\documentclass[reqno]{amsart}
\usepackage[mathscr]{eucal}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 141, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/141\hfil Regularity of weak solutions]
{Regularity of weak solutions to the
Landau-Lifshitz system in bounded \\ regular domains}

\author[K. Santugini-Repiquet\hfil EJDE-2007/141\hfilneg]
{K\'evin Santugini-Repiquet}

\address{K\'evin Santugini-Repiquet \newline
Institut Math\'ematique de Bordeaux \\
Universit\'e Bordeaux 1 \\
351 cours de la lib\'eration \\
33405 Talence Cedex, France}
\email{Kevin.Santugin@math.u-bordeaux1.fr}
\urladdr{www.unige.ch/$\sim$santugin}

\thanks{Submitted July 18, 2005. Published October 24, 2007.}
\thanks{Work done at Institut Galil\'ee, Universit\'e Paris 13. France}
\subjclass[2000]{35D10, 35K55, 35K60}
\keywords{Micromagnetism; Landau-Lifshitz equation; weak solutions;
\hfill\break\indent   boundary regularity}

\begin{abstract}
 In this paper, we study the regularity, on the boundary,
 of weak solutions to the Landau-Lifshitz system in the
 framework of the micromagnetic model in the quasi-static
 approximation. We establish the existence of
 global weak solutions to the Landau-Lifshitz system
 whose tangential space gradient on the boundary is square integrable.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\maketitle

\section{Introduction}

The Landau-Lifshitz equation models the behavior
of ferromagnetic materials, and it is  used in crystallography.
This nonlinear systems of partial differential equations is
\begin{equation}
 \frac{\partial \mathbf{m}}{\partial t}=
-\mathbf{m}\wedge\mathbf{h}
-\alpha\mathbf{m}\wedge(\mathbf{m}\wedge\mathbf{h}).
\label{LL}
\end{equation}
with $\mathbf{h}$ (magnetic excitation) depending on $\mathbf{m}$
(magnetization), and $\alpha\in\mathbb{R}^+_*$ (dampening parameter).

Existence of global weak solutions to the Landau-Lifshitz equation
have been proved in \cite{Visintin:Landau,Alouges.Soyeur:OnGlobal,
labbe:mumag}. However, these solutions are only required
to belong to $\mathrm{L}^\infty(\mathbb{R}^+;\mathbb{H}^1(O))$ and to
$\mathbb{H}^1(O\times(0,T))$ for all time $T>0$. Besides, they are
not unique. There  are existence  results for
of more regular solutions in the case when
$\mathbf{h}=\Delta\mathbf{m}$,
see \cite{Melcher:ExistencePartiallyRegular,Moser:PartialRegularity}.
For 3D and a full characterization of the unique ``good'' solution
in 2D has been obtained by Harpes\cite{Harpes:UniquenessBubblingLandauLifshitz}.
These results are not easily generalized to more complicated
forms of $\mathbf{h}$ as they often rely on harmonic analysis.
Some authors have studied the regularity
of stationary maps when the excitation $\mathbf{h}$ has 
a more complicated form: Critical points of the energy
are regular away from a set of zero two-dimensional Hausdorff measure;
see Carbou \cite{Carbou:Regularity} and Hardt and
Kinderlehrer \cite{Hardt.Kinderlehrer:SomeRegularity}.
Using standard analysis, one can easily prove that
any weak solution $\mathbf{m}$ belongs to
$\mathcal{C}([0,+\infty);\mathbb{H}^1_w(O))$,
and to the Nikol'skii space
$\mathrm{L}^2(0,T;(B^2_{1,\infty}(O))^3)$, and satisfy
$\Delta\mathbf{m}\in\mathrm{L}^2(0,T;\mathbb{L}^1(O))$, see \cite[\S 6.2]
{Santugini:These}.

In this paper, we establish
that for any initial condition, at least
one weak solution to the Landau-Lifshitz
system has a trace that belongs to
$\mathrm{L}^2(0,T;\mathbb{H}^1(\partial O))$.
This result is not limited to the
case $\mathbf{h}=A\Delta\mathbf{m}$ and can be generalized to a wide
range of forms of $\mathbf{h}$.

In section \ref{sect:Notations}, we introduce a common notation
used throughout this paper. Then, in section \ref{sect:MicromModel},
we recall briefly the micromagnetic model. We recall known results
concerning the existence of weak solutions in
section \ref{sect:KnownResults}. We state and prove our main
theorem in section \ref{sect:MainTheorem}.

\subsection*{Notation}\label{sect:Notations}

Given an open set $O$, we denote by $\mathrm{L}^p(O)$
the set of all measurable functions $u$ over $O$
such that $\int_O\lvert u\rvert^p\,\mathrm{d}\mathbf{x}<+\infty$.
This is a Banach space for the norm
\begin{equation*}
\lVert u\rVert_{\mathrm{L}^p(O)}=\Big(\int_O\lvert u\rvert^p
  \,\mathrm{d}\mathbf{x}\Big)^{1/p}.
\end{equation*}
For any integer $m\geq0$, we denote by $\mathrm{H}^m(O)$,
the space of all measurable functions $u$
over $O$ such that for any multi-indices $\alpha$,
$\lvert\alpha\rvert<m$, $\mathrm{D}^\alpha u$ belongs to
$\mathrm{L}^2(O)$. This is an Hilbert space for the norm
\begin{equation*}
\lVert u\rVert_{\mathrm{H}^m(O)}=\Big(\vphantom{\sum}
\smash{\sum_{\lvert\alpha\rvert\leq m}
\lVert \,\mathrm{D}^\alpha u\rVert_{\mathrm{L}^2(O)}^2}
\Big)^{1/2}.
\end{equation*}
We set $\mathbb{H}^m(O)=(\mathrm{H}^m(O))^3$ and
$\mathbb{L}^p(O)=(\mathrm{L}^p(O))^3$. By $\lvert O\rvert$,
we denote the Lebesgue measure of set $O$.

Given a smooth surface $\partial O$, $\mathrm{\nu}$ represents
the unit outward normal vector to the surface,
 $\frac{\partial u}{\partial \mathrm{\nu}}$
the normal trace of $u$ on $\partial O$, and
$\nabla_\mathrm{T} u$ the tangential gradient of $u$
on $\partial O$.

In all this paper, $\Omega$ is a bounded open set of
$\mathbb{R}^3$ with a smooth boundary. We also define
$\Omega_T=\Omega\times(0,T)$.


\section{The micromagnetic model}\label{sect:MicromModel}

In this section, we recall briefly the micromagnetic model.
We begin by introducing the more common energies and excitations
that model completely the static behavior, then we introduce the
nonlinear PDE that models the evolution problem. From now on,
$\Omega$ represents the domain filled with a
ferromagnetic material.

\subsection{The static problem}
The magnetic state of a ferromagnetic material
is represented by two vector fields: the magnetization $\mathbf{m}$
and the magnetic excitation $\mathbf{h}$.
In the micromagnetic model the magnetization must verify a non
convex constraint: $\lvert\mathbf{m}\rvert=1$ in $\Omega$ and
be null outside $\Omega$. The excitation $\mathbf{h}$
depends on $\mathbf{m}$.

To each interaction $p$
is associated an energy $\mathrm{E}_p(\mathbf{m})$ and
an operator $\mathscr{H}_p$ related by:
\[
\mathrm{D}\mathrm{E}_p(\mathbf{m})\cdot\mathrm{v}
= -\int_\Omega \mathscr{H}_p(\mathbf{m})\cdot\mathrm{v}\,\mathrm{d}\mathbf{x},
\quad \mathrm{E}_p(\mathbf{0})=0.
\]
The excitation contributed by $p$ is
$\mathbf{h}_p=\mathscr{H}_p(\mathbf{m})$ and the total excitation is
$\mathbf{h}=\sum_p\mathbf{h}_p$.
In this paper we consider three interactions, see
Brown \cite{Brown:Microm} for details:

\noindent\textbf{Exchange:} The exchange energy and its associated excitation
are given by:
\[ %2.1a
\mathrm{E}_e(\mathbf{m})=\frac{A}{2}
\int_\Omega\lvert\nabla\mathbf{m}\rvert^2\,\mathrm{d}\mathbf{x}, \quad
\mathscr{H}_e(\mathbf{m})=A\Delta\mathbf{m},
\]
where $A>0$.

