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Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Viscosity Sub-solutions in the Theory of M-hessian Equations

Author(s):

Nina Mihajlovna Ivochkina

Saint Petersburg State University
Professor of Department of Mathematical Physics
St. Petersburg, Peterhof, Universitetsky prospekt, 28
Professor. Doctor of Physical and Mathematical Sciences

mail@NI1570.spb.edu

Svetlana Ivanovna Prokof'eva

The St. Petersburg State University of Architecture and Civil Engineering
math. dept. Assoc.Prof.
St. Petersburg, 2-nd Krasnoarmeiskaia St. 4
Assoc.Prof. PhD in Physics and Mathematics

svetlanaprokof@yandex.ru

Galina Vladimirovna Yakunina

The St. Petersburg State University of Architecture and Civil Engineering
Associate Professor of Department of Mathematics
St. Petersburg, 2-nd Krasnoarmeiskaia St. 4
Assoc.Prof. PhD in Physics and Mathematics

yakuninagalina@yandex.ru

Abstract:

We show that possible non-smoothness of viscosity sub-solutions is of no interest in the theory of m-Hessian operators. It is crucial that the set of viscosity C2-sub-solutions coincides with the set of correct setting of the Dirichlet problem. Moreover, we present an example to demonstrate that on the set of ellipticity of 5-Hessian operator the setting of the Dirichlet problem is incorrect because our problem has two infinitely differentiable solutions but only one of them is viscosity sub-solution.

Keywords

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