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On the Differential Structure of Metric Measure Spaces and Applications

Author: Nicola Gigli

Publisher:

Published: 2015

Total Pages: 91

ISBN-13: 9781470422790

The main goals of this paper are: (i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Radon and non-negative. (ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like Δg=μ, where g is a function and μ is a measure. (iii) To show that on spaces with Ricci curvature bounded from below and dimension bounded from above, the Laplacian of the distance function is always a measure and that this measure has the standard sharp comparison properties. This result requires an additional assumption on the space, which reduces to strict convexity of the norm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail.

On the Differential Structure of Metric Measure Spaces and Applications

Author: Nicola Gigli

Publisher: American Mathematical Soc.

Published: 2015-06-26

Total Pages: 104

ISBN-13: 1470414201

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The main goals of this paper are: (i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Radon and non-negative. (ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like , where is a function and is a measure. (iii) To show that on spaces with Ricci curvature bounded from below and dimension bounded from above, the Laplacian of the distance function is always a measure and that this measure has the standard sharp comparison properties. This result requires an additional assumption on the space, which reduces to strict convexity of the norm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail.

Sobolev Spaces on Metric Measure Spaces

Author: Juha Heinonen

Publisher: Cambridge University Press

Published: 2015-02-05

Total Pages: 447

ISBN-13: 1107092345

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This coherent treatment from first principles is an ideal introduction for graduate students and a useful reference for experts.

Metric In Measure Spaces

Author: James J Yeh

Publisher: World Scientific

Published: 2019-11-18

Total Pages: 308

ISBN-13: 9813200421

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Measure and metric are two fundamental concepts in measuring the size of a mathematical object. Yet there has been no systematic investigation of this relation. The book closes this gap.

Gradient Flows

Author: Luigi Ambrosio

Publisher: Springer Science & Business Media

Published: 2008-10-29

Total Pages: 333

ISBN-13: 376438722X

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The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.

Irreducible Geometric Subgroups of Classical Algebraic Groups

Author: Timothy C. Burness,

Publisher: American Mathematical Soc.

Published: 2016-01-25

Total Pages: 100

ISBN-13: 1470414945

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Let be a simple classical algebraic group over an algebraically closed field of characteristic with natural module . Let be a closed subgroup of and let be a non-trivial irreducible tensor-indecomposable -restricted rational -module such that the restriction of to is irreducible. In this paper the authors classify the triples of this form, where is a disconnected maximal positive-dimensional closed subgroup of preserving a natural geometric structure on .

Stability of Line Solitons for the KP-II Equation in $\mathbb {R}^2$

$Stability of Line Solitons for the KP-II Equation in $\mathbb {R}^2$ PDF$

Author: Tetsu Mizumachi

Publisher: American Mathematical Soc.

Published: 2015-10-27

Total Pages: 110

ISBN-13: 1470414244

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The author proves nonlinear stability of line soliton solutions of the KP-II equation with respect to transverse perturbations that are exponentially localized as . He finds that the amplitude of the line soliton converges to that of the line soliton at initial time whereas jumps of the local phase shift of the crest propagate in a finite speed toward . The local amplitude and the phase shift of the crest of the line solitons are described by a system of 1D wave equations with diffraction terms.

Faithfully Quadratic Rings

Author: M. Dickmann

Publisher: American Mathematical Soc.

Published: 2015-10-27

Total Pages: 148

ISBN-13: 1470414686

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In this monograph the authors extend the classical algebraic theory of quadratic forms over fields to diagonal quadratic forms with invertible entries over broad classes of commutative, unitary rings where is not a sum of squares and is invertible. They accomplish this by: (1) Extending the classical notion of matrix isometry of forms to a suitable notion of -isometry, where is a preorder of the given ring, , or . (2) Introducing in this context three axioms expressing simple properties of (value) representation of elements of the ring by quadratic forms, well-known to hold in the field case.

Stability of KAM Tori for Nonlinear Schrödinger Equation

Author: Hongzi Cong

Publisher: American Mathematical Soc.

Published: 2016-01-25

Total Pages: 100

ISBN-13: 1470416573

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The authors prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrödinger equation subject to Dirichlet boundary conditions , where is a real Fourier multiplier. More precisely, they show that, for a typical Fourier multiplier , any solution with the initial datum in the -neighborhood of a KAM torus still stays in the -neighborhood of the KAM torus for a polynomial long time such as for any given with , where is a constant depending on and as .

On the Singular Set of Harmonic Maps into DM-Complexes

Author: Georgios Daskalopoulos

Publisher: American Mathematical Soc.

Published: 2016-01-25

Total Pages: 102

ISBN-13: 1470414600

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The authors prove that the singular set of a harmonic map from a smooth Riemammian domain to a Riemannian DM-complex is of Hausdorff codimension at least two. They also explore monotonicity formulas and an order gap theorem for approximately harmonic maps. These regularity results have applications to rigidity problems examined in subsequent articles.