\noindent\textbf{Anisotropy:}
\[
\mathrm{E}_a(\mathbf{m})=\frac{1}{2}
\int_\Omega \mathbf{m}\cdot\mathbf{K}\mathbf{m}\,\mathrm{d}\mathbf{x}, \quad
\mathscr{H}_a(\mathbf{m})=-\mathbf{K}\mathbf{m},
\]
where $\mathbf{K}$ is a symmetric positive
quadratic form.

\noindent\textbf{Demagnetization field:}
\[
\mathrm{E}_d(\mathbf{m})=\frac{1}{2}\int_{\mathbb{R}^3}
\lvert\mathscr{H}_d(\mathbf{m})\rvert^2\,\mathrm{d}\mathbf{x},\quad
\mathscr{H}_d(\mathbf{m})=\mathbf{h}_d,
\]
where
$\mathbf{h}_d$ is the only solution in $\mathrm{L}^2(\mathbb{R}^3)$
to the magnetostatic system
\begin{align*}
\mathop{\rm div}(\mathbf{h}_d+\mathbf{m})&=0,&
\mathop{\rm curl}(\mathbf{h}_d)&=0,
\end{align*}
in the sense of distributions.
\smallskip

The demagnetization field operator has been extensively studied by
Friedman \cite{Friedman:MathematI,Friedman:MathematII,Friedman:MathematIII}.
This operator is symmetric negative, is linear continuous from
$\mathbb{L}^p(\mathbb{R}^3)$ to $\mathbb{L}^p(\mathbb{R}^3)$, $1<p<+\infty$,
and satisfies
\begin{equation*}
-\int_\Omega\mathscr{H}_d(\mathbf{m})\cdot\mathbf{m}\,\mathrm{d}\mathbf{x}=
\int_{\mathbb{R}^3}\lvert\mathscr{H}_d(\mathbf{m})\rvert^2\,\mathrm{d}\mathbf{x}.
\end{equation*}
The $\mathcal{L}(\mathbb{L}^2(\mathbb{R}^3),\mathbb{L}^2(\mathbb{R}^3))$
norm of $\mathscr{H}_d$ is less than $1$.
We also define
\begin{gather*}
\mathscr{H}=\mathscr{H}_e+\mathscr{H}_a+\mathscr{H}_d, \quad
\mathscr{H}_{d,a}=\mathscr{H}_d+\mathscr{H}_a, \\
\mathscr{H}_{e,a}=\mathscr{H}_e+\mathscr{H}_a, \quad
\mathrm{E}=\mathrm{E}_e+\mathrm{E}_a+\mathrm{E}_d.
\end{gather*}
The solutions to the static problem are the local minimizers of the
total energy $\mathrm{E}$ satisfying the constraint
$\lvert\mathbf{m}\rvert=1$
a.e. in $\Omega$.

\subsection{The evolution system}

The Landau-Lifshitz system models the evolution
of ferromagnetic materials. Let $\alpha>0$
be a dampening parameter. The Landau-Lifshitz system
comprises the nonlinear system
\begin{subequations}\label{subeq:LLSyst}
\begin{equation}
\frac{\partial \mathbf{m}}{\partial t}=
-\mathbf{m}\wedge\mathscr{H}(\mathbf{m})
-\alpha\mathbf{m}\wedge\big(\mathbf{m}\wedge\mathscr{H}(\mathbf{m})\big)
\quad\text{in }\Omega\times\mathbb{R}^+,
\end{equation}
the initial condition
\begin{equation}
\mathbf{m}(\cdot,0)=\mathbf{m}_0\quad\text{in }\Omega,
\end{equation}
the non convex constraint
\begin{equation}
\lvert\mathbf{m}\rvert=
1\quad\text{in }\Omega\times\mathbb{R}^+,
\end{equation}
and the  Neumann boundary conditions
\begin{equation}
\frac{\partial \mathbf{m}}{\partial \mathrm{\nu}}=0
\quad\text{on } \partial\Omega\times\mathbb{R}^+.
\end{equation}
\end{subequations}

\section{Known results}\label{sect:KnownResults}

We give the definition of weak solutions to the
Landau-Lifshitz system \eqref{subeq:LLSyst}.

\begin{definition}\label{def:WeakSolLL} \rm
Given $\mathbf{m}_0$ in $\mathbb{H}^1(\Omega{})$,
$\lvert \mathbf{m}_0\rvert=1$ a.e. in $\Omega{}$, we call $\mathbf{m}$
a weak solution to the Landau-Lifshitz system \eqref{subeq:LLSyst} if
\begin{enumerate}
\item For all $T>0$, $\mathbf{m}$ belongs to
$\mathbb{H}^1(\Omega{}\times(0,T))$, and $\lvert \mathbf{m}\rvert=1$ a.e.
in $\Omega{}\times\mathbb{R}^+$.
\item For all $\mathbf{\phi}$ in $\mathbb{H}^1(\Omega{}\times(0,T))$,
\begin{subequations}\label{subeq:LLWeakSol}
\begin{equation}\label{eq:LLWeakSolVariat}
\begin{aligned}
&\iint_{\Omega{T}} \frac{\partial \mathbf{m}}{\partial t}\cdot
 \mathbf{\phi}\,\mathrm{d}\mathbf{x}\,
\mathrm{d} t -\alpha\iint_{\Omega_T}
    \Big(\mathbf{m}\wedge\frac{\partial \mathbf{m}}{\partial t}\Big)
    \cdot\mathbf{\phi}\,\mathrm{d}\mathbf{x}\,\mathrm{d} t \\
&=(1+\alpha^2) A\iint_{\Omega_T}\sum_{i=1}^3
    \Big(\mathbf{m}\wedge\frac{\partial \mathbf{m}}{\partial x_i}\Big)
    \cdot\frac{\partial \mathbf{\phi}}{\partial x_i}\,\mathrm{d}\mathbf{x}\,
    \mathrm{d} t    \\
&\quad -(1+\alpha^2)\iint_{\Omega_T}
    \left(\mathbf{m}\wedge\mathscr{H}_{d,a}(\mathbf{m})\right)
    \cdot\mathbf{\phi}\,\mathrm{d}\mathbf{x}\,\mathrm{d} t.
\end{aligned}
\end{equation}

\item $\mathbf{m}(\cdot,0)=\mathbf{m}_0$ in the sense of traces.

\item For all $T>0$,
\begin{equation}\label{eq:LLWeakSolEnerg}
\mathrm{E}(\mathbf{m}(T))+\frac{\alpha}{1+\alpha^2}\iint_{\Omega_T}
\big| \frac{\partial \mathbf{m}}{\partial t}\big|^2\,\mathrm{d} t
\,\mathrm{d}\mathbf{x} \leq \mathrm{E}(\mathbf{m}(0)),
\end{equation}
\end{subequations}
where
\begin{equation*}
\mathrm{E}(\mathbf{u})=\frac{A}{2}\lVert\nabla\mathbf{u}\rVert_{\mathrm{L}^2(\Omega)}^2
+\frac{1}{2}\lVert\mathbf{K}\mathbf{u}\rVert_{\mathrm{L}^2(\Omega)}^2
+\frac{1}{2}\lVert\mathscr{H}_d(\mathbf{m})\rVert_{\mathrm{L}^2(\Omega)}^2.
\end{equation*}
\end{enumerate}
\end{definition}

