Trapping Regions for Elliptic Systems with Discontinuous Coupling Vector
Fields
By: Siegfried Carl and Joseph W. Jerome
We consider boundary value problems for
elliptic systems
in a bounded domain, where
the elliptic operators are in divergence form. The boundary conditions are of
mixed Dirichlet-Robin type.
The coupling vector fields
may be discontinuous with respect to all their arguments.
The main goal is to provide conditions on the vector fields
that
allow the identification of
regions of existence of solutions (so called trapping regions). To this
end the problem is transformed to a discontinuously coupled system of
variational inequalities. Assuming
a generalized outward pointing vector field on the boundary of a rectangle
of the dependent variable space, the system of variational inequalities
can be solved via a fixed point problem for some
increasing operator in an appropriate ordered Banach space. The main tools
used in the proof are variational inequalities, truncation and comparison
techniques, and fixed point results in ordered Banach spaces.
This paper will appear in:
Nonsmooth/Nonconvex Mechanics, with Applications in Engineering
(C.C. Banagiotopoulos, editor), Ziti, Thessaloniki, 2002, pp. 15--22.
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