Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems

Georgi Boyadzhiev and Nikolai Kutev
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Georgi Boyadzhiev: Institute of Mathematics and Informatics, 8 Acad. Georgi Bonchev Str., 1113 Sofia, Bulgaria
Nikolai Kutev: Institute of Mathematics and Informatics, 8 Acad. Georgi Bonchev Str., 1113 Sofia, Bulgaria

Mathematics, 2021, vol. 9, issue 22, 1-9

Abstract: In this paper, we consider the validity of the strong maximum principle for weakly coupled, degenerate and cooperative elliptic systems in a bounded domain. In particular, we are interested in the viscosity solutions of elliptic systems with fully nonlinear degenerated principal symbol. Applying the method of viscosity solutions, introduced by Crandall, Ishii and Lions in 1992, we prove the validity of strong interior and boundary maximum principle for semi-continuous viscosity sub- and super-solutions of such nonlinear systems. For the first time in the literature, the strong maximum principle is considered for viscosity solutions to nonlinear elliptic systems. As a consequence of the strong interior maximum principle, we derive comparison principle for viscosity sub- and super-solutions in case when on of them is a classical one. The main novelty of this work is the reduction of the smoothness of the solution. In the literature the strong maximum principle is proved for classical C 2 or generalized C 1 solutions, while we prove it for semi-continuous ones.

Keywords: strong maximum principle; degenerate fully non-linear elliptic systems; viscosity solutions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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