Commit 8a7de7d8 authored by Felix Hilsky's avatar Felix Hilsky
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remove text of first version that got replaced by longer introduction to the mathematical ideas

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......@@ -258,65 +258,4 @@ Unfortunately this result relies heavily on another work \parencite{q4-q6-vortic
This work specialises on two-dimensional domains with holes and extending it is far beyond the scope of this thesis.
In \cref{sec:contraexampleTorus} we show though that there exist a condition on the torus similar to a boundary condition for which the line field minimizer of the harmonic energy is not orientable.
Therefore the question of orientability of harmonic energy minimizers on surfaces remains open.
\vskip 5cm
The mentioned density results are exploited in \cref{sec:orientability_of_sobolev_line_fields}.
If the domain is simply connected it was shown that all Sobolev functions can be weakly approximated by smooth functions.
We have this by embedding the domain manifold and
Secondly it does not take into account how close the molecules are to the preferred direction.
This is also related to the fact that liquid crystals are in some cases forced to have \enquote{defects} where no director can be properly defined.
Thirdly the Frank-Oseen model assumes uniaxiality but biaxial nematic liquid crystals were found later.
In 1974 \textcite{q48-original-de-gennes} Pierre-Gilles de Gennes therefore formulated another model of so-called \newTerm{$Q$-tensors} that are described in \cref{sec:qtensor}.
It is based on the solid-liquid phase transition theory of Lev Landau. \autocite[Section 10][162]{q45-LCD}
Instead of a unit vector, the orientation is described by a trace-free symmetric tensor.
One parameter of the $Q$-tensor is the order parameter.
It describes how close the molecule directions are to the preferred direction.
In the following we restrict ourselves to the simpler case of constant order parameter and uniaxiality.
Therefore our restricted $Q$-tensor model treats opposite directors as equal but does not fix the other shortcomings of the Frank-Oseen model.
We therefore call these restricted $Q$-tensor fields \newTerm{line fields}.
\textbf{What was the work of Ball and Zarnescu?}
The Landau-de Gennes model is more accurate but harder to deal with than the Frank-Oseen model.
\textcite{q4-Ball2011} discusses under which conditions the Frank-Oseen model can be used. They limit their study to
\item uniaxial nematic liquid crystals,
\item constant order parameter, no defects,
\item two-dimensional domain with holes,
\item continuous and Sobolev $W^{1,q}$ line fields (some results only for $q=2$),
\item harmonic energy which is a special case of the Frank-Oseen energy.
They show that a continuous or $W^{1,q}$ line field is orientable if and only if it is orientable on the boundaries of the holes of the domain.
Orientable here means that we can replace the line field by a unit vector director field.
Furthermore they give a characterisation of the domains in which the line field minimizer of the harmonic energy is orientable.
\textbf{What is my work about? Why is that an interesting question?}
The following work is based on the ideas \textcite{q4-Ball2011} and generalises them to manifolds.
Many applications and numerical studies are taking place on curved surfaces as for example the mentioned \textcite{q5-q3-vesicles}.
The question of orientability is closely related to the topology of the underlying domain.
Since manifolds have a different topology than flat domains work is required to generalise the results of \textcite{q4-Ball2011}.
\footnote{2 people asked if this paragraph is actually needed. Should I remove it or move it somewhere else?}
\textbf{What are my (main) results?}
There are three main results of this thesis.
Firstly we%
\footnote{is \enquote{we} actually suitable here?}
show that on simply-connected manifolds on any dimension all $W^{1,q}$ line fields are orientable in \cref{thm:sobolev_orientability_on_simply_connected_manifolds}.
Secondly we give a characterisation of orientability of $W^{1,q}$ line fields on surfaces, \ie two-dimensional manifolds in \cref{thm:characterisation_of_orientability_of_sobolev_line_fields}.
Note that due to the topology there are no unit vector fields on surfaces without boundary except of the torus.
Thirdly we present on example of a torus with \enquote{boundary}\footnote{the torus has no boundary, so it's a different type of condition. But how to call it?} condition that has a line field minimizer of the harmonic energy that is not orientable.
\textbf{What is the structure of my thesis?}
In \cref{sec:qtensor} the $Q$-tensor model is adapted to manifolds.
In order to define what Sobolev ($W^{1,q}$) line and unit vector fields are, we define them intrinsically in \cref{sec:sobolev_tensor_fields}.
For \cref{thm:sobolev_orientability_on_simply_connected_manifolds} and \cref{thm:characterisation_of_orientability_of_sobolev_line_fields} we need two different definitions of Sobolev fields though and therefore need to show that those three definitions are equivalent.
This is done in \cref{sec:embedding_M} via an embedding of the base manifold and
\cref{sec:sobolev_field_spaces_on_parallelizable_manifolds} via a global frame respectively.
The rest of the thesis is structured similarly to \textcite{q4-Ball2011}.
Since the results about Sobolev fields are achieved via approximation with continuous fields, the continuous case is studied in \cref{sec:continuous_q_lift}.
\cref{thm:sobolev_orientability_on_simply_connected_manifolds} and \cref{thm:characterisation_of_orientability_of_sobolev_line_fields} are then proven in
Finally the example for non-orientability of the line field minimizer has its own \cref{sec:harmonic_energy_minimizer_orientation}.
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