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Nonlinear Elliptic Equations and Nonassociative Algebras

Nikolai Nadirashvili Aix-Marseille University, Marseille, France
Serge Vlăduţ Aix-Marseille University, Marseille, France
Available Formats:
Hardcover ISBN: 978-1-4704-1710-9
Product Code: SURV/200
List Price: $96.00 MAA Member Price:$86.40
AMS Member Price: $76.80 Electronic ISBN: 978-1-4704-2045-1 Product Code: SURV/200.E List Price:$90.00
MAA Member Price: $81.00 AMS Member Price:$72.00
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List Price: $144.00 MAA Member Price:$129.60
AMS Member Price: $115.20 Click above image for expanded view Nonlinear Elliptic Equations and Nonassociative Algebras Nikolai Nadirashvili Aix-Marseille University, Marseille, France Vladimir Tkachev Linköping University, Sweden Serge Vlăduţ Aix-Marseille University, Marseille, France Available Formats:  Hardcover ISBN: 978-1-4704-1710-9 Product Code: SURV/200  List Price:$96.00 MAA Member Price: $86.40 AMS Member Price:$76.80
 Electronic ISBN: 978-1-4704-2045-1 Product Code: SURV/200.E
 List Price: $90.00 MAA Member Price:$81.00 AMS Member Price: $72.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$144.00 MAA Member Price: $129.60 AMS Member Price:$115.20
• Book Details

Mathematical Surveys and Monographs
Volume: 2002014; 240 pp
MSC: Primary 17; 35; Secondary 16; 49; 53;

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.

Graduate students and research mathematicians interested in non-linear partial differential equations.

• Chapters
• Chapter 1. Nonlinear elliptic equations
• Chapter 2. Division algebras, exceptional Lie groups, and calibrations
• Chapter 3. Jordan algebras and the Cartan isoparametric cubics
• Chapter 4. Solutions from trialities
• Chapter 5. Solutions from isoparametric forms
• Chapter 6. Cubic minimal cones
• Chapter 7. Singular solutions in calibrated geometries

• Reviews

• This is a very well written book. Through explicit examples and (at times elaborate) calculations, the authors are able to provide answers to some important questions in the theory of elliptic equations. It is a remarkable feat that the seemingly different worlds of nonassociative algebras and that of nonlinear elliptic equations can be combined so effectively in a self-contained book of this size.

Florin Catrina, Zentralblatt Math
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2002014; 240 pp
MSC: Primary 17; 35; Secondary 16; 49; 53;

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.

Graduate students and research mathematicians interested in non-linear partial differential equations.

• Chapters
• Chapter 1. Nonlinear elliptic equations
• Chapter 2. Division algebras, exceptional Lie groups, and calibrations
• Chapter 3. Jordan algebras and the Cartan isoparametric cubics
• Chapter 4. Solutions from trialities
• Chapter 5. Solutions from isoparametric forms
• Chapter 6. Cubic minimal cones
• Chapter 7. Singular solutions in calibrated geometries
• This is a very well written book. Through explicit examples and (at times elaborate) calculations, the authors are able to provide answers to some important questions in the theory of elliptic equations. It is a remarkable feat that the seemingly different worlds of nonassociative algebras and that of nonlinear elliptic equations can be combined so effectively in a self-contained book of this size.

Florin Catrina, Zentralblatt Math
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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