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Throughout this lesson series students will be introduced to the concept of energy usage and ways to reduce energy consumption. They will consider their own energy usage and develop a plan to reduce consumption that can be extended to a make a difference to the global setting.
Consider ways to make the wider and global community more energy efficient.
Implement a campaign to influence the energy consumption of the wider community.


text:
Q: Checking string for various (different) patterns I have a function that will do various regex-matches to find out which courier belongs to the provided tracking-id.
Right now, I'm using many elif statements which works fine but I got the feeling that this isn't very scalable in terms of maintanance.
import re

def get_courier(tracking_id):
    """ Retrieves the courier from a given tracking-id """

    if re.match(r'^ABC\d*$', f'{tracking_id}'):
        courier = "Courier 1"
    elif re.match(r'^\d*-\d$', f'{tracking_id}'):
        courier = "Courier 2"
    elif re.match(r'^PREF\d*$', f'{tracking_id}'):
        courier = "Courier 3"
    # ...
    else:
        courier = None

    return courier

Would there be a possibility to maybe "outsource" the patterns into a file or database where I can define the patterns and their related courier?
I can't get my head around a solution that could work like this
import re

def get_courier(tracking_id):
    """ Retrieves the courier from a given tracking-id """

    patterns = database_select_function()

    courier = re.match(patterns, tracking_id)

    return courier

A: A simple better solution for regex pattern matching in python is the following:
import re

courier_patterns={r'^ABC\d*$': 'Courier 1',
                  r'^\d*-\d$': 'Courier 2',
                  r'^PREF\d*$': 'Courier 3'}

def get_courier(tracking_id):
    """ Retrieves the courier from a given tracking-id """

    courier=None

    for pattern, courier_name in courier_patterns.items():
        if re.match(pattern):
            courier=courier_name
            break

    return courier

As you can see, I'm not loading courier patterns from file or DB,
but adapting this solution is simple.

redpajama_set_name: RedPajamaStackExchange


text:
December 20, 2020	| By Amitava Nag
Soumitra Chatterjee is a maverick genius who dabbled the different streams of performing and creative arts with ease and a rare poise. For six decades he remained a harbinger of hope for Bengalis all around the globe. He was a way of life, a pride in the collective racial identity. Only his mentor Satyajit Ray and the universal gurudev Rabindranath Tagore precede him in adoration and reverence.
But Soumitra Chatterjee was never a star in the glamour sky. He was a daily sustenance in the mundane. Firmly rooted, in his private spaces he was a curious mind free of inhibitions. Silhouette editor Amitava Nag had the privilege of engaging with him in numerous discussions over months and years. Not interviews in the formal sense. But exchanges – of ideas, experiences and reflections.
Blue Pencil is set to release a short and succinct account of those interactions as the book Murmurs: Moments with Soumitra Chatterjee, on 19th January 2021 to celebrate the legend's 87th birthday..
For every Sunday till then, there will be individual episodes of the book.
Standing tall, from the private collection of the author
'Siempre que te pregunto
Que cuándo cómo y dónde
Tú siempre me respondes
Quizás, quizás, quizás'
I was playing Andrea Bocelli's 'Quizas' on YouTube for him. Andrea is a favourite of mine. He seemed to like the timber of Andrea's voice as well. Soft yet vibrating, lilting without being sonorous. I told him that Andrea was blind since he was a teenager. How tragic it is.
We soon shifted to people with disabilities, how the dysfunction of one sense organ sharpens another. He describes how he approached the character of Sashi Bhushan in Goutam Ghosh's Dekha, an old man who was blind from glaucoma at an advanced age. It is one of his finest renderings on screen, subtle, complex and very complicated.
'It is not easy. If one starts losing an ability and it's a gradual decay, it is difficult to develop another one extraordinarily. I know it. My hearing problem has aggravated slowly. And at an advanced age, it didn't help me to train other senses in that way.'
'Yes, but you are just a bit short of hearing. Who knows what happens to one who turns deaf totally?' I argue.
He looks grumpy. Some mornings are worse than others the way they tear us apart. May be his sleep last night wasn't deep or sufficient. May be, he could hear the sound of his own breathing that woke him up for long. Not all hearings need a functioning ear drum.
Just as all seeing don't need eyes.
'Don't romanticize all the moments in art. That is basic in any form,' he suddenly says in a low soliloquy. Not looking up, not inviting a debate or a discussion even. We have been for some time now discussing casually, the art forms and our own experiences and enjoyments.
'I don't, life Is not a platter,' I still reply. He doesn't look up even with a provocation like that.
'You write poems. Which one to you is first – writing or being alive?' he tosses one back at me still looking down slightly.
'Being alive has always been for me,' I know this for sure, only a few that I afford to. Writing is staying alive, not the other way around, probably.
'Amitava, one has to be rude at times, to be creative. Rude with others, rude to the self,' his hands have prominent veins jutting out blue. The skin is pale, broken almost at places with dark clots – so many syringes and tubes, so many medical tests. So many histories. After the successes, there had always been storms.
I don't answer him. For there has to be silence now. Inside our heads music plays on, we don't need ears ever to tap and trap them blind within our skulls. Once we produced art, now is the time to consume, some of it.
'You won't admit you love me
And so how am I ever to know?
You always tell me
Perhaps, perhaps, perhaps
A million times I've asked you,
And then I ask you over again
You only answer
If you can't make your mind up
We'll never get started
And I don't wanna wind up
Being parted, broken-hearted
So if you really love me
Say yes, but if you don't dear, confess
And please don't tell me
Perhaps, perhaps, perhaps.'
This is a preview collection of interactions that is set to release as the book Murmurs: Moments with Soumitra Chatterjee by Amitava Nag, on 19th January 2021 to celebrate the legend's 87th birthday.
Click here to read all published episodes of Murmurs: Moments with Soumitra Chatterjee.
Catherine Berge's Gaach (The Tree, 1998) is a rare documentary on Soumitra Chatterjee. Silhouette is grateful to Catherine and producer James Ivory for providing permission to make it available to the Silhouette readers.
More to read on Soumitra Chatterjee
Soumitra Chatterjee on Acting in Satyajit Ray's Films – Exclusive Interview (Part 1)
Why Feluda Works
Whether you are new or veteran, you are important. Please contribute with your articles on cinema, we are looking forward for an association. Send your writings to amitava@silhouette-magazine.com
Amitava Nag (68 Posts)
Amitava Nag is an independent film critic based in Kolkata and editor of Silhouette. His most recent books on cinema are 16 Frames and Smriti Sattwa o Cinema. His earlier writings include the acclaimed books Satyajit Ray's Heroes and Heroines published by Rupa and Beyond Apu: 20 Favourite film roles of Soumitra Chatterjee published by Harper Collins India. He also writes poetry and short fiction in Bengali and English – observing life in a platter. He can be reached at amitavanag.net.
All Posts of Amitava Nag
Tags: Amitava Nag, Amitava Nag book, Beyond Apu: 20 Favourite film roles of Soumitra Chatterjee, Book on Soumitra Chatterjee, interview of Soumitra Chatterjee, Murmurs Moments with Soumitra Chatterjee, Soumitra Chatterjee, Soumitra Chatterjee interview, Soumitra Chatterjee's films
Silhouette Magazine publishes articles, reviews, critiques and interviews and other cinema-related works, artworks, photographs and other publishable material contributed by writers and critics as a friendly gesture. The opinions shared by the writers and critics are their personal opinion and does not reflect the opinion of Silhouette Magazine. Images on Silhouette Magazine are posted for the sole purpose of academic interest and to illuminate the text. The images and screen shots are the copyright of their original owners. Silhouette Magazine strives to provide attribution wherever possible. Images used in the posts have been procured from the contributors themselves, public forums, social networking sites, publicity releases, YouTube, Pixabay and Creative Commons. Please inform us if any of the images used here are copyrighted, we will pull those images down.

