Partial regularity of minimum energy configurations in ferroelectric liquid crystals (Q1948758)

The paper studies regularity properties of equilibrium configurations of ferroelectric liquid crystals (i.e., smectic liquid crystals possessing polarizations) occupying a bounded open domain \(\Omega\) corresponding to minimizers of the governing energy functional for such a system. The energy functional subject to Maxwell's equations is studied. The polarization is described by the Oseen-Frank energy which for a special case of elastic constants reduces to the Dirichlet integral for harmonic maps. The paper considers a simplified version of the energy functional with special bookshelf geometry which smectic layers are uniform and parallel to the \(xy\)-plane. First, the simplified version of the governing energy functional and Maxwell's equations are presented. Based on the Hardt-Kinderlehrer idea of \((C, \beta)\)-almost energy minimizers, the partial regularity of the minimizers for the simplified problem is studied. Here, \(C = C(\beta)\) is the generic constant depending on the prescribed quantity \(\beta\). In the proof of partial regularity, a hybrid inequality and an energy decay estimate followed by Morrey's lemma are used. In a particular case of elastic constants, a weaker version of the interior monotonicity inequality is established which enables one to prove that there exist only a finite number of singularities inside \(\Omega\) for a minimizer. For the general case of elastic constants, the authors prove that a minimizing pair of the molecular director and polarization vector is locally Hölder continuous on \(\Omega\) except a subset \(Z\) with one dimensional Hausdorff measure zero. The set \(Z\) of singularities is defined as the set of points a from \(\Omega\), for which the integral of the sum of the squares of the gradient of the molecular director and the gradient of the polarization vector over the ball of radius \(r\), centered at a, multiplied by \(1/r\), does not approach zero as \(r\to 0\).