Savaré, Giuseppe
Overview
Works:  33 works in 104 publications in 2 languages and 2,079 library holdings 

Genres:  Conference papers and proceedings 
Roles:  Editor, Author, 958, Other, Opponent 
Publication Timeline
.
Most widely held works by
Giuseppe Savaré
Gradient flows : in metric spaces and in the space of probability measures by
Luigi Ambrosio(
)
47 editions published between 2004 and 2008 in English and German and held by 1,475 WorldCat member libraries worldwide
"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 of two parts, the first one concerning gradient flows in metric spaces and the second one devoted to gradient flows in the space of probability measures on a separable Hilbert space, endowed with the KantorovichRubinsteinWasserstein distance."Jacket
47 editions published between 2004 and 2008 in English and German and held by 1,475 WorldCat member libraries worldwide
"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 of two parts, the first one concerning gradient flows in metric spaces and the second one devoted to gradient flows in the space of probability measures on a separable Hilbert space, endowed with the KantorovichRubinsteinWasserstein distance."Jacket
Nonlinear PDE's and applications : C.I.M.E. Summer School, Cetraro, Italy, 2008 by
Stefano Bianchini(
)
14 editions published in 2011 in English and held by 487 WorldCat member libraries worldwide
This volume collects the notes of the CIME course "Nonlinear PDE’s and applications" held in Cetraro (Italy) on June 23–28, 2008. It consists of four series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), and Cédric Villani (Ecole Normale Superieure de Lyon). They presented a broad overview of farreaching findings and exciting new developments concerning, in particular, optimal transport theory, nonlinear evolution equations, functional inequalities, and differential geometry. A sampling of the main topics considered here includes optimal transport, HamiltonJacobi equations, Riemannian geometry, and their links with sharp geometric/functional inequalities, variational methods for studying nonlinear evolution equations and their scaling properties, and the metric/energetic theory of gradient flows and of rateindependent evolution problems. The book explores the fundamental connections between all of these topics andpoints to new research directions in contributions by leading experts in these fields
14 editions published in 2011 in English and held by 487 WorldCat member libraries worldwide
This volume collects the notes of the CIME course "Nonlinear PDE’s and applications" held in Cetraro (Italy) on June 23–28, 2008. It consists of four series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), and Cédric Villani (Ecole Normale Superieure de Lyon). They presented a broad overview of farreaching findings and exciting new developments concerning, in particular, optimal transport theory, nonlinear evolution equations, functional inequalities, and differential geometry. A sampling of the main topics considered here includes optimal transport, HamiltonJacobi equations, Riemannian geometry, and their links with sharp geometric/functional inequalities, variational methods for studying nonlinear evolution equations and their scaling properties, and the metric/energetic theory of gradient flows and of rateindependent evolution problems. The book explores the fundamental connections between all of these topics andpoints to new research directions in contributions by leading experts in these fields
Gradient Flows in Metric Spaces and in the Space of Probability Measures by
Luigi Ambrosio(
)
5 editions published between 2005 and 2008 in English and Undetermined and held by 16 WorldCat member libraries worldwide
This book is devoted to a theory of gradient ?ows in spaces which are not nec sarily endowed with a natural linear or di?erentiable structure. It is made of two parts, the ?rst one concerning gradient ?ows in metric spaces and the second one 2 1 devoted to gradient ?ows in the L Wasserstein space of probability measures on p a separable Hilbert space X (we consider the L Wasserstein distance, p? (1,?), as well). The two parts have some connections, due to the fact that the Wasserstein space of probability measures provides an important model to which the "metric" theory applies, but the book is conceived in such a way that the two parts can be read independently, the ?rst one by the reader more interested to NonSmooth Analysis and Analysis in Metric Spaces, and the second one by the reader more oriented to theapplications in Partial Di?erential Equations, Measure Theory and Probability
5 editions published between 2005 and 2008 in English and Undetermined and held by 16 WorldCat member libraries worldwide
This book is devoted to a theory of gradient ?ows in spaces which are not nec sarily endowed with a natural linear or di?erentiable structure. It is made of two parts, the ?rst one concerning gradient ?ows in metric spaces and the second one 2 1 devoted to gradient ?ows in the L Wasserstein space of probability measures on p a separable Hilbert space X (we consider the L Wasserstein distance, p? (1,?), as well). The two parts have some connections, due to the fact that the Wasserstein space of probability measures provides an important model to which the "metric" theory applies, but the book is conceived in such a way that the two parts can be read independently, the ?rst one by the reader more interested to NonSmooth Analysis and Analysis in Metric Spaces, and the second one by the reader more oriented to theapplications in Partial Di?erential Equations, Measure Theory and Probability
