Overview

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.

Full Product Details

9781470417109

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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

Author Information

Nikolai Nadirashvili, Aix-Marseille University, France. Vladimir Tkachev, Linkoping University, Sweden. Serge Vladut, Aix-Marseille University, France.

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