It was proved by Alouges and Soyeur \cite{Alouges.Soyeur:OnGlobal}
that the Landau-Lifshitz system \eqref{subeq:LLSyst}
has at least one weak solution
when $\mathbf{h}=A\Delta\mathbf{m}$. This result was generalized
to the full $\mathbf{h}$ in \cite{labbe:mumag}:

\begin{theorem}
Let $\mathbf{m}_0$ belongs to
$\mathbb{H}^1(\Omega)$, such that $\lvert\mathbf{m}_0\rvert=1$ a.e.
in $\Omega$. Then, there exists at least one solution to the Landau-Lifshitz
system $\mathbf{m}$ in the sense of
definition \ref{def:WeakSolLL}.
\end{theorem}

\begin{proof}
The proof is based on the study of a penalized system whose solution
converges to a weak solution to the Landau-Lifshitz system.
The penalized system was:
\begin{subequations}\label{subeq:OldPenalizedLLSyst}
\begin{gather}
\alpha\frac{\partial \mathbf{m}^k}{\partial t}+\mathbf{m}^k
\wedge\frac{\partial \mathbf{m}^k}{\partial t}
=(1+\alpha^2)\big(\mathscr{H}(\mathbf{m}^k)-k(\lvert\mathbf{m}^k\rvert^2-1)\mathbf{m}\big),
\label{eq:OldPenalizedLLEq}\\
\frac{\partial \mathbf{m}^k}{\partial\mathrm{\nu}}=0,\\
\mathbf{m}^k(\cdot,0)=\mathbf{m}_0.
\end{gather}
\end{subequations}
See \cite{Alouges.Soyeur:OnGlobal,labbe:mumag}
for details. See also \cite{Santugini:SolutionsLL} for a generalization
with surface energies.
\end{proof}

We will prove a slightly stronger result. However,
we need a uniform  $\mathbb{L}^\infty(\Omega\times\mathbb{R}^+)$
bound of $\mathbf{m}^k$. But,
the non locality of the $\mathscr{H}_d$ operator
prevents us from proving this result.
For this reason, we introduce another
penalized, less simple, system \eqref{subeq:PenalizedLLSyst}
to prove the existence of solutions more regular on the boundary.

\section{Existence of a solution with $\mathrm{H}^1$ regularity
 in space on the boundary}\label{sect:MainTheorem}

We state our main result as follows.

\begin{theorem}\label{theo:ExistWeakSolLLRegBound}
Let $\mathbf{m}_0$ belong to
$\mathbb{H}^1(\Omega)$, such that $\lvert\mathbf{m}_0\rvert=1$ a.e.
in $\Omega$. Then, there exists at least one solution to the Landau-Lifshitz
system $\mathbf{m}$ in the sense of
definition \ref{def:WeakSolLL} such that
$\gamma\mathbf{m}\in\mathrm{L}^2(0,T;\mathbb{H}^1(\partial\Omega))$.
\end{theorem}

This theorem is a consequence of propositions \ref{prop:ConvPenalizedSystem}
and \ref{prop:UniformL2H1BoundaryBoundedness}. The rest
of this section is dedicated to the proof of this theorem.
We first introduce a penalized system:
\begin{subequations}\label{subeq:PenalizedLLSyst}
\begin{gather}
\begin{split}
\alpha\frac{\partial \mathbf{m}^k}{\partial t}+\mathbf{m}^k
\wedge\frac{\partial \mathbf{m}^k}{\partial t}
&=(1+\alpha^2)\Big(\mathscr{H}_{e,a}(\mathbf{m}^k)
+1_{\{\mathbf{m}^k\neq0\}}\mathscr{H}_d(\mathbf{m}^k)
\\&\phantom{=}
-\big(\mathscr{H}_d(\mathbf{m}^k)\cdot\frac{\mathbf{m}^k}{\lvert\mathbf{m}^k\rvert}\big)
\frac{\mathbf{m}^k}{\lvert\mathbf{m}^k\rvert}\\
&\phantom{=}-k(\lvert\mathbf{m}^k\rvert^2-1)\mathbf{m}^k\Big)
\quad\text{in $\Omega\times\mathbb{R}^+$},
\label{eq:PenalizedLLEq}
\end{split}\\
\frac{\partial \mathbf{m}^k}{\partial\mathrm{\nu}}=0\quad
\text{on $\partial\Omega\times\mathbb{R}^+$},\\
\mathbf{m}^k(\cdot,0)=\mathbf{m}_0
\quad\text{in $\Omega$}.
\end{gather}
\end{subequations}

We first show that this penalized
system has a weak solution
that converges to a weak solution to the Landau-Lifshitz system
as $k$ tends to $+\infty$. We then show that
the $\mathrm{L}^2(0,T;\mathbb{H}^1(\partial\Omega))$ norm
of $\mathbf{m}^k$ is bounded independently of $k$.
This requires a uniform $\mathbb{L}^\infty(\Omega\times\mathbb{R}^+)$ bound of
$\mathbf{m}^k$ first: something we could not establish for
system \eqref{subeq:OldPenalizedLLSyst}, hence the modification
of the penalized system.

\subsection{Properties of solutions to the penalized
system \eqref{subeq:PenalizedLLSyst}}

One can easily adapt the proof
of existence of solution to the
original penalized system \eqref{subeq:OldPenalizedLLSyst}
found in \cite{labbe:mumag,Alouges.Soyeur:OnGlobal}
to the modified penalized system \eqref{subeq:PenalizedLLSyst}:

\begin{proposition}
Let $\mathbf{m}_0$ in $\mathbb{H}^1(\Omega)$, $\lvert\mathbf{m}_0\rvert=1$
a.e. in $\Omega$. Then,
there exists a solution $\mathbf{m}^k$ to
system \eqref{subeq:PenalizedLLSyst}
in $\mathrm{L}^\infty(\mathbb{R}^+;\mathbb{H}^1(\Omega))
\cap\mathrm{H}^1(\Omega\times(0,T))$, \textit{i.e.} satisfying:
\begin{subequations}
\begin{equation}\label{eq:LLPenVariat}
\begin{aligned}
&\iint_{\Omega_T}\left(\alpha\frac{\partial \mathbf{m}^k}{\partial t}
    +\mathbf{m}^{k}\wedge\frac{\partial \mathbf{m}^k}{\partial t}\right)
      \cdot\mathbf{\psi}
      \,\mathrm{d}\mathbf{x}\,\mathrm{d} t \\
&= -(1+\alpha^2)A\iint_{\Omega_T}
\nabla\mathbf{m}^k\cdot\nabla\mathbf{\psi}
 \,\mathrm{d}\mathbf{x}\,\mathrm{d} t
+(1+\alpha^2)\iint_{\Omega_T}
1_{\{\mathbf{m}^k\neq0\}}\mathscr{H}_{d,a}(\mathbf{m}^k)\cdot\mathbf{\psi} \,\mathrm{d}\mathbf{x}\,\mathrm{d} t
\\
&\quad -(1+\alpha^2)\iint_{\Omega_T}\left(
        \frac{(\mathscr{H}_d(\mathbf{m}^k)\cdot\mathbf{m}^k)
          \mathbf{m}^k}{\lvert\mathbf{m}^k\rvert^2}\right)\cdot\mathbf{\psi}
        \,\mathrm{d}\mathbf{x}\,\mathrm{d} t \\
&\quad -(1+\alpha^2)
    k\iint_{\Omega_T}(\lvert\mathbf{m}^{k}\rvert^2-1)\mathbf{m}^{k}
    \cdot\mathbf{\psi}\,\mathrm{d}\mathbf{x}\,\mathrm{d} t,
\end{aligned}
\end{equation}
for all $\mathbf{\psi}$ in
$\mathbb{H}^1(\Omega\times(0,T))$, and for all time $T>0$,
all $\eta>0$:
\begin{equation}\label{eq:LLPenEner}
\begin{aligned}
&\mathrm{E}(\mathbf{m}^k(\cdot,T))
+(\frac{\alpha}{1+\alpha^2}-\eta)\int_0^T\big\|
\frac{\partial \mathbf{m}^k}{\partial t}
  \big\| _{\mathbb{L}^2(\Omega)}^2\,\mathrm{d} t
+\frac{k}{4}\lVert\lvert\mathbf{m}^k(\cdot,T)\rvert^2-1\rVert_{\mathrm{L}^2(\Omega)}^2
\\
&\leq \big(\lvert\Omega\rvert+\mathrm{E}(\mathbf{m}_0)\big)
\exp(\frac{T}{2k\eta})-\lvert\Omega\rvert.
\end{aligned}
\end{equation}
\end{subequations}
\end{proposition}

\begin{proof}
The proof is the same as the one in
\cite{Alouges.Soyeur:OnGlobal,labbe:mumag}
with a minor complication arising from the supplementary term.

