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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.

Book Synopsis Nonlinear Elliptic Equations and Nonassociative Algebras by : Nikolai Nadirashvili

Download or read book Nonlinear Elliptic Equations and Nonassociative Algebras written by Nikolai Nadirashvili and published by American Mathematical Soc.. This book was released on 2014-12-03 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations

Author: Vicentiu D. Radulescu

Publisher: Hindawi Publishing Corporation

Published: 2008

Total Pages: 205

ISBN-13: 9774540395

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This book provides a comprehensive introduction to the mathematical theory of nonlinear problems described by elliptic partial differential equations. These equations can be seen as nonlinear versions of the classical Laplace equation, and they appear as mathematical models in different branches of physics, chemistry, biology, genetics, and engineering and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on the calculus of variations and functional analysis. Concentrating on single-valued or multivalued elliptic equations with nonlinearities of various types, the aim of this volume is to obtain sharp existence or nonexistence results, as well as decay rates for general classes of solutions. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including bifurcation, stability, asymptotic analysis, and optimal regularity of solutions. The method of presentation should appeal to readers with different backgrounds in functional analysis and nonlinear partial differential equations. All chapters include detailed heuristic arguments providing thorough motivation of the study developed later on in the text, in relationship with concrete processes arising in applied sciences. A systematic description of the most relevant singular phenomena described in this volume includes existence (or nonexistence) of solutions, unicity or multiplicity properties, bifurcation and asymptotic analysis, and optimal regularity. The book includes an extensive bibliography and a rich index, thus allowing for quick orientation among the vast collection of literature on the mathematical theory of nonlinear phenomena described by elliptic partial differential equations.

Book Synopsis Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations by : Vicentiu D. Radulescu

Download or read book Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations written by Vicentiu D. Radulescu and published by Hindawi Publishing Corporation. This book was released on 2008 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive introduction to the mathematical theory of nonlinear problems described by elliptic partial differential equations. These equations can be seen as nonlinear versions of the classical Laplace equation, and they appear as mathematical models in different branches of physics, chemistry, biology, genetics, and engineering and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on the calculus of variations and functional analysis. Concentrating on single-valued or multivalued elliptic equations with nonlinearities of various types, the aim of this volume is to obtain sharp existence or nonexistence results, as well as decay rates for general classes of solutions. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including bifurcation, stability, asymptotic analysis, and optimal regularity of solutions. The method of presentation should appeal to readers with different backgrounds in functional analysis and nonlinear partial differential equations. All chapters include detailed heuristic arguments providing thorough motivation of the study developed later on in the text, in relationship with concrete processes arising in applied sciences. A systematic description of the most relevant singular phenomena described in this volume includes existence (or nonexistence) of solutions, unicity or multiplicity properties, bifurcation and asymptotic analysis, and optimal regularity. The book includes an extensive bibliography and a rich index, thus allowing for quick orientation among the vast collection of literature on the mathematical theory of nonlinear phenomena described by elliptic partial differential equations.

Elliptic Regularity Theory by Approximation Methods

Author: Edgard A. Pimentel

Publisher: Cambridge University Press

Published: 2022-09-29

Total Pages: 203

ISBN-13: 1009096664

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A modern account of elliptic regularity theory, with a rigorous presentation of recent developments for fundamental models.

Book Synopsis Elliptic Regularity Theory by Approximation Methods by : Edgard A. Pimentel

Download or read book Elliptic Regularity Theory by Approximation Methods written by Edgard A. Pimentel and published by Cambridge University Press. This book was released on 2022-09-29 with total page 203 pages. Available in PDF, EPUB and Kindle. Book excerpt: A modern account of elliptic regularity theory, with a rigorous presentation of recent developments for fundamental models.

Fully Nonlinear Elliptic Equations

Author: Luis A. Caffarelli

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 118

ISBN-13: 9780821804377

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The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations. This class of equations often arises in control theory, optimization, and other applications. The authors give a detailed presentation of all the necessary techniques. Instead of treating these techniques in their greatest generality, they outline the key ideas and prove the results needed for developing the subsequent theory. Topics discussed in the book include the theory of viscosity solutions for nonlinear equations, the Alexandroff estimate and Krylov-Safonov Harnack-type inequality for viscosity solutions, uniqueness theory for viscosity solutions, Evans and Krylov regularity theory for convex fully nonlinear equations, and regularity theory for fully nonlinear equations with variable coefficients.

Book Synopsis Fully Nonlinear Elliptic Equations by : Luis A. Caffarelli

Download or read book Fully Nonlinear Elliptic Equations written by Luis A. Caffarelli and published by American Mathematical Soc.. This book was released on 1995 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations. This class of equations often arises in control theory, optimization, and other applications. The authors give a detailed presentation of all the necessary techniques. Instead of treating these techniques in their greatest generality, they outline the key ideas and prove the results needed for developing the subsequent theory. Topics discussed in the book include the theory of viscosity solutions for nonlinear equations, the Alexandroff estimate and Krylov-Safonov Harnack-type inequality for viscosity solutions, uniqueness theory for viscosity solutions, Evans and Krylov regularity theory for convex fully nonlinear equations, and regularity theory for fully nonlinear equations with variable coefficients.