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What is the right price for a security vulnerability?
TL;DR: Vendors should focus on vulnerabilities, not on exploits. Vulnerabilities should be priced based on how difficult they are to find, not just on their exploitability.
I've been searching for an answer to this question for a while now. And this blog post is my attempt at answering it from my personal opinion.
The first answer is the economics from the security researchers perspective. Given that vendors do bug bounties as a way to interact with and give back to the security community, the rewards are mostly targeted towards compensating and showing appreciation. As a result, for these researchers, getting 5k USD for what they did over a few days as a research project or personal challenge is pretty neat.
In contrast, the "grey market" for those looking for vulnerabilities to exploit them (let's call them "exploiters"), the priorities are focused around the vulnerability reliability and stability.
As an "exploiter", you want good, simple, dumb, reliable bugs. For bug hunters, finding these *consistently* is difficult. It's not about giving yourself a challenge to find a bug in Chrome this month, but rather you seek to be able to create a pipeline of new bugs every month and if possible, even grow the pipeline over time. This is way more expensive than "bug hunting for fun".
Now, of course, there is an obvious profit opportunity here. Why not buy the bugs from those security researchers that find them in their spare time for fun, and resell them to "exploiters" for 10x the price? Well, that happens! Bug brokers do precisely that. So what happens is that then the prices from these "bug brokers" are just limited by how much the "exploiters" want to pay for them (which is a lot, more on that below).
However, and very importantly. We haven't discussed the cost of going from vulnerability to exploit. Depending on the vulnerability type, that might either be trivial (for some design/logic flaw issues) or very difficult (for some memory corruption issues).
Now, surprisingly, this key difference might give vendors a fighting chance. Software vendors in their mission to make their software better, actually don't care (or at least shouldn't care) about the difficulty to write a reliable exploit. Vendors want the vulnerability to fix it, learn from it, and find ways to prevent it from happening again.
This means that a software vendor should be able to get and find value from a vulnerability immediately, while if you wanted to make an exploit and sell it to those that want to exploit it, that would cost a significant amount of additional research and effort if there are a lot of mitigations along the way (sandboxes, isolation, etc).
So, it seems that the vendor's best chance in the "vendor" vs. "exploiter" competition is twofold: (1) to focus on making exploits harder and more expensive to write, and (2) to focus on making vulnerabilities as profitable to find and to report as possible. With the goal that eventually the cost of "weaponizing" a vulnerability is higher than the cost for finding the next bug.
The second answer to this question is the economics from the "exploiters" and the vendors perspective.
For the vendors, software engineering is so imperfect that if you have a large application, you will have a lot of bugs and you will introduce more the more you code.
So for software vendors, learning of a lot of vulnerabilities isn't as valuable as preventing those many from happening in the first place. In other words, being notified of a vulnerability is not useful except if that knowledge is used to prevent the next one from happening.
Prices then (for vendors) should be, first of all, set to match the traffic these vendors can handle not just the response but the corresponding remediation work. So if the vendor has 2 full time engineers staffed to respond to security vulnerabilities, the prices should be set to approximately 2 full time engineers time.
And then, on top of that, as many engineering resources as possible should be focused on prevention (to make vulnerabilities harder to introduce), optimizing processes (to be able to handle a larger number of reports), and finally making exploits harder to write (to make finding the next bug cheaper than writing an exploit).
For the "exploiters", if they didn't have these vulnerabilities, their alternative would be to do dangerous and expensive human intelligence operations. Bribing, spying, interrogating, sabotaging etc.. all of these expensive and limited by the amount of people you can train, the amount of assets you can turn, and the amount of money they will ask for, and then your ability to keep all of this secret. Human intelligence is really very expensive.
On the other hand, they could use these security holes - super attractive capabilities that allow them to spy on those they want to spy on, reducing (or sometimes eliminating) the need for any of the human intelligence operations and their associated risks. How can it get better than that?
So they have the budget and ability to pay large sums of money. However, the vulnerability market isn't as efficient as it should be for the larger price to matter as much.
What the market inefficiency means is that if someone can make $X00,000 a year by just finding vulnerabilities (and ignoring the exploit writing), then the risk of spending a month or two writing a reliable exploit, it's at the cost of the lost opportunity on the would have been found vulnerabilities. And vendors could be able to take advantage of this opportunity.
(1) Identifying those vulnerabilities in the critical path of an exploit.
(2) Ignore mitigations as much as possible, for the purpose of vulnerability reward decisions.
(a) Vendors have to have a proper investment in remediation, prevention and mitigation, as otherwise one doesn't get any value of buying these vulnerabilities.
(b) Our reliance on requiring full PoCs from security researchers will need to change if we want to get vulnerabilities to learn from them.
Thank you for reading, and please comment below or on Twitter if you disagree with anything or have any comments.


text:
\section{Proof of Theorem \ref{boundary_regularity}}
\subsection{Euler-Lagrange Equation.}
In this subsection, we will derive the Euler-Lagrange equation for energy minimizing maps in $\mathcal A$.

Assume that  $u\in\mathcal A$ is an energy minimizing map. For a sufficiently small $\delta>0$,
let $u(t,\cdot)\in \mathcal A$, $t\in (-\delta,\delta)$, be a family of comparison maps for $u$, i.e. $u(0,\cdot)=u(\cdot)$. For $t\in (-\delta, \delta)$,
let $u^\pm(t, x)$  denote the two one-sided trace value of $u(t,x)$ for $x\in\Gamma$. Then for $t\in (-\delta,\delta)$, we have
$$u(t, x)=g(x) \ {\rm{for}}\ x\in\Sigma;\  u(t, x)\in N^\pm \ {\rm{for}}\ x\in \Omega^\pm;\  u^\pm(t,x)\in M^\pm\ {\rm{for}}\  x\in\Gamma,$$
and
$$\Phi^+(u^+(t, x))=u^-(t, x)\ \ {\rm{for}}\ H^{n-1}\ {\rm{a.e.}}\ x\in\Gamma.$$
Set $\phi(x)=\frac{d}{dt}\big|_{t=0}u(t, x)$ for $x\in\overline\Omega$. Then we have
\begin{eqnarray*}
0&=&\frac{d}{dt}\big|_{t=0}\big(\frac12\int_{\Omega^+}|\nabla u_t|^2+\frac12\int_{\Omega^-}|\nabla u_t|^2\big)\\
&=&\int_{\Omega^+}\nabla u\cdot\nabla\phi+\int_{\Omega^-}\nabla u\cdot\nabla\phi.
\end{eqnarray*}
For the test function $\phi$, if we denote by $\phi^\pm(x)$ the two one-sided trace value of $\phi$ on $\Gamma$ from $\Omega^\pm$,
then
$$\phi(x)\in T_{u(x)}N^\pm \ {\rm{for\ a.e.}}\ x\in\Omega^\pm;\ \phi^\pm(x)\in T_{u^\pm(x)} M^\pm
\ \ {\rm{for}}\ H^{n-1} \ {\rm{a.e.}}\ x\in\Gamma, $$
and
$$D\Phi^+(u^+(x))(\phi^+(x))=\phi^-(x) \  \ {\rm{for}}\ H^{n-1} \ {\rm{a.e.}}\ x\in\Gamma.$$
Let $A^\pm$ denote the second fundamental form of $N^\pm$ in $\mathbb R^k$ and denote $u^\pm=u\big|_{\Omega^\pm}$.
Then by integration by parts $u$ satisfies
\begin{equation}\label{HM}
\begin{cases}
-\Delta u^+=A^+(u^+)(\nabla u^+, \nabla u^+)  & {\rm{in}}\ \Omega^+,\\
-\Delta u^-=A^-(u^-)(\nabla u^-, \nabla u^-)  & {\rm{in}}\ \Omega^-,\\
\ \ \ \ \ \ \ u= g  & {\rm{on}}\ \partial\Omega,\\
u^\pm(x)\in M^\pm, \Phi^+(u^+(x))=u^-(x)  & {\rm{on}}\ \Gamma,\\
\big(\frac{\partial u^+}{\partial\nu}\big)^T(x)=(D\Phi^+(u^+(x)))^t\big[\big(\frac{\partial u^-}{\partial\nu}\big)^T(x)\big]
 & {\rm{on}}\ \Gamma.
\end{cases}
\end{equation}
Here $(\cdot)^T(x): T_{u^+(x)}N^+\mapsto T_{u^+(x)}M^+$ (and $(\cdot)^T(x):T_{u^-(x)}N^-\mapsto T_{u^-(x)}M^-$)
denotes the orthogonal projection map for $x\in\Gamma$, and
$$P^t: T_{u^-(x)}M^-\mapsto T_{u^+(x)} M^+ \ ({\rm{or}}\ T_{u^+(x)}M^+\mapsto T_{u^-(x)} M^-)$$
denotes the adjoint of the linear map
$$P: T_{u^+(x)}M^+\mapsto T_{u^-(x)} M^- \ ({\rm{or }}\ T_{u^-(x)}M^-\mapsto T_{u^+(x)} M^+).$$
It is not hard to see that (2.2)$_5$ can also be written as
$$
\big(\frac{\partial u^-}{\partial\nu}\big)^T(x)=(D\Phi^{-}(u^-(x)))^t\big[\big(\frac{\partial u^+}{\partial\nu}\big)^T(x)\big]
\  {\rm{on}}\ \Gamma.
$$

\subsection{Boundary Monotonicity Inequality}

In order to establish the partial boundary regularity for energy minimizing maps in $\mathcal A$, we need a version of boundary monotonicity inequality.

