A maximum principle for some nonlinear cooperative elliptic PDE systems with mixed boundary conditions

Abstract

One of the classical maximum principles state that any nonnegative solution of a proper elliptic PDE attains its maximum on the boundary of a bounded domain. We suitably extend this principle to nonlinear cooperative elliptic systems with diagonally dominant coupling and with mixed boundary conditions. One of the consequences is a preservation of nonpositivity, i.e. if the coordinate functions or their fluxes are nonpositive on the Dirichlet or Neumann boundaries, respectively, then they are all nonpositive on the whole domain as well. Such a result essentially expresses that the studied PDE system is a qualitatively reliable model of the underlying real phenomena, such as proper reaction-diffusion systems in chemistry.

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

“…. , N, where constant β satisfies conditions (9)- (12). In the annulus U = {x ∈ Ω : r < |x| < R} the function v(x) is a classical super-solution to (1).…”

“…norm of the solutions. MP for nonlinear cooperative elliptic systems with mixed boundary conditions is proved in [12]. Furthermore, the strong MP is proved in [13] for vector bundles on Riemannian manifolds.…”