Commit ca3338e6 authored by Felix Hilsky's avatar Felix Hilsky
Browse files

continue introduction

more explanation of why we do this mathematics and techniques in the mathematics

A lot more explanation for chapter 6 is suddenly missing. I must have
not saved it. Aaaaargggh!!!!
parent 614a0bac
......@@ -50,7 +50,7 @@ Liquid crystals can move like a liquid but in many cases with a high viscosity.
% there is no order in the orientation and position of the molecules.
% Furthermore the position and orientation easily change over time.
% Liquid crystals are matter in a phase between the phases \enquote{solid} and \enquote{liquid}.
% They do have a long-range order in the orientation and some have a partial long-range order of the position.
% They do have a long-range order in the orientation and some have a partial long-range order of the position.
Since Reinitzer's discovery, many substances between liquid and crystal
......@@ -148,7 +148,7 @@ Therefore it is more appropriate to use a \newTerm{line field} instead of a unit
\includegraphics[width=0.8\textwidth,height=5cm,keepaspectratio]{.maindir/zeichnungen/unorientable-line-field}
\caption{An unorientable line field on a domain with one hole.
Intuitively it is clear that a director field that points upwards on the left of the hole has to point downwards on the right.
Therefore it cannot agree with itself below the hole, creating an artificial discontinuity.
Therefore it cannot agree with itself below the hole, creating an discontinuity where the line field is continuous.
From \parencite[Fig.~1][]{q4-Ball2011}.}
\label{fig:unorientable_line_field}
\end{figure}
......@@ -162,36 +162,61 @@ As a main preparation we investigate:
\begin{quote}
Is a given line field orientable?
\end{quote}
We call a line field \newTerm{orientable} if a unit vector exists that points in every point in one of the two directions of the line and is continuous if the line field is continuous or weakly differentiable if the line field is weakly differentiable.
We call a line field \newTerm{orientable} if a unit vector field exists that points in every point in one of the two directions of the line and is continuous if the line field is continuous or weakly differentiable if the line field is weakly differentiable.
\cref{fig:unorientable_line_field} shows an example of a continuous line field that is not orientable as explained in the caption.
In this example we already see that the hole in the domain plays a crucial role.
Without it we could not construct a non-orientable continuous line field.
More generally the question of orientability of continuous line fields depends purely on the topology of the domain and is solved by the algebraic topology result about liftings.
More generally the question of orientability of continuous line fields depends purely on the topology of the domain and is solved by a algebraic topology result about liftings.
This is discussed in \cref{sec:continuous_q_lift}.
The idea is that the sphere in which the unit vector fields map is a double covering of the real projective space in which the line fields map.
This is the term from algebraic topology to describe that the two opposite directions are identified.
The important theorem states that in every point one of the two vectors can be chosen in a continuous way (\enquote{a lifting}) if and only if this is possible on a set of loops that generate the fundamental group of the domain.
The idea is that the sphere in which the unit vector fields map is a double \newTerm{covering} of the real projective space in which the line fields map.
\imp{Covering} is the term from algebraic topology to describe that the two opposite directions are identified.
The theorem we use states that in every point one of the two vectors can be chosen in a continuous way (\enquote{a lifting}) if and only if this is possible on a set of loops that generate the fundamental group of the domain.
As mentioned above the goal is to calculate how the molecules in a liquid crystal are oriented under given circumstances.
These circumstances include among others the shape and topology of the domain, boundary values and electrical or magnetic fields.
They are modeled with the Frank-Oseen energy, a functional of the line or director field.
The orientation of the molecules than minimizes this energy.
These minimization problems are usually mathematically not solved in the class of continuous functions but weakly differentiable (Sobolev) functions.
The orientations of the molecules than minimize this energy.
These minimization problems are usually not solved in the class of continuous functions but weakly differentiable (Sobolev) functions.
Therefore we need to transfer the results about orientability of continuous line fields to weakly differentiable fields.