For $R>0$, denote by $B_R\subset\mathbb R^n$ the ball of radius $R$ and center $0$,
$B_R^\pm=B_R\cap \mathbb R^n_\pm$.
Since $\Gamma$ is smooth,  there exists $r_0=r_0(\Gamma)>0$
such that for any $x_0\in\Gamma$, $0<r\le r_1:=\frac12\min\{r_0, {\rm{dist}}(x_0,\partial\Omega)\}$,
there exist  $C>0$ and $C^1$-diffeomorphism $\Psi: B_{r}(x_0)=B_r(x_0)\cap\Omega\mapsto B_{r}$ so that
\begin{equation}\label{almost_mono}
\Psi\big(\Omega^\pm\cap B_{r}(x_0)\big)=B_{r}^\pm,
\ \left|D\Psi(x)-\mathbb{I}_n\right|\le C|x-x_0| \ {\rm{for}}\  x\in B_{r}(x_0).
\end{equation}
Here $\mathbb{I}_n$ is the identity matrix of order $n$.
By Fubini's theorem,  $u\in H^1(\partial B_r(x_0)\cap \Omega^\pm, N^\pm)$ for almost all $r\in (0, r_1)$
so that if we define
$$
\widetilde{u}(x)=\begin{cases} u(x)  & x\in\Omega\setminus B_r(x_0),\\
u\big(\Psi^{-1}\big[r\frac{\Psi(x)}{|\Psi(x)|}\big]\big)
& x\in \Omega\cap B_r(x_0),
\end{cases}
$$
then $\widetilde u\in\mathcal A$ is a comparison map for $u$. Thus by the energy minimality,
we have
$$\int_{\Omega^+\cap B_r(x_0)}
|\nabla u|^2 +
\int_{\Omega^-\cap B_r(x_0)}
|\nabla u|^2\le \int_{\Omega^+\cap B_r(x_0)}
|\nabla \widetilde {u}|^2
+\int_{\Omega^-\cap B_r(x_0)}
|\nabla \widetilde{u}|^2.$$
Utilizing (\ref{almost_mono}) and direct calculations, we have that
\begin{eqnarray*}
&&(n-2-Cr) \big(\int_{\Omega^+\cap B_r(x_0)}|\nabla u|^2
+\int_{\Omega^-\cap B_r(x_0)}|\nabla u|^2\big)\\
&\le& r\big(\int_{\Omega^+\cap \partial B_r(x_0)}|\nabla u|^2
+\int_{\Omega^-\cap \partial B_r(x_0)}|\nabla u|^2\big)\\
&-&r\big(\int_{\Omega^+\cap \partial B_r(x_0)}\big|\frac{\partial u}{\partial |x-x_0|}\big|^2
+\int_{\Omega^-\cap \partial B_r(x_0)}\big|\frac{\partial u}{\partial |x-x_0|}\big|^2\big).
\end{eqnarray*}
Therefore,  for any $x_0\in\Gamma$ and $r\in (0, r_1)$, we have that
\begin{eqnarray}\label{almost_mono1}
&&\frac{d}{dr}\big[e^{Cr} r^{2-n}(\int_{\Omega^+\cap B_r(x_0)}|\nabla u|^2
+\int_{\Omega^-\cap B_r(x_0)}|\nabla u|^2)\big]\nonumber\\
&&\ge r^{2-n}\big[\int_{\Omega^+\cap \partial B_r(x_0)}\big|\frac{\partial u}{\partial |x-x_0|}\big|^2
+\int_{\Omega^-\cap \partial B_r(x_0)}\big|\frac{\partial u}{\partial |x-x_0|}\big|^2\big]
\end{eqnarray}
holds, provided $u\in\mathcal{A}$ is an energy minimizing map. In particular, by integrating \eqref{almost_mono1} with respect to $r$, we obtain that
for any $x_0\in\Gamma$ and $0<R_1\le R_2<r_1$,
\begin{eqnarray}\label{almost_mono2}
\displaystyle
&e^{CR_1} R_1^{2-n}\displaystyle\big(\int_{\Omega^+\cap B_{R_1}(x_0)}|\nabla u|^2+\int_{\Omega^-\cap B_{R_1}(x_0)}|\nabla u|^2\big)\nonumber\\
&\ \ +\displaystyle\int_{\Omega\cap (B_{R_2}(x_0)\setminus B_{R_1}(x_0))} |x-x_0|^{2-n}\big |\frac{\partial u}{\partial |x-x_0|}\big |^2\nonumber\\
&\ \le e^{CR_2} R_2^{2-n}\displaystyle\big(\int_{\Omega^+\cap B_{R_2}(x_0)}|\nabla u|^2+\int_{\Omega^-\cap B_{R_2}(x_0)}|\nabla u|^2\big)
\end{eqnarray}
holds for any energy minimizing map $u\in\mathcal{A}$.

\subsection{Boundary Extension Lemma}

A crucial ingredient to prove Theorem \ref{boundary_regularity} is the following boundary extension lemma,
similar to \cite{HL3} Lemma 3.1.