Let $(w_1,\dots,w_n, \dots)$ be the orthonormal hilbertian basis
of $\mathrm{L}^2(\Omega)$ comprising the eigenfunctions
of the Laplace operator with
homogeneous Neumann boundary conditions. The $w_i$ belongs to
$\mathcal{C}^\infty(\overline{\Omega})$. Let $V_n$ be the subspace of
$\mathrm{L}^2(\Omega)$ spanned by functions $(w_1,\ldots,w_n)$.
Let $\mathcal{P}_n$ be the orthogonal projector on $V_n$ in
$\mathrm{L}^2(\Omega)$. The basis $(w_1,\dots,w_n, \dots)$ is also
an hilbertian orthogonal basis of $\mathrm{H}¹(\Omega)$ and $\mathcal{P}_n$
is also an orthogonal projector in $\mathrm{H}^1(\Omega)$.
We search for
$\mathbf{m}^k_{n}$ with the form
$\mathbf{m}^k_{n}=
\sum_{i=1}^{n}\mathbf{\phi}_i(t)w_i(\mathbf{x})$,
where the $\mathbf{\phi}_i$ are in
$\mathcal{C}^\infty(\mathbb{R}^+;\mathbb{R}^3)$ such that
\begin{subequations}
\begin{gather}
\begin{split}\label{eq:LLPenGalerkinProj}
 \alpha\frac{\partial \mathbf{m}^k_{n}}{\partial t}+
 \mathcal{P}_n(\mathbf{m}^k_{n}\wedge
 \frac{\partial \mathbf{m}^k_{n}}{\partial t})
 &=(1+\alpha^2)\mathcal{P}_n\left(\mathscr{H}(\mathbf{m}^k_{n})
-\frac{
 (\mathscr{H}_d(\mathbf{m}^k_{n})\cdot\mathbf{m}^k_{n})
 \mathbf{m}^k_{n}}{\lvert\mathbf{m}^k_n\rvert^2+n^{-1}}
\right)\\
 &\phantom{=}-(1+\alpha^2)k \mathcal{P}_n\big(
 (\lvert\mathbf{m}^k_n\rvert^2-1)\mathbf{m}^k_n\big)
\end{split}\\
\mathbf{m}^k_{n}(\cdot,0)=
\mathcal{P}_n(\mathbf{m}_0).
\end{gather}
\end{subequations}
Let $\mathbf{\Phi}_n=(\mathbf{\phi}_1,\ldots,\mathbf{\phi}_n)$.
Equation \eqref{eq:LLPenGalerkinProj} is equivalent to
\[
\frac{\,\mathrm{d}\mathbf{\Phi}_n}{\,\mathrm{d} t}
-\mathbf{A}(\mathbf{\Phi}_n(t))\frac{\,\mathrm{d}\mathbf{\Phi}_n}{\,\mathrm{d}
t} = F(\mathbf{\Phi}_n(t)),
\]
where $F$ is of class $\mathcal{C}^\infty$ (thanks to the $n^{-1}$
in the denominator which serves no other purpose) and
$\mathbf{\Phi}_n(t)\mapsto\mathbf{A}(\mathbf{\Phi}_n(t))$ is linear
continuous, thus smooth. Moreover,
$\mathbf{A}(\mathbf{\Phi})$ is an antisymmetric matrix for all
$\mathbf{\Phi}$. So
the matrix $\mathbf{I}-\mathbf{A}(\mathbf{\Phi})$ is nonsingular
and the function $\mathbf{\Phi}\;\mapsto\; (\mathbf{I}-\mathbf{A}(\mathbf{\Phi}))^{-1}$
is of class $\mathcal{C}^\infty$. By Cauchy-Lipshitz,
$\mathbf{\Phi_i}(t)$ exists locally in time.
Equation \eqref{eq:LLPenGalerkinProj} can be expressed as
\begin{equation}\label{eq:LLPenGalerkinVariat}
\begin{aligned}
&\iint_{\Omega_T}\left(\alpha\frac{\partial \mathbf{m}^k_n}{\partial t}
    +\mathbf{m}^{k}_n\wedge
    \frac{\partial \mathbf{m}^k_n}{\partial t}\right)
      \cdot\mathbf{\psi}
      \,\mathrm{d}\mathbf{x}\,\mathrm{d} t \\
&=-(1+\alpha^2)A\iint_{\Omega_T}
\nabla\mathbf{m}^k_n\cdot\nabla\mathbf{\psi}
 \,\mathrm{d}\mathbf{x}\,\mathrm{d} t
+(1+\alpha^2)\iint_{\Omega_T}
\mathscr{H}_{d,a}(\mathbf{m}^k_n)\cdot\mathbf{\psi} \,\mathrm{d}\mathbf{x}\,
\mathrm{d} t\\
&\quad -(1+\alpha^2)\iint_{\Omega_T}\left(
        \frac{(\mathscr{H}_d(\mathbf{m}^k_{n})\cdot\mathbf{m}^k_{n})
          \mathbf{m}^k_{n}}{\lvert\mathbf{m}^k_n\rvert^2+n^{-1}}\right)\cdot\mathbf{\psi}
        \,\mathrm{d}\mathbf{x}\,\mathrm{d} t \\
&\quad -(1+\alpha^2)
    k\iint_{\Omega_T}(\lvert\mathbf{m}^{k}_n\rvert^2-1)\mathbf{m}^{k}_n
    \mathbf{\psi}\,\mathrm{d}\mathbf{x}\,\mathrm{d} t,
\end{aligned}
\end{equation}
for all $\mathbf{\psi}$ in $V_n \otimes\mathcal{C}^\infty(\mathbb{R}^+;
\mathbb{R}^3)$. In \eqref{eq:LLPenGalerkinVariat}, we take
$\mathbf{\psi}=\frac{\partial \mathbf{m}^k_n}{\partial t}$,
we obtain for all $T>0$:
\begin{align*}%\label{eq:LLPenGalerkinProtoEner}
&\mathrm{E}(\mathbf{m}^k_n(\cdot,T))
+\frac{\alpha}{1+\alpha^2}\int_0^T\big\|
\frac{\partial \mathbf{m}^k_n}{\partial t}
  \big\| _{\mathbb{L}^2(\Omega)}^2\,\mathrm{d} t
+\frac{k}{4}\lVert\lvert\mathbf{m}^k_n(\cdot,T)\rvert^2-
1\rVert_{\mathrm{L}^2(\Omega)}^2 \\
&\leq \mathrm{E}(\mathbf{m}_0)
+\frac{k}{4}\lVert\lvert\mathcal{P}_n(\mathbf{m}_0)\rvert^2
 -1\rVert_{\mathrm{L}^2(\Omega)}^2
-\iint_{\Omega_T}
\left( \frac{(\mathscr{H}_d(\mathbf{m}^k_{n})\cdot\mathbf{m}^k_{n})
\mathbf{m}^k_{n}}{\lvert\mathbf{m}^k_n\rvert^2+n^{-1}}\right)\cdot
\frac{\partial \mathbf{m}^k_n}{\partial t}\,\mathrm{d}\mathbf{x}\,
\mathrm{d} t.
\end{align*}
Therefore, for all $\eta>0$, all time $T>0$:
\begin{align*}%\label{eq:LLPenGalerkinBeforeGromwallEner}
&\mathrm{E}(\mathbf{m}^k_n(\cdot,T))
+(\frac{\alpha}{1+\alpha^2}-\eta)\int_0^T
 \big\|\frac{\partial \mathbf{m}^k_n}{\partial t}
  \big\|_{\mathbb{L}^2(\Omega)}^2\,\mathrm{d} t
+\frac{k}{4}\lVert\lvert\mathbf{m}_n^k(\cdot,T)\rvert^2
-1\rVert_{\mathrm{L}^2(\Omega)}^2 \\
&\leq \mathrm{E}(\mathbf{m}_0)
+\frac{k}{4}\lVert\lvert\mathcal{P}_n(\mathbf{m}_0)\rvert^2-1
\rVert_{\mathrm{L}^2(\Omega)}^2
+\frac{1}{8\eta}\int_0^T
\int_\Omega\big((\lvert\mathbf{m}^k_n\rvert^2-1)^2+1\big)
\,\mathrm{d}\mathbf{x}\,\mathrm{d} t.
\end{align*}
By Gronwall, for all $\eta>0$, all time $T>0$:
\begin{equation}\label{eq:LLPenGalerkinEner}
\begin{aligned}
&\mathrm{E}(\mathbf{m}^k_n(\cdot,T))
+(\frac{\alpha}{1+\alpha^2}-\eta)\int_0^T\big\|
\frac{\partial \mathbf{m}^k_n}{\partial t}
  \big\|_{\mathbb{L}^2(\Omega)}^2\,\mathrm{d} t
+\frac{k}{4}\lVert\lvert\mathbf{m}_n^k(\cdot,T)\rvert^2-1\rVert_{\mathrm{L}^2(\Omega)}^2
\\
&\leq \big(\lvert \Omega\rvert+\mathrm{E}(\mathbf{m}_0)+
\frac{k}{4}\lVert\lvert\mathcal{P}_n(\mathbf{m}_0)\rvert^2
-1\rVert_{\mathrm{L}^2(\Omega)}^2\big)
\exp(\frac{T}{2k\eta})-\lvert\Omega\rvert.
\end{aligned}
\end{equation}
Thus, $\mathbf{m}^k_n$ exists in global time.
Since, $\mathcal{P}_n(\mathbf{m}_0)$ tends to $\mathbf{m}_0$ in
$\mathbb{H}^1(\Omega)$,
$\frac{k}{4}\lVert\lvert\mathcal{P}_n(\mathbf{m}_0)\rvert^2-1
\rVert_{\mathrm{L}^2(\Omega)}^2$ tends to $0$ as  $n$ tends to $+\infty$.
Therefore, $\mathbf{m}^k_n$ is bounded in $\mathbb{H}^1(\Omega_T)$
and in $\mathrm{L}^\infty(0,T;\mathbb{H}^1(\Omega))$,
independently of $n$.
There exists $\mathbf{m}^k$ in $\mathrm{L}^\infty(0,T;\mathbb{H}^1(\Omega))$
and in $\mathbb{H}^1(\Omega\times(0,T))$ and $\mathrm{v}^k$ in $\mathrm{L}^2(\Omega)$,
such that, modulo
a subsequence, as $n$ tends to $+\infty$:
\begin{subequations}
\begin{gather}
\mathbf{m}^k_n\to\mathbf{m}^k\quad\text{strongly in
 $\mathbb{L}^2(\Omega\times(0,T))$},\\
\mathbf{m}^k_n\to\mathbf{m}^k\quad\text{weakly in
 $\mathbb{H}^1(\Omega\times(0,T))$},\\
\mathbf{m}^k_n\to\mathbf{m}^k\quad\text{weakly--\textasteriskcentered{}
  in $\mathrm{L}^\infty(0,T;\mathbb{H}^1(\Omega))$}, \\
1_{\{\mathbf{m}^k=0\}}\frac{(\mathscr{H}_d(\mathbf{m}_n^k)
 \cdot\mathbf{m}_n^k)\mathbf{m}_n^k}
{\lvert\mathbf{m}^k_n\rvert^2+n^{-1}}
\to \mathrm{v}^k\quad\text{weakly in $\mathbb{L}^2(\Omega\times(0,T))$}.
\end{gather}
Also, by Aubin's lemma, for all $1<p<+\infty$,
$1<q<6$, $T>0$,
\begin{equation}
\mathbf{m}^k_n\to\mathbf{m}^k\quad\text{strongly in
  $\mathrm{L}^p(0,T;\mathbb{L}^q(\Omega))$}.
\end{equation}
\end{subequations}

