

Globus Seminar
June 26, 2014 15:40, Moscow, IUM (Bolshoi Vlas'evskii per., 11)






Nonlinear elliptic equations and nonassociative algebras
S. G. Vlăduţ^{ab} ^{a} AixMarseille Université
^{b} A. A. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow

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Abstract:
The first serious research in the theory of nonlinear elliptic partial differential equations was the thesis of Serge Bernstein (1903). He proved that solutions of an elliptic Lagrangian (variational) equation with analytic coefficients are analytic, which is the Hilbert 19th Problem. More generally, L. Nirenberg in 1953 proved that in 2 dimensions solutions of uniformly elliptic equations are classical, i.e. smooth.
That raised the problem whether in higher dimensions there exist nonclassical solutions to uniformly elliptic equations. This problem was open until 2007, when the first fully nonlinear uniformly elliptic equation without classical solution was constructed using the quaternions. Afterwards, applications of nonassociative algebras: the Cayley algebra and Jordan algebras gave a substantial progress towards a classification of nonclassical solutions of fully nonlinear uniformly elliptic equations.
In the talk which is based on joint work with N. Nadirashvili and V. Tkachev, I will give an exposition of these results and methods of their proofs.
The talk will be held in English
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