\begin{lemma}\label{boundary_extension}
There exist positive constants $\delta$, $q$, and $C$ such that, if
$0<\epsilon<1$, $x_0\in\Gamma$, and
$0<r_0<{\rm{dist}}(x_0,\partial\Omega)$,
if $\eta^\pm\in H^1(\partial B_{r_0}(x_0)\cap \Omega^\pm, N^\pm)$
satisfies
\begin{eqnarray}\label{small_osc}
&&\int_{\partial B_{r_0}(x_0)\cap\Omega^\pm}
|\nabla_{\rm{tan}}\eta^\pm|^2\,dH^{n-1}
\big[\int_{\partial B_{r_0}(x_0)\cap \Omega^\pm}
|\eta^\pm-p^\pm|^2\,dH^{n-1}
+\int_{\partial B_{r_0}(x_0)\cap\Gamma} |\eta^\pm-p^\pm|^2\,dH^{n-2}\big]\nonumber\\
&&\le\delta^2\epsilon^q
\end{eqnarray}
for some $p^\pm\in\mathbb R^k$, and if $\eta^\pm: {\partial B_{r_0}(x_0)\cap\Gamma} \mapsto M^\pm$  satisfies $$\eta^-(x)=\Phi^+(\eta^+(x))
\ {\rm{for}}\ H^{n-2} \ {\rm{a.e.}}\ x\in \partial B_{r_0}(x_0)\cap\Gamma,$$
then
there exist maps $\omega^\pm\in H^1(B_{r_0}(x_0)\cap\Omega^\pm, N^\pm)$ such that
$\omega^\pm=\eta^\pm$ on $\partial B_{r_0}(x_0)\cap\Omega^\pm$,
and $\omega^\pm: {B_{r_0}(x_0)\cap\Gamma}\mapsto M^\pm$
satisfies
$$\omega^-(x)=\Phi^+(\omega^+(x)) \  {\rm{for}}\  H^{n-1} \ {\rm{a.e.}}\ x\in B_{r_0}(x_0)\cap\Gamma.$$
Furthermore, it holds that
\begin{eqnarray}\label{bdry_extension_estimate}
&&\int_{B_{r_0}(x_0)\cap\Omega^\pm}|\nabla\omega^\pm|^2\,dx
\le \epsilon \int_{\partial B_{r_0}(x_0)\cap\Omega^\pm}
|\nabla_{\rm{tan}}\eta^\pm|^2\,dH^{n-1}\nonumber\\
&&+C\epsilon^{-q}\big[\int_{\partial B_{r_0}(x_0)\cap \Omega^\pm}
|\eta^\pm-p^\pm|^2\,dH^{n-1}
+\int_{\partial B_{r_0}(x_0)\cap\Gamma} |\eta^\pm-p^\pm|^2\,dH^{n-2}\big].
\end{eqnarray}
Here $\nabla_{\rm{tan}}$ denotes the tangential gradient on $\partial B_{r_0}(x_0)$.
\end{lemma}
\noindent{\bf Proof.\quad}}\def\endpf{\hfill$\Box$ The proof can be done by suitable modifications of the arguments from \cite{HL2,HL3} and \cite{SU1}. It is based on an induction of the dimension $n$.
There are two crucial ingredients of the construction:
\begin{itemize}
\item[(i)]construction in dimension $n=2$; and
\item[(ii)] homogeneous of degree zero extension for $n\ge 3$.
\end{itemize}
For simplicity, we will only indicate how to implement these two ingredients in our situation.
The interested readers can consult with \cite{HL2, HL3, SU1}
for more details.\\
{\it Case} 1: $n=2$ (linear interpolation). Since the problem is invariant under bi-Lipschitz transformations, we may assume that
$x_0=0$, $r_0=1$, $\Omega=B_1$, and $\Gamma=\Gamma_1 (=B_1^2\cap\{x_2=0\})$.  Denote by $S_1^\pm\subset \partial B_1^2$
the half unit circles. Choose $\theta_0^\pm\in S_1^\pm$ so that
$$|\eta^\pm(\theta_0^\pm)-p^\pm|=\inf\Big\{|\eta^\pm(\theta)-p^\pm|: \ \theta\in S_1^\pm\Big\}.$$
Then it is easy to see that
$$\begin{cases}\displaystyle
|\eta^\pm(\theta_0^\pm)-p^\pm|^2\le c\int_{S^\pm_1}|\eta^\pm-p^\pm|^2,\\
\displaystyle\int_{S^\pm_1}|\eta^\pm-\eta^\pm(\theta_0)|^2\le c \int_{S^\pm_1}|\eta^\pm-p^\pm|^2.
\end{cases}
$$
By Sobolev's embedding inequality
$H^1(S_1^\pm)\subset C^\frac12(S_1^\pm)$, we have that
\begin{eqnarray*}
\max_{\theta\in S_1^\pm}\big\{|\eta^\pm(\theta)-\eta^\pm(\theta_0)|^2\big\}
&\le& c\big(\int_{S_1^\pm}|\nabla_{\rm{tan}}\eta^\pm|^2\big)^\frac12
\big(\int_{S_1^\pm}|\eta^\pm-\eta(\theta_0)|^2\big)^\frac12\\
&\le& c\delta\epsilon^{\frac{q}2}.
\end{eqnarray*}
Set
$$w^+(t,0)=\frac{(1-t)}2 \eta^+(-1,0)+\frac{(1+t)}2 \eta^+(1,0), \ -1\le t\le 1.$$
Then we have
$$\max_{-1\le t\le 1}{\rm{dist}}(w^+(t,0), M^+)\le c|\eta^+(1,0)-\eta^+(-1,0)|\le c\delta\epsilon^{\frac{q}2}.$$
Recall that there exists $\delta_0=\delta_0(M^\pm)>0$ such that for any $0<\delta<\delta_0$, the nearest point
projection maps $\Pi_{M^\pm}: (M^\pm)_\delta\mapsto M^\pm$ and $\Pi_{N^\pm}: (N^\pm)_\delta\mapsto N^\pm$
are smooth,  where $(M^\pm)_\delta$ (or $(N^\pm)_\delta$ respectively)
denotes the $\delta$-neighborhood of $M^\pm$  (or $N^\pm$ respectively) in $\mathbb R^k$.
Let $v^+: B_1^+\mapsto\mathbb R^k$ solve
\begin{eqnarray*}
\begin{cases}
\Delta v^+= 0 , &  \ {\rm{in}}\ B_1^+,\\
\ \ v^+= \eta^+, & \ {\rm{on}}\  S_1^+,\\
\ \ v^+=\Pi_{M^+}(w^+), &  \ {\rm{on}}\  \Gamma_1.
\end{cases}
\end{eqnarray*}
Since $\displaystyle\max\big\{{\rm{osc}}_{S_1^+}\eta^+, {\rm{osc}}_{\Gamma_1}\Pi_{M^+}(w^+)\big\}\le C\delta\epsilon^{\frac{q}2}$,
it follows from the maximum principle  that
$$\max_{x\in B_1^+}{\rm{dist}}(v^+(x), N^+)\le c\delta\epsilon^{\frac{q}2}.$$
Thus we can define
$$\omega^+(x)=\Pi_{N^+}(v^+(x)), \ x\in \overline{B_1^+}.$$
To construct $\omega^-$, first let
$$w^-(t,0)=\Phi^+(\Pi_{M^+}(w^+(t,0))), \ -1\le t\le 1,$$
so that $w^-(\Gamma_1)\subset M^-$. Let $v^-: B_1^-\mapsto\mathbb R^k$ solve
\begin{eqnarray*}
\begin{cases}
\Delta v^-= 0 , &  \ {\rm{in}}\ B_1^-,\\
\ \ v^-= \eta^-, & \ {\rm{on}}\  S_1^-,\\
\ \ v^-=w^-, &  \ {\rm{on}}\  \Gamma_1.
\end{cases}
\end{eqnarray*}
Then we also have
$$\max_{x\in B_1^-}{\rm{dist}}(v^-(x), N^-)\le c\delta\epsilon^{\frac{q}2},$$
so that we can define
$$\omega^-(x)=\Pi_{N^-}(v^-(x)), \ x\in \overline{B_1^-}.$$
It follows directly from the above construction that
$\omega^-(x)=\Phi^+(\omega^+(x))$ for $x\in\Gamma_1$, and (\ref{bdry_extension_estimate})
follows from the standard estimate on harmonic functions.

