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. It can be viewed in the following format: