| Description |
1 online resource |
| Series |
Lecture notes in mathematics. C.I.M.E. foundation subseries ; volume 2296 |
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Lecture notes in mathematics (Springer-Verlag). CIME Foundation subseries.
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Lecture notes in mathematics (Springer-Verlag) ; 2296.
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| Contents |
Intro -- Contents -- Introduction to the Notes of the School on Analysis and Geometry in Metric Spaces -- Geometric Inequalities on Riemannian and Sub-Riemannian Manifolds by Heat Semigroups Techniques -- 1 Introduction -- 2 Subelliptic Diffusion Operators -- 2.1 Diffusion Operators -- 2.2 Subelliptic Diffusion Operators -- 2.3 The Distance Associated to Subelliptic Diffusion Operators -- 2.4 Essentially Self-Adjoint Subelliptic Operators -- 2.5 The Heat Semigroup Associated to a Subelliptic Diffusion Operator |
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3 The Heat Semigroup on a Complete Riemannian Manifold and Its Geometric Applications -- 3.1 The Laplace-Beltrami Operator -- 3.2 The Heat Semigroup on a Compact Riemannian Manifold -- 3.3 Bochner's Identity -- 3.4 The Curvature Dimension Inequality -- 3.5 Stochastic Completeness -- 3.6 Convergence to Equilibrium, Poincaré and Log-Sobolev Inequalities -- 3.7 The Li-Yau Inequality -- 3.8 The Parabolic Harnack Inequality -- 3.9 The Gaussian Upper Bound -- 3.10 Volume Doubling Property -- 3.11 Upper and Lower Gaussian Bounds for the Heat Kernel -- 3.12 The Poincaré Inequality on Domains |
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6 Weak Besicovitch Covering Property -- References -- Sobolev Spaces in Extended Metric-Measure Spaces -- 1 Introduction -- 1.1 Main Notation -- 2 Topological and Metric-Measure Structures -- 2.1 Metric-Measure Structures -- 2.1.1 Topological and Measure Theoretic Notions -- 2.1.2 Extended Metric-Topological (Measure) Spaces -- 2.1.3 Examples -- 2.1.4 The Kantorovich-Rubinstein Distance -- 2.1.5 The Asymptotic Lipschitz Constant -- 2.1.6 Compatible Algebra of Functions -- 2.1.7 Embedding and Compactification of Extended Metric-Measure Spaces -- 2.1.8 Notes |
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2.2 Continuous Curves and Nonparametric Arcs -- 2.2.1 Continuous Curves -- 2.2.2 Arcs -- 2.2.3 Rectifiable Arcs -- 2.2.4 Notes -- 2.3 Length and Conformal Distances -- 2.3.1 The Length Property -- 2.3.2 Conformal Distances -- 2.3.3 Duality for Kantorovich-Rubinstein Cost Functionals Induced by Conformal Distances -- 2.3.4 Notes -- 3 The Cheeger Energy -- 3.1 The Strongest Form of the Cheeger Energy -- 3.1.1 Relaxed Gradients and Local Representation of the Cheeger Energy -- 3.1.2 Invariance w.r.t. Restriction and Completion -- 3.1.3 Notes |
| Summary |
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 |
| Notes |
Includes index |
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Online resource; title from PDF title page (SpringerLink, viewed February 11, 2022) |
| Subject |
Geometric measure theory.
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Calculus of variations.
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Differential equations, Partial.
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Geometry, Differential.
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Calculus of variations
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Differential equations, Partial
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Geometric measure theory
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Geometry, Differential
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Teoria de la mesura geomètrica.
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Càlcul de variacions.
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Equacions en derivades parcials.
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Geometria diferencial.
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| Genre/Form |
Llibres electrònics.
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| Form |
Electronic book
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| Author |
Rigot, Séverine, author.
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Savaré, Giuseppe, author.
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Shanmugalingam, Nageswari, author.
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Ambrosio, Luigi, editor.
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Franchi, Bruno (Mathematician), editor.
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Markina, Irina (Mathematician), editor.
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Serra Cassano, Francesco, editor
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| ISBN |
9783030841416 |
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3030841413 |
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