\noindent{\it Case} 2: $n\ge 3$ (homogeneous of degree zero extension). For
$0<\delta<1$, let $B^{\pm,n-1}_{\delta}$ be  $(n-1)$-dimensional half balls of radius
$\delta>0$, and $C^{\pm, n}_\delta=B^{\pm, n-1}_\delta\times [-\delta,\delta]$ be the
$n$-dimensional half cylinders of size $\delta$. Let $S_\delta^{\pm, n-2}$ be the $(n-2)$-dimensional half spheres of radius $\delta$ so that
$\partial B_{\delta}^{\pm, n-1}=S_\delta^{\pm,n-2}\cup B_\delta^{n-2}$.
\begin{lemma} \label{homogeneous} For
$u^\pm\in H^1\big((B_\delta^{\pm, n-1}\times\{\pm\delta\})
\cup (S^{\pm, n-2}_\delta\times [-\delta, \delta]), N^\pm\big)$, if
 $u_1^\pm(x)=u^\pm(x,-\delta)$ and $u_2^\pm(x)=u^\pm(x,\delta)$, $x\in B^{\pm, n-1}_\delta$,
 satisfies $u_1^\pm, u_2^\pm \in H^1(B_\delta^{\pm, n-1}, N^\pm)$, if $u^\pm(x,t)=u^\pm_0(x)$ for $
(x,t)\in S^{\pm, n-2}_\delta\times [-\delta,\delta]$,
with $u^\pm_0\in H^1(S^{\pm, n-2}_\delta, N^\pm)$, and  if
\begin{equation}\label{bdry_trace}
u^\pm(x,t)\in M^\pm \ {\rm{satisfies}}\ u^-(x,t)=\Phi^+(u^-(x,t)) \ \ {\rm{for}}\ H^{n-2} \ {\rm{a.e.}}\
x\in B_\delta^{n-2}, \ {\rm{and}}\  t=-\delta, \delta.
\end{equation}
Then there exist extension maps $\overline{u^\pm}\in H^1(C_\delta^{\pm, n}, N^\pm)$ such that
$$\overline{u^\pm}=u^\pm
\ {\rm{ on }}\  (B_\delta^{\pm, n-1}\times\{\pm\delta\})
\cup (S^{\pm, n-2}_\delta\times [-\delta, \delta]),$$
\begin{equation}\label{bdry_trace1}
\overline{u^\pm}(x,t)\in M^\pm,
\ \overline{u^-}(x,t)=\Phi^+\big(\overline{u^+}(x,t)\big) \ \ {\rm{for}}\ H^{n-1} \ {\rm{a.e.}}\
(x,t)\in B_\delta^{n-2}\times [-\delta, \delta],
\end{equation}
and
\begin{eqnarray}\label{extension1}
E\big(\overline{u^\pm}; C_\delta^{\pm, n}\big)
&\le& c\delta\big[E_\delta(u^\pm_1)+E_\delta(u^\pm_2)+\delta E(u^\pm_0)\big],\\
W\big(\overline{u^\pm}; C_\delta^{\pm, n}\big)
&\le& c\delta\big[W_\delta(u^\pm_1)+W_\delta(u^\pm_2)+\delta W(u^\pm_0)\big].
\label{extension2}
\end{eqnarray}
Here $$E_\delta(u_i^\pm)=\int_{B_\delta^{\pm, n-1}}|\nabla u_i^\pm|^2\,dH^{n-1} (i=1,2);
\ E(u_0^\pm)=\int_{S_\delta^{\pm, n-2}}|\nabla_{\rm{tan}}u_0^\pm|^2\,dH^{n-2},$$
and
$$W_\delta(u_i^\pm)=\int_{B_\delta^{\pm, n-1}}|u_i^\pm-p^*|^2\,dH^{n-1} (i=1,2);
\ W(u_0^\pm)=\int_{S_\delta^{\pm, n-2}}|u_0^\pm-p^*|^2\,dH^{n-2}$$
for some fixed $p^*\in \mathbb R^L$.
\end{lemma}
{\bf Proof of Lemma 2.3}. By scaling, we may assume $\delta=1$.  There exists
a bi-Lipschitz homeomorphism $f^\pm: \partial B_1^{\pm, n}\mapsto \partial C_1^{\pm, n}$
such that $\overline{f^\pm}(x)=|x|f^\pm(\frac{x}{|x|}): B_1^{\pm,n}\mapsto C_1^{\pm,n}$ is also
a bi-Lipschitz homeomorphism. Let $\Pi(x)=\frac{x}{|x|}: B_1^n\setminus\{0\}\mapsto \partial B_1^n$ be the radial projection map.
Define the projection map
$\Pi^\pm: C_1^{\pm, n}\setminus\{0\}\mapsto \partial C_1^{\pm,n}$ by
$\Pi^\pm=f^\pm\circ\Pi\circ \big(\overline{f^\pm}\big)^{-1}$. Then define
$$\overline{u^\pm}(x)=u^\pm\circ \Pi^\pm(x), \ \ x\in C_1^{\pm,n}.$$
It is easy to see that (\ref{bdry_trace}) implies that $\overline{u^\pm}$ satisfies
the trace condition (\ref{bdry_trace1}) on $\Gamma_1$. It is also easy to see that
$$ E\big(\overline{u^\pm}; C_1^{\pm, n}\big)\le K E\big(\overline{u^\pm}\circ \overline{f^\pm}; B_1^{\pm, n}\big)
\le \frac{K}{n-2} E\big(u^\pm\circ f^\pm; \partial B_1^{\pm, n}\big)
\le C(K) E(u^\pm; \partial C_1^{\pm, n}),
$$
where $K$ is a constant depending on the Lipschitz constants of $f^\pm$ and
$(\overline{f^\pm})^{-1}$. This implies (\ref{extension1}). Similar argument for $W$
also yields (\ref{extension2}).  \qed

\begin{corollary}
There is a constant $c>0$ such that under the same assumptions of Lemma \ref{boundary_extension},
if $u\in H^1(\Omega^\pm, N^\pm)\cap\mathcal A$ is energy minimizing among all maps
in $\mathcal A$, and for any $x_0\in\Gamma$ and $0<r_0<{\rm{dist}}(x_0,\partial\Omega)$,
$$r_0^{2-n}\big(\int_{\Omega^+\cap B_{r_0}(x_0)}|\nabla u^+|^2+\int_{\Omega^-\cap B_{r_0}(x_0)}|\nabla u^-|^2\big)
\le c^{-1}\lambda^{-\frac{q}2},$$
then
\begin{eqnarray}\label{bdry_extension_estimate1}
&&(\frac{r_0}2)^{2-n}\big(\int_{\Omega^+\cap B_{\frac{r_0}2}(x_0)}|\nabla u^+|^2
+\int_{\Omega^-\cap B_{\frac{r_0}2}(x_0)}|\nabla u^-|^2\big)\nonumber\\
&&\le \lambda r_0^{2-n}\big(\int_{\Omega^+\cap B_{r_0}(x_0)} |\nabla u^+|^2
+\int_{\Omega^-\cap B_{r_0}(x_0)} |\nabla u^-|^2\big)\nonumber\\
&&+ c\lambda^{-q}r_0^{-n}\big[\int_{\Omega^+\cap B_{r_0}(x_0)}|u^+ - \widehat{u^+}|^2
+\int_{\Omega^-\cap B_{r_0}(x_0)}|u^- - \widehat{u^-}|^2\big]\nonumber\\
&&+ c\lambda^{-q}r_0^{2-n}\big[\int_{\partial B_{r_0}(x_0)\cap\Gamma}|u^+ - \widehat{u^+}|^2\,dH^{n-2}
+\int_{\partial B_{r_0}(x_0)\cap\Gamma}|u^- - \widehat{u^-}|^2\,dH^{n-2}\big],
\end{eqnarray}
where $u^\pm=u|_{\Omega^\pm}$ denotes the restriction of $u$ on $\Omega^\pm$, and
$\displaystyle\widehat{u^\pm}=\frac{1}{|B_{r_0}(x_0)\cap\Gamma|}\int_{B_{r_0}(x_0)\cap\Gamma} u^\pm\,dH^{n-1}$
is the average of the one-side trace of $u^\pm$ in $B_{r_0}(x_0)\cap\Gamma$.
\end{corollary}
\noindent{\bf Proof.\quad}}\def\endpf{\hfill$\Box$  For simplicity, we assume $r_0=1$. Since $u^\pm:\Omega^\pm\to N^\pm$ and $N^\pm$ is compact, it follows
$$\big|\int_{B_1(x_0)\cap\Gamma} u^\pm\,dH^{n-1}\big|\le C.$$
From the Poincar\'e inequality, we have that
$$\int_{\Omega^\pm\cap B_1(x_0)}|u^\pm-\widehat{u^\pm}|^2\le c\int_{\Omega^\pm\cap B_1(x_0)}|\nabla u^\pm|^2.$$
From the trace estimate and the Poincar\'e inequality, we also have that
$$
\int_{B_1(x_0)\cap\Gamma}|u^\pm-\widehat{u^\pm}|^2\,dH^{n-1}
\le c\big\|u^\pm-\widehat{u^\pm}\big\|_{H^\frac12(B_1(x_0)\cap \Gamma)}^2
\le c\int_{\Omega^\pm\cap B_1(x_0)}|\nabla u^\pm|^2.$$
Applying Fubini's theorem, we can choose $r\in [\frac12,1]$ such that
$$
\int_{\Omega^\pm\cap \partial B_r(x_0)}|\nabla u^\pm|^2\,dH^{n-1}
\le c\int_{\Omega^\pm\cap B_1(x_0)}|\nabla u^\pm|^2
$$
and
\begin{eqnarray*}
&&\big[\int_{\Omega^\pm\cap \partial B_{r}(x_0)}|u^\pm - \widehat{u^\pm}|^2\,dH^{n-1}
+\int_{\partial B_{r}(x_0)\cap\Gamma}|u^\pm - \widehat{u^\pm}|^2\,dH^{n-2}\big]\nonumber\\
&&\le c
\big[\int_{\Omega^\pm\cap B_{1}(x_0)}|u^\pm - \widehat{u^\pm}|^2
+\int_{\Gamma\cap B_{1}(x_0)}|u^\pm - \widehat{u^\pm}|^2\big]\\
&&\le c\int_{\Omega^\pm\cap B_1(x_0)}|\nabla u^\pm|^2.
\end{eqnarray*}
By choosing a sufficiently small $c>0$, we can apply Lemma \ref{boundary_extension}
with $\eta^\pm=u^\pm\big|{\partial B_r(x_0)\cap\Omega^\pm}$ and $p^\pm=\widehat{u^\pm}$
to obtain an extension map $\omega^\pm\in H^1(B_{r}(x_0)\cap\Omega^\pm, N^\pm)$ such that
$\omega^\pm=u^\pm$ on $\partial B_{r}(x_0)\cap\Omega^\pm$,
$\omega^\pm\big|_{B_{r}(x_0)\cap\Gamma}$ has image in $M^\pm$ that
satisfies
$$\omega^-(x)=\Phi^+(\omega^+(x))
\ {\rm{ for}}\  H^{n-1} \ {\rm{a.e.}}\ x\in B_{r}(x_0)\cap\Gamma,$$
and the estimate (\ref{bdry_extension_estimate}).  If we define $\widetilde u:\Omega\to N$ by
$$\widetilde{u}(x)=
\begin{cases} \omega^\pm(x) & x\in B_r(x_0)\cap\Omega^\pm\\
u(x) & x\in \Omega\setminus B_r(x_0).
\end{cases}
$$
Then $\widetilde u\in\mathcal A$ is a comparison map of $u$. Hence
the energy minimality of $u$ implies that
$$
\int_{\Omega^+\cap B_{r}(x_0)}|\nabla u^+|^2
+\int_{\Omega^-\cap B_{r}(x_0)}|\nabla u^-|^2
\le \int_{\Omega^+\cap B_{r}(x_0)}|\nabla \omega^+|^2
+\int_{\Omega^-\cap B_{r}(x_0)}|\nabla \omega^-|^2,
$$
which, combined with (\ref{bdry_extension_estimate}),  then implies (\ref{bdry_extension_estimate1}). This completes the proof.
\qed

