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Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces

2019 Differential calculus
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Release : 2019
Genre : Differential calculus
Kind : eBook
Book Rating : 132/5 ( reviews)

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Book Synopsis Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces by : Luigi Ambrosio

Download or read book Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces written by Luigi Ambrosio. This book was released on 2019. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces

2020-02-13 Education
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Release : 2020-02-13
Genre : Education
Kind : eBook
Book Rating : 131/5 ( reviews)

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Book Synopsis Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces by : Luigi Ambrosio

Download or read book Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces written by Luigi Ambrosio. This book was released on 2020-02-13. Available in PDF, EPUB and Kindle. Book excerpt: The 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, the authors' 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, the authors' 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.

New Trends on Analysis and Geometry in Metric Spaces

2022-02-04 Mathematics
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Release : 2022-02-04
Genre : Mathematics
Kind : eBook
Book Rating : 413/5 ( reviews)

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Book Synopsis New Trends on Analysis and Geometry in Metric Spaces by : Fabrice Baudoin

Download or read book New Trends on Analysis and Geometry in Metric Spaces written by Fabrice Baudoin. This book was released on 2022-02-04. Available in PDF, EPUB and Kindle. Book excerpt: This book includes four courses on geometric measure theory, the calculus of variations, partial differential equations, and differential geometry. Authored by leading experts in their fields, the lectures present different approaches to research topics with the common background of a relevant underlying, usually non-Riemannian, geometric structure. In particular, the topics covered concern differentiation and functions of bounded variation in metric spaces, Sobolev spaces, and differential geometry in the so-called Carnot–Carathéodory spaces. The text is based on lectures presented at the 10th School on "Analysis and Geometry in Metric Spaces" held in Levico Terme (TN), Italy, in collaboration with the University of Trento, Fondazione Bruno Kessler and CIME, Italy. The book is addressed to both graduate students and researchers.

Lectures on Nonsmooth Differential Geometry

2020-02-10 Mathematics
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Release : 2020-02-10
Genre : Mathematics
Kind : eBook
Book Rating : 139/5 ( reviews)

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Book Synopsis Lectures on Nonsmooth Differential Geometry by : Nicola Gigli

Download or read book Lectures on Nonsmooth Differential Geometry written by Nicola Gigli. This book was released on 2020-02-10. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to some aspects of the flourishing field of nonsmooth geometric analysis. In particular, a quite detailed account of the first-order structure of general metric measure spaces is presented, and the reader is introduced to the second-order calculus on spaces – known as RCD spaces – satisfying a synthetic lower Ricci curvature bound. Examples of the main topics covered include notions of Sobolev space on abstract metric measure spaces; normed modules, which constitute a convenient technical tool for the introduction of a robust differential structure in the nonsmooth setting; first-order differential operators and the corresponding functional spaces; the theory of heat flow and its regularizing properties, within the general framework of “infinitesimally Hilbertian” metric measure spaces; the RCD condition and its effects on the behavior of heat flow; and second-order calculus on RCD spaces. The book is mainly intended for young researchers seeking a comprehensive and fairly self-contained introduction to this active research field. The only prerequisites are a basic knowledge of functional analysis, measure theory, and Riemannian geometry.

Recent Advances in Alexandrov Geometry

2022-10-27 Mathematics
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Release : 2022-10-27
Genre : Mathematics
Kind : eBook
Book Rating : 985/5 ( reviews)

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Book Synopsis Recent Advances in Alexandrov Geometry by : Gerardo Arizmendi Echegaray

Download or read book Recent Advances in Alexandrov Geometry written by Gerardo Arizmendi Echegaray. This book was released on 2022-10-27. Available in PDF, EPUB and Kindle. Book excerpt: This volume is devoted to various aspects of Alexandrov Geometry for those wishing to get a detailed picture of the advances in the field. It contains enhanced versions of the lecture notes of the two mini-courses plus those of one research talk given at CIMAT. Peter Petersen’s part aims at presenting various rigidity results about Alexandrov spaces in a way that facilitates the understanding by a larger audience of geometers of some of the current research in the subject. They contain a brief overview of the fundamental aspects of the theory of Alexandrov spaces with lower curvature bounds, as well as the aforementioned rigidity results with complete proofs. The text from Fernando Galaz-García’s minicourse was completed in collaboration with Jesús Nuñez-Zimbrón. It presents an up-to-date and panoramic view of the topology and geometry of 3-dimensional Alexandrov spaces, including the classification of positively and non-negatively curved spaces and the geometrization theorem. They also present Lie group actions and their topological and equivariant classifications as well as a brief account of results on collapsing Alexandrov spaces. Jesús Nuñez-Zimbrón’s contribution surveys two recent developments in the understanding of the topological and geometric rigidity of singular spaces with curvature bounded below.

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