Strong Maximum Principle Degenerate Fully Non-linear Elliptic Systems
Issue Date:
22-Nov-2021
Publisher:
MDPI
Citation:
Boyadzhiev, G.; Kutev, N. Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems. Mathematics, 2021, 9, 2985. https://doi.org/10.3390/math9222985
Series/Report no.:
Mathematics;9, 2985
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.