\subsection{Small Energy Regularity}

Another crucial step to prove Theorem \ref{boundary_regularity} is the following energy improvement
property.
\begin{lemma}\label{energy_improve}
There exist positive constants $\epsilon, C$, and $\theta<1$ such that if $u\in\mathcal A$
is an energy minimizing map that satisfies, for $x_0\in\Gamma$ and some $0<r_0<{\rm{dist}}(x_0,\partial\Omega)$,
\begin{equation}\label{bdry_small_cond}
r_0^{2-n}\big(\int_{\Omega^+\cap B_{r_0}(x_0)} |\nabla u|^2
+\int_{\Omega^-\cap B_{r_0}(x_0)} |\nabla u|^2\big)\leq\epsilon^2,
\end{equation}
then
\begin{eqnarray}\label{bdry_energy_improve}
&&(\theta r_0)^{2-n}\big(\int_{\Omega^+\cap B_{\theta r_0}(x_0)} |\nabla u|^2
+\int_{\Omega^-\cap B_{\theta r_0}(x_0)} |\nabla u|^2\big)\nonumber\\
&\le&\frac12\max\big\{ r_0^{2-n}\big(\int_{\Omega^+\cap B_{r_0}(x_0)} |\nabla u|^2
+\int_{\Omega^-\cap B_{r_0}(x_0)} |\nabla u|^2\big),
\ C{\rm{Lip}}(\Gamma)\big\}.
\end{eqnarray}
\end{lemma}

\medskip
The proof of Lemma \ref{energy_improve} is based on a blowing up argument, similar to \cite{HL3} Theorem 3.3. Before presenting it,
we need the following regularity estimate on the linear equation, resulting from the blow-up process of the nonlinear harmonic map equation (2.2).

Denote by $B^+_1$ and $B^-_1$ the upper half and lower half unit ball, and
set $\Gamma_1=B_1\cap \{x_n=0\}$. For $a^+\in M^+$, let $a^-=\Phi^+(a^+)\in M^-$.
Let ${\rm{Tan}}({a^\pm}, M^\pm)$ denote the tangent space of $M^\pm$ at $a^\pm$, and
${\rm{Nor}}({a^\pm}, M^\pm)$ denote the normal space of $M^\pm\subset N^\pm$ at $a^\pm$, i.e.,
$${\rm{Tan}}({a^\pm}, M^\pm) \oplus {\rm{Nor}}({a^\pm}, M^\pm)
={\rm{Tan}}({a^\pm}, N^\pm).$$
For any vector $v_\pm\in {\rm{Tan}}({a^\pm}, N^\pm)$, we decompose it as
$$v_\pm =v_\pm^t+v_\pm^n,$$
where $v_\pm^t$ denotes the orthogonal projection of $v_\pm$ into ${\rm{Tan}}({a^\pm}, M^\pm)$, and $v_\pm^n$ denotes the orthogonal projection of $v_\pm$ into ${\rm{Nor}}({a^\pm}, M^\pm)$.

\begin{lemma} \label{linear_regularity} Suppose that $v_\pm\in H^1(B_1^\pm, {\rm{Tan}}(a^\pm, N^\pm))$ are two harmonic functions, with traces
$v_\pm\big|_{\Gamma_1}\in H^\frac12(\Gamma_1,
{\rm{Tan}}(a^\pm, M^\pm))$, satisfying
\begin{equation}\label{trace_match}
\begin{cases} v_-=D\Phi^+(a^+)(v_+) &\ {\rm{on}}\ \Gamma_1,\\
\big(\frac{\partial v_+}{\partial x_n}\big)^T= (D\Phi^+(a^+))^t\big(\frac{\partial v_-}{\partial x_n}\big)^T
&\ {\rm{on}}\ \Gamma_1.
\end{cases}
\end{equation}
Then $v_\pm\in C^\infty\big(B_\frac12^\pm\cup \Gamma_\frac12\big)$, and
for any $l\ge 1$, it holds
\begin{equation}\label{trace_estimate1}
\big\|v_\pm\big\|_{C^l\big(B_\frac12^\pm\cup\Gamma_\frac12\big)}
\le C\big(l, \big\|\Phi^+\big\|_{C^1(M^+)}, \big\|v_\pm\big\|_{H^1(B_1^\pm)}\big).
\end{equation}
\end{lemma}
\noindent{\bf Proof.\quad}}\def\endpf{\hfill$\Box$ Since $a^\pm \in M^\pm$, we can decompose
$v_\pm=v_\pm^t+v_\pm^n$ so that
\begin{equation} \label{harm_1}
\Delta v_\pm^t=0 \ \ \ {\rm{in}}\ B_1^\pm,
\end{equation}
and
\begin{equation}\label{harm_2}
\Delta v_\pm^n=0 \ \ \ {\rm{in}}\ B_1^\pm.
\end{equation}
Since $v_\pm(x)\in {\rm{Tan}}(a^\pm, M^\pm)$
for $H^{n-1}$ a.e. $x\in\Gamma_1$, we have that
\begin{equation}\label{trace_cond}
v_\pm^n =0 \ {\rm{on}} \ \Gamma_1.
\end{equation}
It is readily seen that by (\ref{harm_2}) and (\ref{trace_cond}),
$v_\pm^n \in C^\infty\big(B_\frac12^\pm\cup\Gamma_\frac12\big)$, and
for any $l\ge 1$
\begin{equation}\label{trace_estimate2}
\big\|v_\pm^n\big\|_{C^l\big(B_\frac12^\pm\cup\Gamma_\frac12\big)}
\le C\big(l, \big\|v_\pm^n\big\|_{H^1(B_1^\pm)}\big).
\end{equation}
To show regularity of $v_\pm^t$, we denote
$P=D\Phi^+(a^+)$ and proceed as follows.
Define $\widetilde{v_{-}}:B_1^{+}
\mapsto T_{a^+}N^+$ be an even extension $v_{-}$, i.e.,
$$\widetilde{v_{-}}(x',x_n)=v_{-}(x', -x_n),  \ \ (x', x_n)\in B_1^+.$$
Then it is easy to see that
\begin{equation}\label{odd_harm}
\begin{cases}
\Delta \big(\widetilde{v_{-}}^t-P(v_{+}^t)\big)=0 & \ {\rm{in}}\ B_1^+,\\
\widetilde{v_{-}}^t-P(v_{+}^t)=0 & \ {\rm{on}}\ \Gamma_1,
\end{cases}
\end{equation}
and
\begin{equation}\label{even_harm}
\begin{cases}
\Delta \big(v_{+}^t-P^t(\widetilde{v_{-}}^t)\big)=0 & \ {\rm{in}}\ B_1^+,\\
\frac{\partial}{\partial x_n}\big(v_{+}^t-P^t(\widetilde{v_{-}}^t)\big)=0 & \ {\rm{on}}\ \Gamma_1.
\end{cases}
\end{equation}
From the standard theory of harmonic functions,
we see that (\ref{odd_harm}) and (\ref{even_harm}) imply
$$\big(\widetilde{v_{-}}^t-P(v_{+}^t)\big),\  \big(v_{+}^t+P^t(\widetilde{v_{-}}^t)\big)
\in C^\infty(B_\frac12^-\cup\Gamma_\frac12),$$
and it holds that, for any $l\ge 1$,
\begin{equation}\label{trace_estimate3}
\big\|\widetilde{v_{-}}^t-P(v_{+}^t)\big\|_{C^l\big(B_\frac12^-\cup\Gamma_\frac12\big)}
+\big\|v_{+}^t+P^t(\widetilde{v_{-}}^t)\big\|_{C^l\big(B_\frac12^-\cup\Gamma_\frac12\big)}
\le C\big(l, \big\|v_{\pm}\big\|_{H^1(B_1^\pm)}\big).
\end{equation}

