Liquid crystals: Relaxed energies, dipoles, singular lines and singular points.

*(English)*Zbl 0760.49026The paper deals with the relaxation problem for the energy functional of liquid crystals. The energy functional of liquid crystals is the following
\[
{\mathcal E}(u)=\int_ \Omega\bigl[\alpha| Du|^ 2+(k_ 1-\alpha)(\text{div }u)^ 2+(k_ 2-\alpha)(u\cdot\text{rot }u)^ 2+(k_ 3-\alpha)(u\times\text{rot }u)^ 2\bigr]
\]
and is defined for mappings from a bounded domain in \(\mathbb{R}^ 3\) into the unit sphere \(S^ 2\), \(\alpha>0\), \(k_ i>\alpha\). Following the ideas of other papers of the same authors the parametric extension of \(\mathcal E\) to a class of Cartesian currents without boundary in \(\Omega\times S^ 2\) is considered and a representation result for such an extension is provided.

The case of the Dirichlet problem with fixed boundary datum \(\varphi\) is also treated and a representation result, in terms of the above parametric extension, for the relaxed functional of the restriction of \(\mathcal E\) to the set of the functions having \(\varphi\) as boundary datum is presented. An explicit computation of the relaxed energy functional in the case of a dipole is also provided, hence the infimum of \(\mathcal E\) for the dipole problem is explicitly computed.

Again in the case of the Dirichlet problem it is proved that the minimizer currents have in general line singularities but no point singularities with non-zero degrees. These line singularities show up, in the approximation by smooth maps, as lines where the energy density concentrates. The relaxed energy of \(\mathcal E\) in Sobolev spaces is also described and some modified energies of liquid crystals on Cartesian currents with fractures are studied.

The case of the Dirichlet problem with fixed boundary datum \(\varphi\) is also treated and a representation result, in terms of the above parametric extension, for the relaxed functional of the restriction of \(\mathcal E\) to the set of the functions having \(\varphi\) as boundary datum is presented. An explicit computation of the relaxed energy functional in the case of a dipole is also provided, hence the infimum of \(\mathcal E\) for the dipole problem is explicitly computed.

Again in the case of the Dirichlet problem it is proved that the minimizer currents have in general line singularities but no point singularities with non-zero degrees. These line singularities show up, in the approximation by smooth maps, as lines where the energy density concentrates. The relaxed energy of \(\mathcal E\) in Sobolev spaces is also described and some modified energies of liquid crystals on Cartesian currents with fractures are studied.

Reviewer: R.De Arcangelis (Napoli)

##### MSC:

49Q20 | Variational problems in a geometric measure-theoretic setting |

##### Keywords:

geometric measure theory; integral representation; relaxation problem; energy functional of liquid crystals; representation result; Dirichlet problem; dipole; minimizer currents; line singularities; Cartesian currents with fractures
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\textit{M. Giaquinta} et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, No. 3, 415--437 (1990; Zbl 0760.49026)

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