Nonlinear Diffusion Equations And Curvature Conditions In Metric Measure Spaces
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Nonlinear Diffusion Equations And Curvature Conditions In Metric Measure Spaces
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Author : Luigi Ambrosio
language : en
Publisher:
Release Date : 2019
Nonlinear Diffusion Equations And Curvature Conditions In Metric Measure Spaces written by Luigi Ambrosio and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019 with Differential calculus categories.
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
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Author : Luigi Ambrosio
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-02-13
Nonlinear Diffusion Equations And Curvature Conditions In Metric Measure Spaces written by Luigi Ambrosio and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-02-13 with Education categories.
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
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Author : Fabrice Baudoin
language : en
Publisher: Springer Nature
Release Date : 2022-02-04
New Trends On Analysis And Geometry In Metric Spaces written by Fabrice Baudoin and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-02-04 with Mathematics categories.
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.
Anisotropic Isoperimetric Problems And Related Topics
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Author : Valentina Franceschi
language : en
Publisher: Springer Nature
Release Date : 2024-12-18
Anisotropic Isoperimetric Problems And Related Topics written by Valentina Franceschi and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-18 with Mathematics categories.
This book contains contributions from speakers at the "Anisotropic Isoperimetric Problems & Related Topics" conference in Rome, held from Sep 5 to 9, 2022. The classic isoperimetric problem has fascinated mathematicians of all eras, starting from the ancient Greeks, due to its simple statement: what are the sets of a given volume with minimal perimeter? The problem is mathematically well understood, and it plays a crucial role in explaining physical phenomena such as soap bubble shapes. Variations of the problem, including weighted counterparts with density dependencies, representing inhomogeneity and anisotropy of the medium, broaden its applicability, even in non-Euclidean environments, and they allow for descriptions, e.g., of crystal shapes. At large, the perimeter's physical interpretation is that of an attractive force; hence, it also appears in describing systems of particles where a balance between attractive and repulsive forces appears. A prominent example is that of Gamow's liquid drop model for atomic nuclei, where protons are subject to the strong nuclear attractive force (represented by the perimeter) and the electromagnetic repulsive force (represented by a nonlocal term). Such a model has been shown to be sound, as it explains the basic characteristics of the nuclei, and it successfully predicts nuclear fission for nuclei with a large atomic number. Similar energy functionals model various physical and biological systems, showcasing the competition between short-range interfacial and long-range nonlocal terms, leading to pattern formation. The authors mention, e.g., the Ohta–Kawasaki model for microphase separation of diblock copolymers and the Yukawa potential for colloidal systems. Despite diverse systems, the emergence of microphases follows similar patterns, although rigorously proving this phenomenon remains a challenge. The book collects several contributions within these topics, shedding light on the current state of the art.
Lectures On Nonsmooth Differential Geometry
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Author : Nicola Gigli
language : en
Publisher: Springer Nature
Release Date : 2020-02-10
Lectures On Nonsmooth Differential Geometry written by Nicola Gigli and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-02-10 with Mathematics categories.
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
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Author : Gerardo Arizmendi Echegaray
language : en
Publisher: Springer Nature
Release Date : 2022-10-27
Recent Advances In Alexandrov Geometry written by Gerardo Arizmendi Echegaray and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-10-27 with Mathematics categories.
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.
Geometric Optics For Surface Waves In Nonlinear Elasticity
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Author : Jean-François Coulombel
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-04-03
Geometric Optics For Surface Waves In Nonlinear Elasticity written by Jean-François Coulombel and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-04-03 with Education categories.
This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. The authors consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic models, which includes in particular the Saint Venant-Kirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives the leading term of an approximate Rayleigh wave solution to the underlying elasticity equations. This evolution equation, which is referred to as “the amplitude equation”, is an integrodifferential equation of nonlocal Burgers type. The authors begin by reviewing and providing some extensions of the theory of the amplitude equation. The remainder of the paper is devoted to a rigorous proof in 2D that exact, highly oscillatory, Rayleigh wave solutions uε to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength ε, and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to uε on a time interval independent of ε. This paper focuses mainly on the case of Rayleigh waves that are pulses, which have profiles with continuous Fourier spectrum, but the authors' method applies equally well to the case of wavetrains, whose Fourier spectrum is discrete.
Global Smooth Solutions For The Inviscid Sqg Equation
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Author : Angel Castro
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-09-28
Global Smooth Solutions For The Inviscid Sqg Equation written by Angel Castro and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-09-28 with Mathematics categories.
In this paper, the authors show the existence of the first non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation.
Propagating Terraces And The Dynamics Of Front Like Solutions Of Reaction Diffusion Equations On R
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Author : Peter Poláčik
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-05-13
Propagating Terraces And The Dynamics Of Front Like Solutions Of Reaction Diffusion Equations On R written by Peter Poláčik and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-13 with Education categories.
The author considers semilinear parabolic equations of the form ut=uxx+f(u),x∈R,t>0, where f a C1 function. Assuming that 0 and γ>0 are constant steady states, the author investigates the large-time behavior of the front-like solutions, that is, solutions u whose initial values u(x,0) are near γ for x≈−∞ and near 0 for x≈∞. If the steady states 0 and γ are both stable, the main theorem shows that at large times, the graph of u(⋅,t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). The author proves this result without requiring monotonicity of u(⋅,0) or the nondegeneracy of zeros of f. The case when one or both of the steady states 0, γ is unstable is considered as well. As a corollary to the author's theorems, he shows that all front-like solutions are quasiconvergent: their ω-limit sets with respect to the locally uniform convergence consist of steady states. In the author's proofs he employs phase plane analysis, intersection comparison (or, zero number) arguments, and a geometric method involving the spatial trajectories {(u(x,t),ux(x,t)):x∈R}, t>0, of the solutions in question.
Explicit Arithmetic Of Jacobians Of Generalized Legendre Curves Over Global Function Fields
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Author : Lisa Berger
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-09-28
Explicit Arithmetic Of Jacobians Of Generalized Legendre Curves Over Global Function Fields written by Lisa Berger and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-09-28 with Mathematics categories.
The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $mathbb F_p(t)$, when $p$ is prime and $rge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $mathbb F_q(t^1/d)$.