A Trapping Principle for Discontinuous Elliptic Systems of Mixed Monotone
Type
By: Joseph W. Jerome
We consider discontinuous semilinear elliptic systems, with
boundary
conditions on
the individual components
of Dirichlet/Neumann type.
The components of the vector field are required to satisfy monotonicity
conditions associated with competitive or cooperative species. The latter
model defines a system of mixed monotone type. We also illustrate the
theory via higher order mixed monotone systems which combine competitive
and cooperative subunits.
We seek solutions on
special intervals defined by lower and upper solutions associated with
outward pointing vector fields.
It had been shown by Heikkila and Lakshmikantham that the general
discontinuous mixed monotone system does not necessarily admit solutions
on an interval defined by lower and upper solutions. Our result, obtained
via the Tarski fixed point theorem, shows that solutions exist
for the models described above in the sense of a
measurable selection (in the principal arguments) from a maximal monotone
multi-valued mapping.
We use intermediate variational
inequalities in the proof. Applications involving quantum confinement
and chemically reacting systems with change of phase are discussed.
These are natural examples of discontinuous systems.
This paper has appeared: J. Math. Anal. Appl. 262 (2001), 700--721, and
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