Hardcover ISBN:  9781470417109 
Product Code:  SURV/200 
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eBook ISBN:  9781470420451 
Product Code:  SURV/200.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470417109 
eBook: ISBN:  9781470420451 
Product Code:  SURV/200.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 
Hardcover ISBN:  9781470417109 
Product Code:  SURV/200 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470420451 
Product Code:  SURV/200.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470417109 
eBook ISBN:  9781470420451 
Product Code:  SURV/200.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 200; 2014; 240 ppMSC: 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.
ReadershipGraduate students and research mathematicians interested in nonlinear partial differential equations.

Table of Contents

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


Additional Material

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 selfcontained book of this size.
Florin Catrina, Zentralblatt Math


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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.
Graduate students and research mathematicians interested in nonlinear 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 selfcontained book of this size.
Florin Catrina, Zentralblatt Math