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\centerline{\large \bf On Singular Limits in Polymer Stabilized Liquid
  Crystals}

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\centerline{P.~Bauman}
\centerline{\it Purdue University, USA}
\centerline{e-mail {\tt bauman@math.purdue.edu}}

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\centerline{D.~Phillips}
\centerline{\it Purdue University, USA}
\centerline{e-mail {\tt phillips@math.purdue.edu}}

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\centerline{Q.~Shen}
\centerline{\it Purdue University, USA}
\centerline{e-mail {\tt qshen@math.purdue.edu}}

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We investigate equilibrium configurations for a polymer stabilized
liquid crystal material subject to an applied magnetic field.  The
configurations are determined by energy minimization where the
energies of the system include those of bulk, surface, and external
field. The Euler-Lagrange equation is a nonlinear PDE with nonlinear
boundary conditions defined on a perforated domain modeling the cross
section of the liquid crystal-polymer fiber composite.  We analyze the
critical values for the external magnetic field representing
Fredericks transitions and describe the equilibrium configurations
under any magnitude of the external field.  We also discuss the limit
of the critical values and configurations as the number of polymer
fibers approaches infinity. In the case where away from the boundary
of the composite, the fibers are part of a periodic array, we prove
that non-constant configurations develop order-one oscillations on the
scale of the array's period. Further more we determine the small-scale
structure of the configurations as the period tends to zero.

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{\bf Presented by Dan Phillips}

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