On the K13 problem in the Oseen-Frank theory of nematic liquid crystals

Saturday, October 24, 2015 - 9:00am - 9:40am
Arghir Zarnescu (University of Sussex)
Some of the most intriguing terms in the energy functionals corresponding to physical theories are the null Lagrangians. For liquid crystals, their effects and experimental determination, remain elusive as noted already in the 1962 seminal paper on nilpotent energies of Professor Ericksen.
One such term is the so-called K13 term. It is well-known that the K13 term in the Oseen-Frank energy functional is compatible with the physical invariances, yet its presence makes the energy unbounded from below. For this reason it is usually ignored by physicists and mathematicians alike.
We study critical points in the presence of the K13 term and show that there are no critical points unless the boundary conditions are either hometropic (normal) or tangential at almost all points. Furthermore for tangential boundary conditions the K13 term overlaps with the tame null Lagrangian the K24 term, but not so for hometropic bundary conditions. One can also obtain some partial regularity results in a suitable setting.
This is joint work with Stuart Day (University of Sussex).
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