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

Author

Listed:
  • Georgi Boyadzhiev

    (Institute of Mathematics and Informatics, 8 Acad. Georgi Bonchev Str., 1113 Sofia, Bulgaria
    University of Architecture, Civil Engineering and Geodesy, 1046 Sofia, Bulgaria
    These authors contributed equally to this work.)

  • Nikolai Kutev

    (Institute of Mathematics and Informatics, 8 Acad. Georgi Bonchev Str., 1113 Sofia, Bulgaria
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Georgi Boyadzhiev & Nikolai Kutev, 2021. "Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems," Mathematics, MDPI, vol. 9(22), pages 1-9, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:22:p:2985-:d:685209
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