The limit $\mathbf{m}^k$ has the required properties:
$\mathcal{P}_n(\mathbf{m}_0)$ converges to $\mathbf{m}_0$
in $\mathbb{H}^1(\Omega)$.
Computing the limit of \eqref{eq:LLPenGalerkinEner},
yields \eqref{eq:LLPenEner}.
Since on the set $\{(\mathbf{x},t)|\mathbf{m}^k=0\}$,
$\frac{\partial \mathbf{m}^k}{\partial t}=0$ and
$\Delta\mathbf{m}^k=0$ a.e.,
computing the limit of \eqref{eq:LLPenGalerkinVariat}
yields first $\mathrm{v}^k=1_{\{\mathbf{m}^k=0\}}\mathscr{H}_d(\mathbf{m}^k)$,
then \eqref{eq:LLPenVariat} for all $\mathbf{\psi}$
in $\bigcup_{n=1}^\infty V_n\otimes\mathcal{C}^\infty([0,T];\mathbb{R}^3)$.
By density,  \eqref{eq:LLPenVariat} holds
for all $\mathbf{\psi}$ in $\mathbb{H}^1(\Omega\times(0,T))$.
\end{proof}

Now, we can prove that $\mathbf{m}^k$ converges
to a weak solution to the Landau-Lifshitz system.

\begin{proposition}\label{prop:ConvPenalizedSystem}
Let $\mathbf{m}_0$ be in $\mathbb{H}^1(\Omega)$, $\lvert\mathbf{m}_0\rvert=1$
a.e. in $\Omega$. Let $\mathbf{m}^k$ be a weak solution to
the penalized system \eqref{subeq:PenalizedLLSyst}. Then,
there exists a subsequence of $\mathbf{m}^k$, that converges to a weak
solution $\mathbf{m}$ to the Landau-Lifshitz system \eqref{subeq:LLSyst}
weakly in $\mathbb{H}^1(\Omega\times(0,T))$.
\end{proposition}

\begin{proof}
By \eqref{eq:LLPenEner}, $\mathbf{m}^k$ is bounded
in $\mathrm{L}^\infty(0,T;\mathbb{H}^1(\Omega))$ and in
$\mathbb{H}^1(\Omega\times(0,T)$. Besides,
$k\lVert\lvert\mathbf{m}^k\rvert^2-1\rVert_{\mathrm{L}^\infty(0,T;\mathrm{L}^2(\Omega))}^2$
is bounded. There exists $\mathbf{m}$ in $\mathrm{H}^1(\Omega\times(0,T)$,
such that, up to a subsequence,
\begin{subequations}
\begin{gather}
\mathbf{m}^k\to\mathbf{m}\quad\text{strongly in $\mathbb{L}^2(\Omega\times(0,T))$},\\
\mathbf{m}^k\to\mathbf{m}\quad\text{weakly in $\mathbb{H}^1(\Omega\times(0,T))$},\\
\mathbf{m}^k\to\mathbf{m}\quad\text{weakly--\textasteriskcentered{}
  in $\mathrm{L}^\infty(0,T;\mathbb{H}^1(\Omega))$},\\
\lvert\mathbf{m}^k\rvert^2-1\to 0 \quad\text{strongly in
  $\mathbb{L}^2(\Omega\times(0,T))$},\label{eq:ConvModule}
\end{gather}
Also, by Aubin's lemma, for all $1<p<+\infty$,
$1<q<6$, $T>0$,
\begin{equation}
\mathbf{m}^k\to\mathbf{m}\quad\text{strongly in
  $\mathrm{L}^p(0,T;\mathbb{L}^q(\Omega))$}.
\end{equation}
\end{subequations}
Obviously, $\mathbf{m}(\cdot,0)=\mathbf{m}_0$.
By \eqref{eq:ConvModule}, $\lvert\mathbf{m}\rvert=1$.

We compute the limit of \eqref{eq:LLPenEner}
as $k$ tends to $+\infty$ as
in \cite{Alouges.Soyeur:OnGlobal,labbe:mumag}. Then, we have
$\eta$ tend to $0$: energy
inequality \eqref{eq:LLWeakSolEnerg}
is satisfied.