The basic idea is to use approximations of weakly differentiable functions by continuous functions.
The main limitation is that the regarded functions are mapping into manifolds, namely the sphere and the real projective space and continuous functions mapping into manifolds are in general not dense in the respective Sobolev spaces.
Thus our results are limited to the cases where they are dense in some way:
Thus we want to approximate weakly differentiable functions by continuous functions.
This is often not possible because the functions are mapping into manifolds, namely the sphere and the real projective space and continuous functions mapping into manifolds are in general not dense in the respective Sobolev spaces.
In two cases we have sufficient density results though:
simply connected domains (\cref{sec:sobolev_orientability_on_simply_connected_manifolds})
and surfaces (two-dimensional domains) (\cref{sec:sobolev_orientability_on_surface}).
For flat domains the presented problems were discussed by \textcite{q4-Ball2011}.
For flat domains \textcite{q4-Ball2011} discusses the presented problems.
They showed that on simply-connected two- and three-dimensional domains for $p ≥ 2$ all weakly differentiable $W^{1,p}$ line fields are orientable \parencite[Thm.~2][]{q4-Ball2011}.
For two-dimensional domains with holes they showed that orientability of a line field is equivalent to orientability on the boundaries of the holes \parencite[Prop.~7][]{q4-Ball2011}.
Furthermore they characterise orientability on a hole boundary with the winding number of an auxiliary unit vector valued map.
Intuitively spoken, this auxiliary map turns twice as fast as the line field and if its winding number is even, the line field is orientable.%
\ask{Vielleicht die letzten beiden Sätze weglassen? Das wird technisch so gemacht und es ist auch ein Grund, warum die Technik nicht auf höhere Dimensionen angewendet werden kann. Aber ich finde es nicht so klar verständlich und vielleicht auch nicht so wichtig.}
Some applications like \parencite{q5-q3-vesicles} study thin films of liquid crystals which are best modeled as two-dimensional surfaces.
Therefore this thesis generalises the results of \parencite{q4-Ball2011} to curved surfaces.
In order to stay as general as possible we also manifolds in higher dimensions.
% As long as possible we do not limit ourselves to surfaces though.
As a consequence the definitions of unit vector fields, line fields and Sobolev spaces need to be reconsidered
because the directions need to be tangent to the manifold.
As the first step we define the unit vector fields and line fields as vector fields tangent to the domain manifold.
The line fields are defined as special 2-tensors because it is a special case of the \newTerm{$Q$-tensor} model discussed below.
\todo{actually discuss it}
% $Q$-tensors are a more general model for liquid crystals
% Here would be a good point to mention Q-Tensors but they and Landau de Gennes were not introduced yet.
% At this point we decide to define them as contravariant since this is the natural way of seeing directions.
% In most papers
Then in \cref{sec:sobolev_spaces} we define Sobolev spaces of tangent vector fields.
Proper discussion of Sobolev tangent vector fields is surprisingly rare in the literature probably because there are no major difficulties.
Nevertheless we need to take care because there several possible definitions.
In \cref{sec:sobolev_tensor_fields} we define them intrinsically.
% while in \cref{sec:embedding_M} and \cref{sec:sobolev_field_spaces_on_parallelizable_manifolds} we use an embedding of the domain manifold or a global frame respectively\ask{do I need a comma here somewhere?} to
Then in \cref{sec:embedding_M} we use an embedding of the domain into a Euclidean space.
This definition is necessary for the proof of orientability of line fields on simply-connected domains.
On surfaces we have a global frame and use this to view tangent vector fields as $^m$-valued function for which Sobolev spaces definitions are clear.
In \cref{sec:sobolev_spaces} we show that all of these definitions are equivalent
which boils down to the equivalence of the norms and the density of smooth functions in all defined Sobolev spaces.
Here the smooth functions are dense because the codomains are in every point vector spaces.
Secondly it does not take into account how close the molecules are to the preferred direction.
......@@ -231,7 +256,7 @@ Since manifolds have a different topology than flat domains work is required to
\textbf{What are my (main) results?}
There are three main results of this thesis.
Firstly we%
\footnote{is \enquote{we} actually suitable here?}
\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.
......
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