Associative and Non-Associative Algebras and Applications

Author: Mercedes Siles Molina

Publisher: Springer Nature

Published: 2020-01-02

Total Pages: 338

ISBN-13: 3030352560

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This book gathers together selected contributions presented at the 3rd Moroccan Andalusian Meeting on Algebras and their Applications, held in Chefchaouen, Morocco, April 12-14, 2018, and which reflects the mathematical collaboration between south European and north African countries, mainly France, Spain, Morocco, Tunisia and Senegal. The book is divided in three parts and features contributions from the following fields: algebraic and analytic methods in associative and non-associative structures; homological and categorical methods in algebra; and history of mathematics. Covering topics such as rings and algebras, representation theory, number theory, operator algebras, category theory, group theory and information theory, it opens up new avenues of study for graduate students and young researchers. The findings presented also appeal to anyone interested in the fields of algebra and mathematical analysis.

Book Synopsis Associative and Non-Associative Algebras and Applications by : Mercedes Siles Molina

Download or read book Associative and Non-Associative Algebras and Applications written by Mercedes Siles Molina and published by Springer Nature. This book was released on 2020-01-02 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gathers together selected contributions presented at the 3rd Moroccan Andalusian Meeting on Algebras and their Applications, held in Chefchaouen, Morocco, April 12-14, 2018, and which reflects the mathematical collaboration between south European and north African countries, mainly France, Spain, Morocco, Tunisia and Senegal. The book is divided in three parts and features contributions from the following fields: algebraic and analytic methods in associative and non-associative structures; homological and categorical methods in algebra; and history of mathematics. Covering topics such as rings and algebras, representation theory, number theory, operator algebras, category theory, group theory and information theory, it opens up new avenues of study for graduate students and young researchers. The findings presented also appeal to anyone interested in the fields of algebra and mathematical analysis.

Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems

Author: Mariano Giaquinta

Publisher: Princeton University Press

Published: 1983-11-21

Total Pages: 312

ISBN-13: 9780691083315

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The description for this book, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105), Volume 105, will be forthcoming.

Book Synopsis Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems by : Mariano Giaquinta

Download or read book Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems written by Mariano Giaquinta and published by Princeton University Press. This book was released on 1983-11-21 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: The description for this book, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105), Volume 105, will be forthcoming.

Integro-Differential Elliptic Equations

Author: Xavier Fernández-Real

Publisher: Springer Nature

Published:

Total Pages: 409

ISBN-13: 3031542428

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Book Synopsis Integro-Differential Elliptic Equations by : Xavier Fernández-Real

Download or read book Integro-Differential Elliptic Equations written by Xavier Fernández-Real and published by Springer Nature. This book was released on with total page 409 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs

Author: Emanuel Indrei

Publisher: American Mathematical Society

Published: 2023-01-09

Total Pages: 148

ISBN-13: 147046652X

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This volume contains the proceedings of the virtual conference on Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs, held from February 28–March 1, 2021, and hosted by Purdue University, West Lafayette, IN. The mathematical content of this volume is at the intersection of viscosity theory, Fourier analysis, mass transport theory, fractional elliptic theory, and geometric analysis. The reader will encounter, among others, the following topics: the principal-agent problem; Maxwell's equations; Liouville-type theorems for fully nonlinear elliptic equations; a doubly monotone flow for constant width bodies; and the edge dislocations problem for crystals that describes the equilibrium configurations by a nonlocal fractional Laplacian equation.

Book Synopsis Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs by : Emanuel Indrei

Download or read book Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs written by Emanuel Indrei and published by American Mathematical Society. This book was released on 2023-01-09 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the virtual conference on Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs, held from February 28–March 1, 2021, and hosted by Purdue University, West Lafayette, IN. The mathematical content of this volume is at the intersection of viscosity theory, Fourier analysis, mass transport theory, fractional elliptic theory, and geometric analysis. The reader will encounter, among others, the following topics: the principal-agent problem; Maxwell's equations; Liouville-type theorems for fully nonlinear elliptic equations; a doubly monotone flow for constant width bodies; and the edge dislocations problem for crystals that describes the equilibrium configurations by a nonlocal fractional Laplacian equation.

A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials

Author: Florica C. Cîrstea

Publisher: American Mathematical Soc.

Published: 2014-01-08

Total Pages: 97

ISBN-13: 0821890220

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In particular, for b = 1 and λ = 0, we find a sharp condition on h such that the origin is a removable singularity for all non-negative solutions of [[eqref]]one, thus addressing an open question of Vázquez and Véron.

Book Synopsis A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials by : Florica C. Cîrstea

Download or read book A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials written by Florica C. Cîrstea and published by American Mathematical Soc.. This book was released on 2014-01-08 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: In particular, for b = 1 and λ = 0, we find a sharp condition on h such that the origin is a removable singularity for all non-negative solutions of [[eqref]]one, thus addressing an open question of Vázquez and Véron.

Introduction to the Theory of Nonlinear Elliptic Equations

Author: Jindric Necas

Publisher:

Published: 1986-12-29

Total Pages: 170

ISBN-13:

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This book is concerned with the study of boundary value problems for nonlinear, second order, elliptic partial differential equations. A short introduction to Sobolev and Morrey-Campanato spaces and to methods of approximation is included.

Book Synopsis Introduction to the Theory of Nonlinear Elliptic Equations by : Jindric Necas

Download or read book Introduction to the Theory of Nonlinear Elliptic Equations written by Jindric Necas and published by . This book was released on 1986-12-29 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is concerned with the study of boundary value problems for nonlinear, second order, elliptic partial differential equations. A short introduction to Sobolev and Morrey-Campanato spaces and to methods of approximation is included.