It only remains to prove that $\mathbf{m}$
satisfy \eqref{eq:LLWeakSolVariat}: in \eqref{eq:LLPenVariat},
we take $\mathbf{\psi}=\mathbf{m}^k\wedge\mathbf{\phi}$, with $\mathbf{\phi}$
in $\mathcal{C}^\infty(\overline{\Omega\times(0,T)};\mathbb{R}^3)$
and take the limit as $k$ tends to $+\infty$.
Since the supplementary term
containing
$(\mathscr{H}_d(\mathbf{m}^k)\cdot\frac{\mathbf{m}^k}{\lvert\mathbf{m}^k\rvert})
\frac{\mathbf{m}^k}{\lvert\mathbf{m}^k\rvert}$
disappears,
we can then conclude as in \cite{Alouges.Soyeur:OnGlobal,labbe:mumag}
and obtain \eqref{eq:LLWeakSolVariat}
for all $\mathbf{\phi}$ in
$\mathcal{C}^\infty(\overline{\Omega\times(0,T)})$.
By density, \eqref{eq:LLWeakSolVariat} also holds
for all $\mathbf{\phi}$ in $\mathbb{H}^1(\Omega\times(0,T))$.
\end{proof}

We establish an important proposition that we
were unable to prove for the original
penalized system \eqref{subeq:OldPenalizedLLSyst}
due to the non locality of $\mathscr{H}_d$.

\begin{proposition}\label{prop:LinftyBoundMomentk}
Let $\mathbf{m}_0$ be in $\mathbb{H}^1(\Omega)$,
$\lvert\mathbf{m}_0\rvert=1$ a.e.
in $\Omega$. Let $\mathbf{m}^k$ be a weak solution
to system \eqref{subeq:PenalizedLLSyst}.
Then, $\mathbf{m}^k$ satisfy
$\lvert\mathbf{m}^k\rvert\leq 1$
a.e. in $\Omega\times\mathbb{R}^+$.
\end{proposition}

\begin{proof}
We follow Alouges-Soyeur \cite{Alouges.Soyeur:OnGlobal},
and introduce the map
$g:\mathbb{R}\to\mathbb{R}$ defined by
\begin{equation*}
g(x)=\begin{cases}
0&\text{if $x<0$},\\
x&\text{if $0\leq x\leq1$},\\
1&\text{if $x\geq 1$},
\end{cases}
\end{equation*}
We set
$\mathbf{\psi}(\mathbf{x},t)=g(\lvert\mathbf{m}^k(\mathbf{x},t)\rvert^2-1)\mathbf{m}^k(\mathbf{x},t)$.
The function $\mathbf{\psi}$ belongs to $\mathbb{H}^1(\Omega\times(0,T))$
and
\begin{equation*}
\frac{\partial \mathbf{\psi}}{\partial x_i}=
2g'(\lvert\mathbf{m}^k\rvert^2-1)
\big(\mathbf{m}^k\cdot\frac{\partial \mathbf{m}^k}{\partial x_i}\big)\mathbf{m}^k
+g(\lvert\mathbf{m}^k\rvert^2-1)\frac{\partial \mathbf{m}^k}{\partial x_i}.
\end{equation*}
Reporting $\mathbf{\psi}$ in \eqref{eq:LLPenVariat} yields
\begin{align*}
&\alpha \iint_{\Omega_T}g(\lvert\mathbf{m}^k(\mathbf{x},t)\rvert^2-1)
\mathbf{m}^k(\mathbf{x},t)\cdot\frac{\partial \mathbf{m}^k}{\partial t}
         \,\mathrm{d}\mathbf{x}\,\mathrm{d} t \\
&= -(1+\alpha^2)A\sum_{i=1}^3\iint_{\Omega_T}
2g'(\lvert\mathbf{m}^k\rvert^2-1)
\Big(\mathbf{m}^k\cdot\frac{\partial \mathbf{m}^k}{\partial x_i}\Big)^2
\,\mathrm{d}\mathbf{x}\,\mathrm{d} t\\
&\quad -(1+\alpha^2) \iint_{\Omega_T}
g(\lvert\mathbf{m}^k\rvert^2-1)
\lvert\nabla\mathbf{m}^k\rvert^2
 \,\mathrm{d}\mathbf{x}\,\mathrm{d} t\\
&\quad -(1+\alpha^2)\iint_{\Omega_T}
g(\lvert\mathbf{m}^k(\mathbf{x},t)\rvert^2-1)
\mathbf{m}^k\cdot\mathbf{K}\mathbf{m}^k \,\mathrm{d}\mathbf{x}\,\mathrm{d} t
\\
&\quad -(1+\alpha^2)
    k\iint_{\Omega_T}(\lvert\mathbf{m}^{k}\rvert^2-1)
        \lvert\mathbf{m}^{k}\rvert^2
        g(\lvert\mathbf{m}^k(\mathbf{x},t)\rvert^2-1)
    \,\mathrm{d}\mathbf{x}\,\mathrm{d} t.
\end{align*}
Should we have used system \eqref{subeq:OldPenalizedLLSyst} instead, then
the term containing the global operator would have been
very difficult, if not outright impossible,
to estimate.

Therefore,
$\iint_{\Omega_T}g(\lvert\mathbf{m}^k(\mathbf{x},t)\rvert^2-1)
\mathbf{m}^k(\mathbf{x},t)\cdot\frac{\partial \mathbf{m}^k}{\partial t}
         \,\mathrm{d}\mathbf{x}\,\mathrm{d} t
\leq 0$.
 Thus,  for all $T>0$,
\[
\int_\Omega G(\lvert\mathbf{m}^k(\cdot,T)\rvert^2-1)\,\mathrm{d}\mathbf{x}
\leq
\int_\Omega G(\lvert\mathbf{m}_0\rvert^2-1)\,\mathrm{d}\mathbf{x}=0,
\]
where $G(x)=\int_0^xg(s)\,\mathrm{d} s$.
Since $G\geq0$,  $G(\lvert\mathbf{m}^k(\cdot,T)\rvert^2-1)=0$
a.e. in $\Omega$ for all $T>0$. Therefore,
$\mathbf{m}^k\leq 1$ a.e. in $\Omega\times\mathbb{R}^+$.
\end{proof}

As a corollary to proposition \ref{prop:LinftyBoundMomentk},
we have

\begin{corollary}\label{coroll:RegMomentk}
Any weak solution $\mathbf{m}^k$ to
system \eqref{subeq:PenalizedLLSyst}
belongs to the space $\mathbb{H}^{2,1}(\Omega\times(0,T))$.
\end{corollary}


\subsection{Uniform $\mathbb{H}^1(\partial\Omega)$ bound
of the penalized
solution}\label{subsect:UniformBoundednessL2H1Boundary}

To prove Theorem \ref{theo:ExistWeakSolLLRegBound},
we only need to prove the following proposition.

\begin{proposition}\label{prop:UniformL2H1BoundaryBoundedness}
Let $\mathbf{m}_0$ be in $\mathbb{H}^1(\Omega)$, $\lvert\mathbf{m}_0\rvert=1$.
Let $\mathbf{m}^k$ be a weak solution to
the penalized system \eqref{subeq:PenalizedLLSyst}. Then,
for all time $T>0$ the quantity
$\lVert\nabla_\mathrm{T}\gamma\mathbf{m}^k
\rVert_{\mathbb{L}^2(\partial\Omega\times(0,T))}^2$
is bounded uniformly in $k$.
\end{proposition}

