Fully Nonlinear PDEs in Real and Complex Geometry and Optics
Fully Nonlinear PDEs in Real and Complex Geometry and Optics

Author: Luca Capogna

Publisher: Springer

Published: 2013-10-07

Total Pages: 224

ISBN-13: 3319009427

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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.

The Golden Anniversary Celebration of the National Association of Mathematicians
The Golden Anniversary Celebration of the National Association of Mathematicians

Author: Omayra Ortega

Publisher: American Mathematical Soc.

Published: 2020-12-10

Total Pages: 184

ISBN-13: 1470451301

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This volume is put together by the National Association of Mathematicians to commemorate its 50th anniversary. The articles in the book are based on lectures presented at several events at the Joint Mathematics Meeting held from January 16–19, 2019, in Baltimore, Maryland, including the Claytor-Woodard Lecture as well as the NAM David Harold Blackwell Lecture, which was held on August 2, 2019, in Cincinnati, Ohio.

Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane
Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane

Author: Kari Astala

Publisher: Princeton University Press

Published: 2008-12-29

Total Pages: 696

ISBN-13: 1400830117

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This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.

Nonlinear Elliptic Equations and Nonassociative Algebras
Nonlinear Elliptic Equations and Nonassociative Algebras

Author: Nikolai Nadirashvili

Publisher: American Mathematical Soc.

Published: 2014-12-03

Total Pages: 250

ISBN-13: 1470417103

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This book presents applications of noncommutative and nonassociative algebras to constructing unusual (nonclassical and singular) solutions to fully nonlinear elliptic partial differential equations of second order. The methods described in the book are used to solve a longstanding problem of the existence of truly weak, nonsmooth viscosity solutions. Moreover, the authors provide an almost complete description of homogeneous solutions to fully nonlinear elliptic equations. It is shown that even in the very restricted setting of "Hessian equations", depending only on the eigenvalues of the Hessian, these equations admit homogeneous solutions of all orders compatible with known regularity for viscosity solutions provided the space dimension is five or larger. To the contrary, in dimension four or less the situation is completely different, and our results suggest strongly that there are no nonclassical homogeneous solutions at all in dimensions three and four. Thus this book gives a complete list of dimensions where nonclassical homogeneous solutions to fully nonlinear uniformly elliptic equations do exist; this should be compared with the situation of, say, ten years ago when the very existence of nonclassical viscosity solutions was not known.

Optical Engineering
Optical Engineering

Author:

Publisher:

Published: 1991

Total Pages: 1236

ISBN-13:

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Publishes papers reporting on research and development in optical science and engineering and the practical applications of known optical science, engineering, and technology.

Geometry in Partial Differential Equations
Geometry in Partial Differential Equations

Author: Agostino Prastaro

Publisher: World Scientific

Published: 1994

Total Pages: 482

ISBN-13: 9789810214074

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This book emphasizes the interdisciplinary interaction in problems involving geometry and partial differential equations. It provides an attempt to follow certain threads that interconnect various approaches in the geometric applications and influence of partial differential equations. A few such approaches include: Morse-Palais-Smale theory in global variational calculus, general methods to obtain conservation laws for PDEs, structural investigation for the understanding of the meaning of quantum geometry in PDEs, extensions to super PDEs (formulated in the category of supermanifolds) of the geometrical methods just introduced for PDEs and the harmonic theory which proved to be very important especially after the appearance of the Atiyah-Singer index theorem, which provides a link between geometry and topology.