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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Spectral properties of nonassociative algebras and breaking regularity for nonlinear elliptic type PDEs


Author: V. G. Tkachev
Original publication: Algebra i Analiz, tom 31 (2019), nomer 2.
Journal: St. Petersburg Math. J. 31 (2020), 223-240
MSC (2010): Primary 53A04; Secondary 52A40, 52A10
DOI: https://doi.org/10.1090/spmj/1593
Published electronically: February 4, 2020
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Abstract: In this paper, we address the following question: Why certain nonassociative algebra structures emerge in the regularity theory of elliptic type PDEs and also in constructing nonclassical and singular solutions? Our aim in the paper is twofold. First, to give a survey of diverse examples on nonregular solutions to elliptic PDEs with emphasis on recent results on nonclassical solutions to fully nonlinear equations. Second, to define an appropriate algebraic formalism, which makes the analytic part of the construction of nonclassical solutions more transparent.


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Additional Information

V. G. Tkachev
Affiliation: Department of Mathematics, Linköping University, SE-581 83, Sweden
Email: vladimir.tkatjev@liu.se

DOI: https://doi.org/10.1090/spmj/1593
Keywords: Viscosity solutions, elliptic type PDEs, compositions algebras
Received by editor(s): November 12, 2018
Published electronically: February 4, 2020
Dedicated: Dedicated to V. G. Maz’ya on the occasion of his $80$th birthday
Article copyright: © Copyright 2020 American Mathematical Society