\begin{proof}
Let $\phi_i$ be in $\mathcal{C}^\infty(\overline{\Omega\times(0,T)})$
for any integer $i$, $1\leq i\leq 3$.
By corollary \ref{coroll:RegMomentk}
and proposition \ref{prop:LinftyBoundMomentk}, we can
multiply \eqref{eq:PenalizedLLEq} by
$\frac{\partial \mathbf{m}^k}{\partial x_i}\phi_i$
and integrate over any open set $O\subset\Omega$
with a smooth boundary:
% \begin{equation}\label{eq:FormuleFaibleI}
\begin{align*}
&\alpha\int_{O\times(0,T)}\frac{\partial \mathbf{m}^k}{\partial t}\cdot
\frac{\partial \mathbf{m}^k}{\partial x_i}
\phi_i\,\mathrm{d}\mathbf{x}\,\mathrm{d} t
+\int_{O\times(0,T)}(\mathbf{m}^k\wedge\frac{\partial \mathbf{m}^k}{\partial t})\cdot
\frac{\partial \mathbf{m}^k}{\partial x_i}\phi_i\,\mathrm{d}\mathbf{x}\,
\mathrm{d} t\\
&=(1+\alpha^2)\int_{O\times(0,T)}1_{\{\mathbf{m}^k\neq0\}}
 \mathscr{H}_{d,a}(\mathbf{m}^k)\cdot
 \frac{\partial \mathbf{m}^k}{\partial x_i}\phi_i\,\mathrm{d}\mathbf{x}\,
 \mathrm{d} t\\
&\quad -(1+\alpha^2)\int_{O\times(0,T)}(\mathscr{H}_{d}(\mathbf{m}^k)\cdot
 \frac{\mathbf{m}^k}{\lvert\mathbf{m}^k\rvert})
 \frac{\mathbf{m}^k}{\lvert\mathbf{m}^k\rvert}\cdot
 \frac{\partial \mathbf{m}^k}{\partial x_i}\phi_i\,\mathrm{d}\mathbf{x}\,
 \mathrm{d} t\\
&\quad -(1+\alpha^2)\underbrace{k\int_{O\times(0,T)}
  \big(\lvert\mathbf{m}^k\rvert^2-1\big) \mathbf{m}^k\cdot
 \frac{\partial \mathbf{m}^k}{\partial x_i}\phi_i\,\mathrm{d}
  \mathbf{x}\,\mathrm{d} t}_{I}\\
&\quad +(1+\alpha^2)A\underbrace{\sum_{j=1}^3
  \int_{O\times(0,T)}\Big(
    \frac{\partial }{\partial x_j}\Big(\frac{\partial \mathbf{m}^k}{\partial x_i}
    \cdot\frac{\partial \mathbf{m}^k}{\partial x_j}
    \Big)
    -\frac{1}{2}\frac{\partial }{\partial x_i}
    \left\lvert\frac{\partial \mathbf{m}^k}{\partial x_j}\right\rvert^2
    \Big)\phi_i
  \,\mathrm{d}\mathbf{x}\,\mathrm{d} t
}_{II}
\end{align*}
However, in the above equality,
\[
I=-\frac{k}{4}\int_{O\times(0,T)}
\big(\lvert\mathbf{m}^k\rvert^2-1\big)^2
\frac{\partial \phi_i}{\partial x_i}\,\mathrm{d}\mathbf{x}\,\mathrm{d} t
+\frac{k}{4}\int_{\partial O\times(0,T)}
\big(\lvert\mathbf{m}^k\rvert^2-1\big)^2\nu_i\phi_i\,\mathrm{d}\sigma(\mathbf{x})\,\mathrm{d} t,
\]
and
\begin{align*}
II&= -\sum_{j=1}^3  \int_{O\times(0,T)}
  \Big(\frac{\partial \mathbf{m}^k}{\partial x_i}
  \cdot\frac{\partial \mathbf{m}^k}{\partial x_j}
  \Big)\frac{\partial \phi_i}{\partial x_j}  \,\mathrm{d}\mathbf{x}\,\mathrm{d} t
+\frac{1}{2}\int_{O\times(0,T)}
    \lvert\nabla\mathbf{m}^k\rvert^2
    \frac{\partial \phi_i}{\partial x_i}
  \,\mathrm{d}\mathbf{x}\,\mathrm{d} t\\
&\quad +\int_{\partial O\times(0,T)}
  \Big(\frac{\partial \mathbf{m}^k}{\partial x_i}
  \cdot\frac{\partial \mathbf{m}^k}{\partial\mathrm{\nu}}
  \Big)\phi_i\,\mathrm{d}\sigma(\mathbf{x})\,\mathrm{d} t
-\frac{1}{2}\int_{\partial O\times(0,T)}
    \lvert\nabla\mathbf{m}^k\rvert^2\nu_i\phi_i
    \,\mathrm{d}\sigma(\mathbf{x})\,\mathrm{d} t.
\end{align*}
Therefore,
\begin{equation}\label{eq:FormuleFaibleII}
\begin{aligned}
&\alpha\int_{O\times(0,T)}\frac{\partial \mathbf{m}^k}{\partial t}
 \cdot\frac{\partial \mathbf{m}^k}{\partial x_i}
 \phi_i\,\mathrm{d}\mathbf{x}\,\mathrm{d} t
 +\int_{O\times(0,T)}(\mathbf{m}^k\wedge
 \frac{\partial \mathbf{m}^k}{\partial t})\cdot
 \frac{\partial \mathbf{m}^k}{\partial x_i}\phi_i\,\mathrm{d}\mathbf{x}\,
 \mathrm{d} t\\
&-(1+\alpha^2)\int_{O\times(0,T)}1_{\{\mathbf{m}^k\neq0\}}
 \mathscr{H}_{d,a}(\mathbf{m}^k)\cdot
 \frac{\partial \mathbf{m}^k}{\partial x_i}\phi_i\,\mathrm{d}\mathbf{x}\,
 \mathrm{d} t\\
&+(1+\alpha^2)\int_{O\times(0,T)}(\mathscr{H}_{d}(\mathbf{m}^k)\cdot
 \frac{\mathbf{m}^k}{\lvert\mathbf{m}^k\rvert})
 \frac{\mathbf{m}^k}{\lvert\mathbf{m}^k\rvert}\cdot
 \frac{\partial \mathbf{m}^k}{\partial x_i}\phi_i\,\mathrm{d}\mathbf{x}\,
 \mathrm{d} t\\
&+(1+\alpha^2)A\sum_{j=1}^3  \int_{O\times(0,T)}
  \Big(\frac{\partial \mathbf{m}^k}{\partial x_i}
  \cdot\frac{\partial \mathbf{m}^k}{\partial x_j}
  \Big)\frac{\partial \phi_i}{\partial x_j}  \,\mathrm{d}\mathbf{x}\,
  \mathrm{d} t\\
&-(1+\alpha^2)\frac{A}{2}\int_{O\times(0,T)}
    \lvert\nabla\mathbf{m}^k\rvert^2
    \frac{\partial \phi_i}{\partial x_i}
  \,\mathrm{d}\mathbf{x}\,\mathrm{d} t\\
&-(1+\alpha^2)\frac{k}{4}\int_{O\times(0,T)}
 \big(\lvert\mathbf{m}^k\rvert^2-1\big)^2
 \frac{\partial \phi_i}{\partial x_i}\,\mathrm{d}\mathbf{x}\,\mathrm{d} t\\
&=-(1+\alpha^2)\frac{k}{4}\int_{\partial O\times(0,T)}
 \big(\lvert\mathbf{m}^k\rvert^2-1\big)^2\nu_i\phi_i\,\mathrm{d}
 \sigma(\mathbf{x})\,\mathrm{d} t\\
&\quad +(1+\alpha^2)A\int_{\partial O\times(0,T)}
  \Big(\frac{\partial \mathbf{m}^k}{\partial x_i}
  \cdot\frac{\partial \mathbf{m}^k}{\partial\mathrm{\nu}}
  \Big)\phi_i\,\mathrm{d}\sigma(\mathbf{x})\,\mathrm{d} t\\
&\quad -(1+\alpha^2)\frac{A}{2}\int_{\partial O\times(0,T)}
    \lvert\nabla\mathbf{m}^k\rvert^2\nu_i\phi_i
    \,\mathrm{d}\sigma(\mathbf{x})\,\mathrm{d} t,
\end{aligned}
\end{equation}
for all $\phi_i$ in
$\mathcal{C}^\infty(\overline{\Omega\times(0,T)})$.
In \eqref{eq:FormuleFaibleII}, we choose $\phi_i$ independent of the
time $t$ such that $\phi_i=\nu_i$ on $\partial O$ and sum over
$i$. We obtain,
denoting by $\mathbf{\phi}$ the vector valued function
$(\phi_1,\phi_2,\phi_3)$,
\begin{align*}%\label{eq:FormuleFaibleIII}
&\alpha\int_{O\times(0,T)}\frac{\partial \mathbf{m}^k}{\partial t}
 \cdot(\mathbf{\phi}\cdot\nabla)\mathbf{m}^k\,\mathrm{d}\mathbf{x}\,
 \mathrm{d} t
 +\int_{O\times(0,T)}(\mathbf{m}^k\wedge\frac{\partial
 \mathbf{m}^k}{\partial t}) \cdot(\mathbf{\phi}\cdot\nabla)\mathbf{m}^k\,
 \mathrm{d}\mathbf{x}\,\mathrm{d} t\\
&-(1+\alpha^2)\int_{O\times(0,T)}1_{\{\mathbf{m}^k\neq0\}}
 \mathscr{H}_{d,a}(\mathbf{m}^k)\cdot
 (\mathbf{\phi}\cdot\nabla)\mathbf{m}^k\,\mathrm{d}\mathbf{x}\,\mathrm{d} t\\
&+(1+\alpha^2)\int_{O\times(0,T)}(\mathscr{H}_{d}(\mathbf{m}^k)\cdot
 \frac{\mathbf{m}^k}{\lvert\mathbf{m}^k\rvert})
 \frac{\mathbf{m}^k}{\lvert\mathbf{m}^k\rvert}\cdot
 (\mathbf{\phi}\cdot\nabla)\mathbf{m}^k\,\mathrm{d}\mathbf{x}\,\mathrm{d} t\\
&+(1+\alpha^2)A\sum_{i,j=1}^3  \int_{O\times(0,T)}
  \Big(\frac{\partial \mathbf{m}^k}{\partial x_i}
  \cdot\frac{\partial \mathbf{m}^k}{\partial x_j}
  \Big)\frac{\partial \phi_i}{\partial x_j}
  \,\mathrm{d}\mathbf{x}\,\mathrm{d} t\\
&-(1+\alpha^2)\frac{A}{2}\int_{O\times(0,T)}
    \lvert\nabla\mathbf{m}^k\rvert^2
    \mathop{\rm div}\mathbf{\phi}
  \,\mathrm{d}\mathbf{x}\,\mathrm{d} t\\
&-(1+\alpha^2)\frac{k}{4}\int_{O\times(0,T)}
 \big(\lvert\mathbf{m}^k\rvert^2-1\big)^2
 \mathop{\rm div}\mathbf{\phi}\,\mathrm{d}\mathbf{x}\,\mathrm{d} t\\
&=-(1+\alpha^2)\frac{k}{4}\int_{\partial O\times(0,T)}
\big(\lvert\mathbf{m}^k\rvert^2-1\big)^2\,\mathrm{d}\sigma(\mathbf{x})\,
 \mathrm{d} t\\
&\quad +(1+\alpha^2)A\int_{\partial O\times(0,T)}
  \big|\frac{\partial \mathbf{m}^k}{\partial\mathrm{\nu}}\big|^2
  \,\mathrm{d}\sigma(\mathbf{x})\,\mathrm{d} t\\
&\quad -(1+\alpha^2)\frac{A}{2}\int_{\partial O\times(0,T)}
    \lvert\nabla\mathbf{m}^k\rvert^2
    \,\mathrm{d}\sigma(\mathbf{x})\,\mathrm{d} t.
\end{align*}
The left hand-side is bounded uniformly in $k$. Therefore,
\begin{align*}
&\Big| \frac{A}{2}\lVert\nabla_\mathrm{T}\gamma\mathbf{m}^k
    \rVert_{\mathbb{L}^2(\partial O\times(0,T))}^2
-\frac{A}{2}\big\|\frac{\partial \mathbf{m}^k}{\partial\mathrm{\nu}}
    \big\|_{\mathbb{L}^2(\partial O\times(0,T))}^2\\
&+\frac{k}{4}\lVert\lvert\gamma\mathbf{m}^k\rvert^2
-1\rVert_{\mathrm{L}^2(\partial O\times(0,T))}^2\Big|
\leq C(O).
\end{align*}
Since $\frac{\partial \mathbf{m}^k}{\partial\mathrm{\nu}}=0$ on
$\partial\Omega$,
taking $O=\Omega$ yields the wanted result.
\end{proof}