If $PP^t=I_k$, i.e. $P\in O(k)$ is an orthogonal matrix, then we have
$$|\widetilde{v_{-}}^t-P(v_{+}^t)|=|P(P^t(\widetilde{v_{-}}^t)-v_{+}^t)|=|P^t(\widetilde{v_{-}}^t)-v_{+}^t|.$$
This and \eqref{trace_estimate3} easily yield \eqref{trace_estimate1}.

If $PP^t\not=I_k$, then $P^{-1}\not=P^t$ and we can also see easily that \eqref{trace_estimate1}
follows from \eqref{trace_estimate3}. This completes the proof. \qed

\bigskip
\noindent{\bf Proof of Lemma 2.5}.  The proof follows from a blow-up argument, Lemma \ref{linear_regularity},
and the boundary extension Lemma 2.2. Here we only sketch the argument.

For simplicity, assume that $x_0=0$, $r_0=1$, $\Omega=B_1$, and $\Gamma=\Gamma_1$ so that ${\rm{Lip}}(\Gamma)=0$. Suppose that the conclusion were false.
Then for any $\theta\in (0,1)$, there would exist $\epsilon_i\rightarrow 0$ and a sequence of minimizing harmonic
maps $u_i\in \mathcal A$ that satisfy
\begin{equation}\label{small_energy_seq}
\int_{B_1^+}|\nabla u_i^+|^2+\int_{B_1^-}|\nabla u_i^-|^2=\epsilon_i^2,
\end{equation}
and
\begin{equation}\label{no_decay}
\theta^{2-n}\big(\int_{B_\theta^+}|\nabla u_i^+|^2
+\int_{B_\theta^-}|\nabla u_i^-|^2\big)>\frac12 \epsilon_i^2.
\end{equation}
Let $\displaystyle\overline{u_i^\pm}=\frac{1}{|\Gamma_1|}\int_{\Gamma_1} u_i^\pm$ denote the average of
the two one-sided traces of $u_i$ on $\Gamma_1$. By the Poincar\'e inequality on $\Gamma_1$
and $H^1$ trace theory, we have
\begin{eqnarray*}
{\rm{dist}}\big(\overline{u_i^+}, M^+\big)^2
\le \frac{1}{|\Gamma_1|}\int_{\Gamma_1}\big|u_i^+-\overline{u_i^+}\big|^2\,dH^{n-1}
\le c\big\|\nabla u_i^+\big\|_{L^2(B_1^+)}^2\le c\epsilon_i^2.
\end{eqnarray*}
Therefore for $i$ sufficiently large there is a unique nearest point $a_i^+=\Pi_{M^+}\big(\overline{u_i^+}\big) \in M^+$ such that
$$\big|\overline{u_i^+}-a_i^+\big|={\rm{dist}}\big(\overline{u_i^+}, M^+\big).$$
Since $u_i^-=\Phi^+(u_i^+)$ on $\Gamma_1$, it is readily seen that
$a_i^-\equiv \Phi^+(a_i^+)\in M^-$ satisfies
\begin{eqnarray*}\big|\overline{u_i^-}-a_i^-\big|^2
&=&\big|\frac{1}{\Gamma_1}\int_{\Gamma_1}\Phi^+(u_i^+)-\Phi^+(a_i^+)\big|^2
\le c{\rm{Lip}}^2(\Phi^+)\int_{\Gamma_1}|u_i^+-a_i^+|^2\\
&\le& c\int_{\Gamma_1}\big|u_i^+-\overline{u_i^+}\big|^2+\big|\overline{u_i^+}-a_i^+\big|^2\le c\epsilon_i^2.
\end{eqnarray*}
Now we define the corresponding blow-up sequence
$v_i:B_1\to\mathbb R^k$ by letting
$$
v_i(x):=\begin{cases}v_i^+(x)=\frac{u_i^+(x)-a_i^+}{\epsilon_i}, & x\in B_1^+,\\
v_i^-(x)=\frac{u_i^-(x)-a_i^-}{\epsilon_i}, & x\in B_1^-.
\end{cases}
$$
It is easy to see that
\begin{equation}\label{H1_bound_vi}
\int_{B_1^+}|\nabla v_i^+|^2+\int_{B_1^-}|\nabla v_i^-|^2=1,
\end{equation}
and
\begin{equation}\label{no_H1_decay}
\theta^{2-n}\big(\int_{B_\theta^+}|\nabla v_i^+|^2+\int_{B_\theta^-}|\nabla v_i^-|^2\big)>\frac12.
\end{equation}
By (\ref{H1_bound_vi}) and the $H^1$-trace theory, we have
$$\displaystyle \|v_i^\pm\|_{H^1(B_1^\pm)}\le c.$$
Hence, after taking a subsequence, there exists $v:B_1\to\mathbb R^k$,
with $v_\pm (=v|_{B_1^\pm}) \in H^1(B_1^\pm,\mathbb R^k)$, such that $v_i^\pm$ converge
to $v_\pm$ weakly in $H^1(B_1^\pm,\mathbb R^k)$. In particular, by (\ref{H1_bound_vi}), we have
\begin{equation}\label{H1_bound_v}
\int_{B_1^+}|\nabla v_+|^2+\int_{B_1^-}|\nabla v_-|^2\le 1.
\end{equation}
Again passing to a subsequence, we assume that
$$\lim_{i\rightarrow\infty} a_i^+=a^+\in M^+
\ {\rm{and}}\ \lim_{i\rightarrow\infty} a_i^-=a^-=\Phi^+(a^+)\in M^-.
$$
It is not hard to verify that $v_{+}(x)\in T_{a^+} N^+$ for a.e. $x\in B_1^+$, and
$v_{-}(x)\in T_{a^-} N^-$ for a.e. $x\in B_1^{-}$.
Since $u_i^\pm(x)\in M^\pm$ for $H^{n-1}$ a.e. $x\in\Gamma_1$, it is also not hard to see that
\begin{equation}\label{trace_cond_v}
v_\pm(x)\in {\rm{Tan}}(a^\pm, M^\pm)
\ {\rm{and}} \ \ v_-(x)=D\Phi^+(a^+)(v_+(x))
\  H^{n-1}\ {\rm{a.e.}} \ x\in\Gamma_1.
\end{equation}
Since $v_i^\pm$ satisfies
$$-\Delta v_i^\pm =\epsilon_iA^\pm(u_i^\pm)(\nabla v_i^\pm, \nabla v_i^\pm)
\ \ {\rm{in}} \ \ B_1^\pm,$$
and
$$\int_{B_1^\pm} |A^\pm(u_i^\pm)(\nabla v_i^\pm, \nabla v_i^\pm)|\le
c\int_{B_1^\pm} |\nabla v_i^\pm|^2\le c,$$
we have, after taking $i$ to infinity, that
\begin{equation}\label{harm_v}
-\Delta v_\pm=0 \ \ {\rm{in}} \ \ B_1^\pm.
\end{equation}
Since $v_i^\pm$ also satisfies the trace condition
$$
\big(\epsilon_i\frac{\partial v_i^{+}}{\partial \nu}\big)^T
=(D\Phi^+(\epsilon_i v_i^+(x)+a_i^+))^t\big(\epsilon_i \frac{\partial v_i^-}{\partial \nu}\big)^T \ \ {\rm{on}}\ \ \Gamma_1,
$$
we obtain, after taking $i$ to infinity,  that
\begin{equation}\label{trace_cond1_v}
\big(\frac{\partial v_{+}}{\partial \nu}\big)^T
=(D\Phi^+(a^+))^t\big(\frac{\partial v_{-}}{\partial \nu}\big)^T
\ \ {\rm{on}}\ \ \Gamma_1.
\end{equation}
Here $(\cdot)^T: T_{a^\pm}N^\pm \mapsto T_{a^\pm}M^\pm$ is  the orthogonal projection map.
Moreover, we claim
\begin{equation}\label{zero_average_trace}
\int_{\Gamma_1}v_\pm\,dH^{n-1}=0.
\end{equation}
Set $\displaystyle w_i^+=\frac{\overline{u_i^+}-a_i^+}{\epsilon_i}$
and $\displaystyle w^+=\lim_{i\rightarrow\infty} w_i^+$. Then we have that $w^+\in {\rm{Nor}}(a^+, M^+)$. Hence for $H^{n-1}$ a.e. $x\in\Gamma_1$,
$$\lim_{i\rightarrow \infty}\frac{u_i^+(x)-a_i^+}{\epsilon_i}\cdot w_i^+=0,$$
since $\frac{u_i^+(x)-a_i^+}{\epsilon_i}$ converges to a vector in ${\rm{Tan}}(a^+, M^+)$.
Thus
$$|w^+|^2=\lim_{i\rightarrow\infty}|w_i^+|^2=\frac{1}{|\Gamma_1|}\lim_{i\rightarrow \infty}\int_{\Gamma_1}
\frac{u_i^+(x)-a_i^+}{\epsilon_i}\cdot w_i^+\,dH^{n-1}=0.$$
This implies
$$\frac{1}{|\Gamma_1|}\int_{\Gamma_1}v_+\,dH^{n-1}
=\frac{1}{|\Gamma_1|}\lim_{i\rightarrow\infty}\int_{\Gamma_1}\frac{u_i^+-a_i^+}{\epsilon_i}\,dH^{n-1}
=\lim_{i\rightarrow\infty}\frac{\overline{u_i^+}-a_i^+}{\epsilon_i}=w^+=0.$$
To see $\displaystyle\int_{\Gamma_1} v_-\,dH^{n-1}=0$, observe that
\begin{eqnarray*}\frac{\overline{u_i^-}-a_i^-}{\epsilon_i}
&=&D\Phi^+(a_i^+)\big(\frac{\overline{u_i^+}-a_i^+}{\epsilon_i}\big)
+o(1)\int_{\Gamma_1}|\frac{u_i^+-a_i^+}{\epsilon_i}|\,dH^{n-1}\\
&=&D\Phi^+(a_i^+)(w_i^+)
+o(1)\|v_i^+\|_{L^1(\Gamma_1)}
\end{eqnarray*}
so that
$$\int_{\Gamma_1} v_-\,dH^{n-1}=\frac{1}{|\Gamma_1|}\lim_{i\rightarrow\infty}
\frac{\overline{u_i^-}-a_i^-}{\epsilon_i}=\frac{1}{|\Gamma_1|}
D\Phi^+(a^+)(w^+)=0.$$

