Nonlinear Elliptic Equations and Nonassociative Algebras
About this Title
Nikolai Nadirashvili, Aix-Marseille University, Marseille, France, Vladimir Tkachev, Linköping University, Sweden and Serge Vlăduţ, Aix-Marseille University, Marseille, France
Publication: Mathematical Surveys and Monographs
Publication Year 2014: Volume 200
ISBNs: 978-1-4704-1710-9 (print); 978-1-4704-2045-1 (online)
MathSciNet review: MR3243534
MSC: Primary 35-02; Secondary 17A99, 17C55, 35A30, 35J60, 53C38, 58J05
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.
Table of Contents
- 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