We derive from the previous proof the following corollary.

\begin{corollary} \label{coro4.7}
Let $\mathbf{m}_0$ be in $\mathbb{H}^1(\Omega)$, $\lvert\mathbf{m}_0\rvert=1$
a.e. in $\Omega$. Let $\mathbf{m}^k$ be a weak solution to
the penalized system \eqref{subeq:PenalizedLLSyst}. Then,
the quantity $k\lVert\lvert\gamma\mathbf{m}^k\rvert^2-1
\rVert_{\mathbb{L}^2(\partial\Omega\times(0,T))}^2$ is bounded
uniformly in $k$.
\end{corollary}

\subsection*{Conclusion}
In this paper, we have proved the existence of weak solutions
to the Landau-Lifshitz system with an $\mathbb{H}^1$ regularity in space
on the boundary of the domain.
This result holds for very general form of
$\mathbf{h}$ and is not limited to the case
$\mathbf{h}=A\Delta\mathbf{m}$. These kinds of results are important
because there is currently no ``perfect'' concept of what is
a good weak solution to the Landau-Lifshitz system. Any criteria
allowing to discriminate among those weak solutions is
always welcome. It is natural to prefer among weak solutions those
that are more regular. This result is also interesting as it opens the
possibility to use first order transmission conditions between
adjacent domains.

\subsection*{Acknowledgments}
The author wansts to thank S. Labb\'e and L. Halpern for their support.

\begin{thebibliography}{10}

\bibitem{Alouges.Soyeur:OnGlobal}
F. Alouges and A. Soyeur.
\newblock On global weak solutions for Landau-Lifshitz equations :
  existence and nonuniqueness.
\newblock {\em Nonlinear Analysis. Theory, Methods \& Applications},
  18(11):1071--1084, 1992.

\bibitem{Brown:Microm}
W. F. Brown.
\newblock {\em Micromagnetics}.
\newblock Interscience Publishers, 1963.

\bibitem{Carbou:Regularity}
G. Carbou.
\newblock Regularity for critical points of a non local energy.
\newblock {\em Calc. Var. PDE}, 5:409--433, 1997.

\bibitem{Friedman:MathematI}
M. J. Friedman.
\newblock Mathematical study of the nonlinear singular integral magnetic field
  equation I.
\newblock {\em SIAM J. Appl. Math.}, 39(1):14--20, Ao\^ut 1980.

\bibitem{Friedman:MathematII}
M. J. Friedman.
\newblock Mathematical study of the nonlinear singular integral magnetic field
  equation II.
\newblock {\em SIAM J. Numer. Math.}, 18(4):644--653, August 1981.

\bibitem{Friedman:MathematIII}
M. J. Friedman.
\newblock Mathematical study of the nonlinear singular integral magnetic field
  equation \mbox{III}.
\newblock {\em SIAM J. Math. Anal.}, 12(4):644--653, July 1981.

\bibitem{Hardt.Kinderlehrer:SomeRegularity}
R. Hardt and D. Kinderlehrer.
\newblock Some regularity results in micromagnetism.
\newblock {\em Communications in Partial Differential Equations},
  25(7--8):1235--1258, 2000.

\bibitem{Harpes:UniquenessBubblingLandauLifshitz}
P. Harpes.
\newblock Uniqueness and bubbling of the 2-dimensional Landau-Lifshitz flow.
\newblock {\em Calc. Var. Partial Differential Equations}, 20(2):213--229,
  2004.

\bibitem{labbe:mumag}
S. Labb\'e.
\newblock {\em Simulation num\'erique du comportement hyperfr\'equence des
  mat\'eriaux ferromagn\'etiques}.
\newblock PhD thesis, Universit\'e Paris 13, D\'ecembre 1998.

\bibitem{Melcher:ExistencePartiallyRegular}
C. Melcher.
\newblock Existence of partially regular solutions for \mbox{Landau-Lifshit}z
  equations.
\newblock preprint, Institute for mathematics and its applications, University
  of Minnesota, November 2003.

\bibitem{Moser:PartialRegularity}
R. Moser.
\newblock Partial regularity for the Landau-Lifshitz equation in small
  dimensions.
\newblock Technical Report~26, Max-Planck-Institut fur Mathematik in den
  Naturwissenschaften, Leipzig, 2002.

\bibitem{Santugini:SolutionsLL}
K. Santugini.
\newblock Solutions to the Landau-Lifshitz system with nonhomogenous
  boundary conditions.
preprint   http://www-math.math.univ-paris13.fr/prepub/pp2004/pp2004-11.ps.gz,
  submitted, March 2004.

\bibitem{Santugini:These}
K. Santugini-Repiquet.
\newblock {\em Mat\'eriaux ferromagnetiques: influence d'un espaceur mince non
  magn\'etique, et homog\'en\'eisation d'agencements multicouches, en pr\'esence de
  couplage sur la fronti\`ere}.
\newblock Th\'ese de doctorat, Universit\'e Paris 13, Villetaneuse, dec 2004.

\bibitem{Visintin:Landau}
A. Visintin.
\newblock On Landau-Lifshitz equations for ferromagnetism.
\newblock {\em Japan J. Appl. Math.}, 2(1):69--84, 1985.

\end{thebibliography}


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