Volume: 200; 2014; 240 pp; Hardcover
MSC: Primary 17; 35; Secondary 16; 49; 53
Print ISBN: 978-1-4704-1710-9
Product Code: SURV/200
List Price: $96.00
AMS Member Price: $76.80
MAA Member Price: $86.40
Electronic ISBN: 978-1-4704-2045-1
Product Code: SURV/200.E
List Price: $90.00
AMS Member Price: $72.00
MAA Member Price: $81.00
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Supplemental Materials
Nonlinear Elliptic Equations and Nonassociative Algebras
Share this pageNikolai Nadirashvili; Vladimir Tkachev; Serge Vlăduţ
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.
Readership
Graduate students and research mathematicians interested in non-linear partial differential equations.
Reviews & Endorsements
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
Table of Contents
Table of Contents
Nonlinear Elliptic Equations and Nonassociative Algebras
- Cover Cover11 free
- Title page i2 free
- Contents iii4 free
- Preface v6 free
- Chapter 1. Nonlinear elliptic equations 110 free
- Chapter 2. Division algebras, exceptional Lie groups, and calibrations 4554
- Chapter 3. Jordon algebras and the Cartan isoparametric cubics 6372
- Chapter 4. Solutions from trialities 8594
- Chapter 5. Solutions from isoparametric forms 117126
- Chapter 6. Cubic minimal cones 155164
- Chapter 7. Singular solutions in calibrated geometries 205214
- Bibliography 223232
- Notation 235244
- Index 239248 free
- Back Cover Back Cover1250