Nonlinear Elliptic Equations and Nonassociative Algebras

Abstract : 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.
Document type :
Complete list of metadatas
Contributor : Aigle I2m <>
Submitted on : Tuesday, January 27, 2015 - 4:45:10 PM
Last modification on : Monday, March 4, 2019 - 2:04:18 PM


  • HAL Id : hal-01110218, version 1



Serge Vladut, Nikolai Nadirashvili, Vladimir Tkachev. Nonlinear Elliptic Equations and Nonassociative Algebras. Nikolai Nadirashvili; Vladimir Tkachev; Serge Vlăduţ. ⟨American Mathematical Society (AMS)⟩, 200, 240 p., 2014, Mathematical Surveys and Monographs, 978-1-4704-1710-9. ⟨hal-01110218⟩



Record views