entropy power, which is the natural extension of Shannon's entropy power and exhibits a nice behavior...

Entropy | Plasmas | Mathematical model | Indexes | Equations | Space heating | nonlinear heat equation | information theory | Rényi entropy | information measure | INEQUALITIES | POROUS-MEDIUM EQUATION | INFORMATION | COMPUTER SCIENCE, INFORMATION SYSTEMS | PROOF | Renyi entropy | CENTRAL-LIMIT-THEOREM | ENGINEERING, ELECTRICAL & ELECTRONIC | Distribution (Probability theory) | Uncertainty (Information theory) | Analysis | Methods | Entropy (Information) | Mathematical analysis | Inequalities | Density | Concavity | Heat equations | Information theory

Entropy | Plasmas | Mathematical model | Indexes | Equations | Space heating | nonlinear heat equation | information theory | Rényi entropy | information measure | INEQUALITIES | POROUS-MEDIUM EQUATION | INFORMATION | COMPUTER SCIENCE, INFORMATION SYSTEMS | PROOF | Renyi entropy | CENTRAL-LIMIT-THEOREM | ENGINEERING, ELECTRICAL & ELECTRONIC | Distribution (Probability theory) | Uncertainty (Information theory) | Analysis | Methods | Entropy (Information) | Mathematical analysis | Inequalities | Density | Concavity | Heat equations | Information theory

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 05/2016, Volume 137, pp. 77 - 134

We introduce the setting of as a general “Wiener like” framework for optimal transport problems and nonsmooth metric analysis in infinite dimension. After a...

Evolution variational inequality | Heat flow | Optimal transport | MATHEMATICS | CONFIGURATION-SPACES | MATHEMATICS, APPLIED | METRIC MEASURE-SPACES | DIRICHLET FORMS | FLOWS | RADEMACHERS THEOREM | GEOMETRY | Radon | Dynamics | Mathematical analysis | Dirichlet problem | Links | Nonlinearity | Transport | Optimization

Evolution variational inequality | Heat flow | Optimal transport | MATHEMATICS | CONFIGURATION-SPACES | MATHEMATICS, APPLIED | METRIC MEASURE-SPACES | DIRICHLET FORMS | FLOWS | RADEMACHERS THEOREM | GEOMETRY | Radon | Dynamics | Mathematical analysis | Dirichlet problem | Links | Nonlinearity | Transport | Optimization

Journal Article

SIAM Journal on Mathematical Analysis, ISSN 0036-1410, 2016, Volume 48, Issue 4, pp. 2869 - 2911

We discuss a new notion of distance on the space of finite and nonnegative measures on Omega subset of R-d, which we call the Hellinger-Kantorovich distance....

Dissipation distance | Geodesic curves | Reaction-diffusion equations | Optimal transport | Cone space | Onsager operator | geodesic curves | MATHEMATICS, APPLIED | ENERGY | EULERIAN CALCULUS | LARGE-DEVIATIONS | cone space | optimal transport | dissipation distance | reaction-diffusion equations

Dissipation distance | Geodesic curves | Reaction-diffusion equations | Optimal transport | Cone space | Onsager operator | geodesic curves | MATHEMATICS, APPLIED | ENERGY | EULERIAN CALCULUS | LARGE-DEVIATIONS | cone space | optimal transport | dissipation distance | reaction-diffusion equations

Journal Article

Archive for Rational Mechanics and Analysis, ISSN 0003-9527, 1/2018, Volume 227, Issue 1, pp. 423 - 475

We study a model for rate-dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the...

Physics, general | Fluid- and Aerodynamics | Theoretical, Mathematical and Computational Physics | Complex Systems | Physics | Classical Mechanics | MICROSTRUCTURES | MATHEMATICS, APPLIED | MECHANICS | EVOLUTION | MINIMIZATION | LOWER SEMICONTINUITY | ELASTOPLASTICITY | FORMULATION | MINIMIZERS | Mathematics - Analysis of PDEs

Physics, general | Fluid- and Aerodynamics | Theoretical, Mathematical and Computational Physics | Complex Systems | Physics | Classical Mechanics | MICROSTRUCTURES | MATHEMATICS, APPLIED | MECHANICS | EVOLUTION | MINIMIZATION | LOWER SEMICONTINUITY | ELASTOPLASTICITY | FORMULATION | MINIMIZERS | Mathematics - Analysis of PDEs

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 09/2019, Volume 277, Issue 6, pp. 1868 - 1957

We investigate a first-order mean field planning problem of the form associated to a convex Hamiltonian with quadratic growth and a monotone interaction term...

Mean field planning | Kantorovich duality | Superposition principle | Optimal transport | SPACE | MATHEMATICS | GAMES | GEODESICS

Mean field planning | Kantorovich duality | Superposition principle | Optimal transport | SPACE | MATHEMATICS | GAMES | GEODESICS

Journal Article

Archive for Rational Mechanics and Analysis, ISSN 0003-9527, 2/2018, Volume 227, Issue 2, pp. 477 - 543

We propose the new notion of Visco-Energetic solutions to rate-independent systems $${(X, \mathcal{E},}$$ ( X , E , d) driven by a time dependent energy...

Physics, general | Fluid- and Aerodynamics | Theoretical, Mathematical and Computational Physics | Complex Systems | Physics | Classical Mechanics | EXISTENCE | MATHEMATICS, APPLIED | MECHANICS | QUASI-STATIC EVOLUTION | GROWTH | SYSTEMS | LIMIT | BRITTLE-FRACTURE | MODEL | ELASTOPLASTICITY | FORMULATION | DAMAGE

Physics, general | Fluid- and Aerodynamics | Theoretical, Mathematical and Computational Physics | Complex Systems | Physics | Classical Mechanics | EXISTENCE | MATHEMATICS, APPLIED | MECHANICS | QUASI-STATIC EVOLUTION | GROWTH | SYSTEMS | LIMIT | BRITTLE-FRACTURE | MODEL | ELASTOPLASTICITY | FORMULATION | DAMAGE

Journal Article