A variational problem for nematic liquid crystals with variable degree of orientation (Q1189744)

This paper examines the problem in which a sample of nematic liquid crystal is contained in a circular cylinder of radius \(R\) and height \(H\). When the unit vector \({\mathbf n}\) that represents the optical axis is aligned everywhere radially on the lateral surface of the cylinder, the classical disclination solution, \({\mathbf n}={\mathbf e}_ r\), of Frank's model [\textit{F. C. Frank}, Discuss. Faraday Soc., 25, 19-28 (1958)], has the unfortunate feature of being associated with an infinite energy. Seeking minimizers of the one constant approximation of Frank's energy integral having the form \({\mathbf n}=(\cos \phi){\mathbf e}_ r+(\sin \phi){\mathbf e}_ z\), where \(\phi\) depends only on the radial distance from the cylinder's axis, Calids and Kléman and Meyer obtained the continuous solution \(\phi=\pi/2-2\arctan(r/R)\) which is associated with a finite energy. Here the authors adopt a simplified form of Ericksen's model [\textit{J. L. Ericksen}, Arch. Rational Mech. Anal., 113, No. 2, 97-120 (1991; Zbl 0729.76008)] by assuming that the energy functional has the form \[ F[s,{\mathbf n}]= \alpha\int_ V\{k|\nabla s|^ 2+s^ 2| \nabla{\mathbf n}|^ 2\}dV, \] where \(\alpha\) and \(k\) are positive constants, \(V\) denotes the volume occupied by the liquid crystal and \(s\) is an additional scalar variable, the degree of orientation, which describes the local microstructure of the nematic material. Assuming that \({\mathbf n}=(\cos \phi){\mathbf e}_ r+(\sin\phi){\mathbf e}_ z\), where the axis of the cylinder is the \(z\)-axis of a set of cylindrical polar coordinates, and that \(s\) and \(\phi\) depend only upon the radial coordinate \(r\), the energy functional takes the relatively simple form \[ F[s,\phi]=\int^ R_ 0\left\{k\left ({ds\over dr}\right )^ 2+s^ 2\left ({d\phi\over dr}\right )^ 2+{\cos^ 2\phi\over r^ 2}\right\}rdr. \] Appropriate boundary conditions on the lateral surface of the cylinder are \[ {\mathbf n}={\mathbf e}_ r\text{ and } s(r)=s_ 0\;(\text{a constant}). \] Hence the authors consider the problem of finding the minimizers of \(F[s,\phi]\) in the class of functions \[ C:=\{(s,\phi)| s\in AC[0,B],\;\phi\in AC_{\text{loc}}]0,R[:\;s(R)=s_ 0>0,\;\phi(R)=0\}. \] After obtaining the appropriate Euler equations for \(F\), away from the singular set, the authors deduce a dichotomy concerning the singular set and the du Bois-Raymond equation. An application of the Hamilton-Jacobi method results in the appropriate Hamilton-Jacobi equation for this problem, a phase plane analysis of which is required to yield the main result of the paper. This states that there is a critical value \(k_ c=1\) such that for values \(0<k\leq k_ c\) the equilibrium state of minimal energy has an \({\mathbf n}\) field like the Frank disclination solution, while for values \(k>k_ c\) the equilibrium state has a continuous \({\mathbf n}\) field like that obtained by Cladis and Kléman and Meyer. However, both solutions are now associated with finite energies. This is an interesting paper for any researcher in liquid crystal theory.