Balanced viscosity (BV) solutions to infinitedimensional rateindependent systems by
Alexander Mielke(
Book
)
2 editions published in 2013 in English and held by 12 WorldCat member libraries worldwide
2 editions published in 2013 in English and held by 12 WorldCat member libraries worldwide
Variational convergence of gradient flows and rateindependent evolutions in metric spaces(
)
1 edition published in 2012 in English and held by 9 WorldCat member libraries worldwide
1 edition published in 2012 in English and held by 9 WorldCat member libraries worldwide
Nonsmooth analysis of doubly nonlinear evolution equations(
)
1 edition published in 2011 in English and held by 9 WorldCat member libraries worldwide
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 timediscretization scheme with variational techniques. Finally, we discuss an application to a material model in finitestrain elasticity
1 edition published in 2011 in English and held by 9 WorldCat member libraries worldwide
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 timediscretization scheme with variational techniques. Finally, we discuss an application to a material model in finitestrain elasticity
Optimal EntropyTransport problems and a new HellingerKantorovich distance between positive measures by
Matthias Liero(
)
1 edition published in 2016 in English and held by 8 WorldCat member libraries worldwide
We develop a full theory for the new class of Optimal EntropyTransport 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 EntropyTransport problems and introduce the new HellingerKantorovich 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 wellknown HellingerKakutani and KantorovichWasserstein distances
1 edition published in 2016 in English and held by 8 WorldCat member libraries worldwide
We develop a full theory for the new class of Optimal EntropyTransport 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 EntropyTransport problems and introduce the new HellingerKantorovich 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 wellknown HellingerKakutani and KantorovichWasserstein distances
Global existence results for viscoplasticity at finite strain by
Alexander Mielke(
)
1 edition published in 2016 in English and held by 8 WorldCat member libraries worldwide
We study a model for ratedependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of globalintime 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 finitestrain elasticity as well as the multiplicative decomposition of finitestrain plasticity. Moreover, the dissipation potential depends on the leftinvariant 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 energydissipationbalance (EDB) and energydissipationinequality (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
1 edition published in 2016 in English and held by 8 WorldCat member libraries worldwide
We study a model for ratedependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of globalintime 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 finitestrain elasticity as well as the multiplicative decomposition of finitestrain plasticity. Moreover, the dissipation potential depends on the leftinvariant 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 energydissipationbalance (EDB) and energydissipationinequality (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
Optimal transport in competition with reaction: The HellingerKantorovich distance and geodesic curves by
Matthias Liero(
)
1 edition published in 2015 in English and held by 7 WorldCat member libraries worldwide
We discuss a new notion of distance on the space of finite and nonnegative measures on Omega C Rd, which we call HellingerKantorovich distance. It can be seen as an infconvolution of the wellknown KantorovichWasserstein distance and the HellingerKakutani 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 some of its properties. In particular, the distance can be equivalently described by an optimal transport problem on the cone space over the underlying space Omega. We give a construction of geodesic curves and discuss examples and their general properties
1 edition published in 2015 in English and held by 7 WorldCat member libraries worldwide
We discuss a new notion of distance on the space of finite and nonnegative measures on Omega C Rd, which we call HellingerKantorovich distance. It can be seen as an infconvolution of the wellknown KantorovichWasserstein distance and the HellingerKakutani 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 some of its properties. In particular, the distance can be equivalently described by an optimal transport problem on the cone space over the underlying space Omega. We give a construction of geodesic curves and discuss examples and their general properties
A posteriori error estimates for variable timestep discretizations of nonlinear evolution equations by
Ricardo H Nochetto(
Book
)
4 editions published between 1998 and 1999 in English and held by 5 WorldCat member libraries worldwide
4 editions published between 1998 and 1999 in English and held by 5 WorldCat member libraries worldwide
A metric approach to a class fo doubly nonlinear evolution euations and applications(
)
1 edition published in 2007 in English and held by 3 WorldCat member libraries worldwide
This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in $L^1$ spaces
