Luca Capogna's Books

Fully Nonlinear PDEs in Real and Complex Geometry and Optics

Cetraro, Italy 2012, Editors: Cristian E. Gutiérrez, Ermanno Lanconelli

By: Luca Capogna,Pengfei Guan,Cristian E. Gutiérrez,Annamaria Montanari

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Fully Nonlinear PDEs in Real and Complex Geometry and Optics

Cetraro, Italy 2012, Editors: Cristian E. Gutiérrez, Ermanno Lanconelli

By: Luca Capogna,Pengfei Guan,Cristian E. Gutiérrez,Annamaria Montanari

The purpose of this CIME summer school was to present current areas of research arising both in the theoretical and applied setting that involve fully nonlinear partial different equations. The equations presented in the school stem from the fields of Conformal Mapping Theory, Differential Geometry, Optics, and Geometric Theory of Several Complex Variables. The school consisted of four courses: Extremal problems for quasiconformal mappings in space by Luca Capogna, Fully nonlinear equations in geometry by Pengfei Guan, Monge-Ampere type equations and geometric optics by Cristian E. Gutiérrez, and On the Levi Monge Ampere equation by Annamaria Montanari.

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Page Count

224

Publisher

Springer

Published Date

ISBN 10

3319009427

ISBN 13

9783319009421

An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem

By: Luca Capogna,Donatella Danielli,Scott D. Pauls,Jeremy Tyson

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An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem

By: Luca Capogna,Donatella Danielli,Scott D. Pauls,Jeremy Tyson

This book gives an up-to-date account of progress on Pansu's celebrated problem on the sub-Riemannian isoperimetric profile of the Heisenberg group. It also serves as an introduction to the general field of sub-Riemannian geometric analysis. It develops the methods and tools of sub-Riemannian differential geometry, nonsmooth analysis, and geometric measure theory suitable for attacks on Pansu's problem.

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Page Count

235

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3764381337

ISBN 13

9783764381332

Harmonic Analysis and Boundary Value Problems

Selected Papers from the 25th University of Arkansas Spring Lecture Series, Recent Progress in the Study of Harmonic Measure from a Geometric and Analytic Point of View, March 2-4, 2000, Fayetteville, Arkansas

By: Luca Capogna,Loredana Lanzani

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Harmonic Analysis and Boundary Value Problems

By: Luca Capogna,Loredana Lanzani

This volume presents research and expository articles by the participants of the 25th Arkansas Spring Lecture Series on ``Recent Progress in the Study of Harmonic Measure from a Geometric and Analytic Point of View'' held at the University of Arkansas (Fayetteville). Papers in this volume provide clear and concise presentations of many problems that are at the forefront of harmonic analysis and partial differential equations. The following topics are featured: the solution of the Kato conjecture, the ``two bricks'' problem, new results on Cauchy integrals on non-smooth curves, the Neumann problem for sub-Laplacians, and a new general approach to both divergence and nondivergence second order parabolic equations based on growth theorems. The articles in this volume offer both students and researchers a comprehensive volume of current results in the field.

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Page Count

170

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821827456

ISBN 13

9780821827451

Harmonic Measure

Geometric and Analytic Points of View

By: Luca Capogna,Carlos E. Kenig,Loredana Lanzani

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Harmonic Measure

Geometric and Analytic Points of View

By: Luca Capogna,Carlos E. Kenig,Loredana Lanzani

Recent developments in geometric measure theory and harmonic analysis have led to new and deep results concerning the regularity of the support of measures which behave "asymptotically" (for balls of small radius) as the Euclidean volume. A striking feature of these results is that they actually characterize flatness of the support in terms of the asymptotic behavior of the measure. Such characterizations have led to important new progress in the study of harmonic measure fornon-smooth domains. This volume provides an up-to-date overview and an introduction to the research literature in this area. The presentation follows a series of five lectures given by Carlos Kenig at the 2000 Arkansas Spring Lecture Series. The original lectures have been expanded and updated to reflectthe rapid progress in this field. A chapter on the planar case has been added to provide a historical perspective. Additional background has been included to make the material accessible to advanced graduate students and researchers in harmonic analysis and geometric measure theory.

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Page Count

170

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821827286

ISBN 13

9780821827284

Hormander Operators

By: Marco Bramanti,Luca Brandolini

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Hormander Operators

By: Marco Bramanti,Luca Brandolini

Hörmander operators are a class of linear second order partial differential operators with nonnegative characteristic form and smooth coefficients, which are usually degenerate elliptic-parabolic, but nevertheless hypoelliptic, that is highly regularizing. The study of these operators began with the 1967 fundamental paper by Lars Hörmander and is intimately connected to the geometry of vector fields.Motivations for the study of Hörmander operators come for instance from Kolmogorov-Fokker-Planck equations arising from modeling physical systems governed by stochastic equations and the geometric theory of several complex variables. The aim of this book is to give a systematic exposition of a relevant part of the theory of Hörmander operators and vector fields, together with the necessary background and prerequisites.The book is intended for self-study, or as a reference book, and can be useful to both younger and senior researchers, already working in this area or aiming to approach it.

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