1 edition published in 2007 in English and held by 3 WorldCat member libraries worldwide
This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in $L^1$ spaces
Variational convergence of gradient flows and rateindependent evolutions in metric spaces by
Alexander Mielke(
Book
)
1 edition published in 2012 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 2012 in English and held by 3 WorldCat member libraries worldwide
Nonsmooth analysis of doubly nonlinear evolution equations by
Alexander Mielke(
Book
)
1 edition published in 2011 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 2011 in English and held by 3 WorldCat member libraries worldwide
Modeling solutions with jumps for rateindependent systems on metric spaces(
)
1 edition published in 2008 in English and held by 3 WorldCat member libraries worldwide
Rateindependent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rateindependent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of generalized gradient flows in metric spaces, and comparison with other notions of solutions is given
1 edition published in 2008 in English and held by 3 WorldCat member libraries worldwide
Rateindependent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rateindependent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of generalized gradient flows in metric spaces, and comparison with other notions of solutions is given
Modeling solutions with jumps for rate independent systems on metric spaces by
Alexander Mielke(
Book
)
1 edition published in 2008 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 2008 in English and held by 3 WorldCat member libraries worldwide
BV solutions and viscosity approximations of rateindependent systems(
)
1 edition published in 2009 in English and held by 3 WorldCat member libraries worldwide
In the nonconvex case solutions of rateindependent 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 finitedimensional case we study the vanishingviscosity 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 rateindependent dissipation potential. The resulting definition of BV solutions' involves, in a nontrivial way, both the rateindependent 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 timecontinuous and for the timediscretized 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 rateindependent systems
1 edition published in 2009 in English and held by 3 WorldCat member libraries worldwide
In the nonconvex case solutions of rateindependent 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 finitedimensional case we study the vanishingviscosity 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 rateindependent dissipation potential. The resulting definition of BV solutions' involves, in a nontrivial way, both the rateindependent 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 timecontinuous and for the timediscretized 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 rateindependent systems
A metric approach to a class of doubly nonlinear evolution equations and applications by
Riccarda Rossi(
Book
)
1 edition published in 2007 in English and held by 3 WorldCat member libraries worldwide
This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in $L1$ spaces
1 edition published in 2007 in English and held by 3 WorldCat member libraries worldwide
This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in $L1$ spaces
Error control of nonlinear evolution equations by
Ricardo H Nochetto(
Book
)
2 editions published in 1998 in English and held by 3 WorldCat member libraries worldwide
2 editions published in 1998 in English and held by 3 WorldCat member libraries worldwide
BV solutions and viscosity approximations of rateindependent systems by
Alexander Mielke(
Book
)
1 edition published in 2009 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 2009 in English and held by 3 WorldCat member libraries worldwide
Gradient Flows : In Metric Spaces and in the Space of Probability Measures by
Nicola Gigli(
)
3 editions published in 2008 in English and held by 2 WorldCat member libraries worldwide
Devoted to a theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure, this book focuses on gradient flows in metric spaces. It covers gradient flows in the space of probability measures on a separable Hilbert space, endowed with the KantorovichRubinsteinWasserstein distance
3 editions published in 2008 in English and held by 2 WorldCat member libraries worldwide
Devoted to a theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure, this book focuses on gradient flows in metric spaces. It covers gradient flows in the space of probability measures on a separable Hilbert space, endowed with the KantorovichRubinsteinWasserstein distance
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Related Identities
 Ambrosio, Luigi Author Editor
 Gigli, Nicola
 Gigli, Nicola Other Opponent Thesis advisor Author Editor
 Centro internazionale matematico estivo Editor
 Bianchini, S. (Stefano)
 Bianchini, Stefano 1970
 Mielke, Alexander Author
 Rossi, Riccarda Author
 WeierstraßInstitut für Angewandte Analysis und Stochastik
 Liero, Matthias Author
Associated Subjects
Differential equations, Nonlinear Differential equations, Parabolic Differential equations, Partial Distribution (Probability theory) Evolution equations, Nonlinear Functional analysis Geometry, Differential Global analysis (Mathematics) Global differential geometry Mathematical optimization Mathematics Measure theory Metric spaces Monotone operators