By (\ref{trace_cond_v}), (\ref{harm_v}), and (\ref{trace_cond1_v}), we can apply Lemma \ref{linear_regularity} to conclude that
$v_\pm\in C^\infty\big(\overline{B_\frac12^\pm}\big)$.
Moreover,  by (\ref{H1_bound_v}) and (\ref{zero_average_trace})
we have that for any $0<\theta<1$,
\begin{eqnarray}
&&\theta^{-n}\big(\int_{B_\theta^+}|v_+-(v_+)_\theta|^2
+\int_{B_\theta^-}|v_{-}-(v_{-})_\theta|^2\big)\nonumber\\
&&\le  c\theta^2
\big(\int_{B_1^+}|\nabla v_+|^2
+\int_{B_1^-}|\nabla v_{-}|^2\big)\le c\theta^2,\label{harm_v_estimate1}
\end{eqnarray}
where $\displaystyle
(v_\pm)_\theta=\frac{1}{|\Gamma_\theta|}\int_{\Gamma_\theta} v_\pm \,dH^{n-1}.$
By the Poincar\'e inequality and the trace theory we also have
\begin{equation}\label{harm_v_estimate2}
\theta^{1-n}\big(\int_{\Gamma_\theta}|v_+-(v_+)_\theta|^2
+\int_{\Gamma_\theta}|v_{-}-(v_{-})_\theta|^2\big)
\le c\theta^2\big(\int_{B_1^+}|\nabla v_+|^2
+\int_{B_1^-}|\nabla v_{-}|^2\big)\le c\theta^2 .
\end{equation}
Since $v_i^\pm\rightarrow v_\pm$ in $L^2(B_1^\pm)$
and $L^2(\Gamma_1)$, it follows from (\ref{harm_v_estimate1}) and (\ref{harm_v_estimate2})
that for $i$ sufficiently large
\begin{eqnarray}
&&\theta^{-n}\big(\int_{B_\theta^+}|u_i^+-(u_i^+)_\theta|^2
+\int_{B_\theta^-}|u_i^--(u_i^-)_\theta|^2\big)
+\theta^{1-n}\int_{\Gamma_\theta}(|u_i^+-(u_i^+)_\theta|^2
+|u_i^--(u_i^-)_\theta|^2)\nonumber\\
&&\le c\theta^2\epsilon_i^2.\label{l2_decay_ui}
\end{eqnarray}
Combining (\ref{bdry_extension_estimate1}) with (\ref{l2_decay_ui}). we can
repeat the argument of \cite{HL2} to get a desired contradiction.

\medskip
\noindent{\bf Proof of Theorem 1.2}. It is well-known that iterations of Lemma \ref{energy_improve}, combined with the interior $\epsilon$-regularity,
implies that there exist $\epsilon_0>0$ and $\alpha_0\in (0,1)$ such that if for $x_0\in\Gamma$, there exists $r_0>0$ such that
$$r_0^{2-n}\big(\int_{\Omega^+\cap B_{r_0}(x_0)} |\nabla u|^2+\int_{\Omega^-\cap B_{r_0}(x_0)} |\nabla u|^2\big)\le \epsilon_0^2,$$
then $u\in C^{\alpha_0}(\overline{\Omega^\pm}\cap B_{\frac{r_0}2}(x_0), N^\pm)$\footnote{Higher order regularity of $u$, e.g, $u\in C^{l,\alpha}(\overline{\Omega^\pm}\cap B_{\frac{r_0}2}(x_0))$,
can be shown, provided that the map $\Phi^+:M^+\to M^{-}$ is assumed to be $C^{l+1,\alpha}$ for some $l\ge 1$ and $0<\alpha<1$.}.
It follows from this property that  the set $\mathcal{S}(u)$ of discontinuity for $u$ in $\Omega^\pm\cup\Gamma$ can be shown to have
$H^{n-2}(\mathcal{S}(u))=0$.
It follows from \cite{SU1} that the Hausdorff dimension of $\mathcal{S}(u)$, ${\rm{dim}}_H(\mathcal{S}(u)\cap(\Omega^+\cup\Omega^{-}))\le n-3$ for $n\ge 3$.
Employing the boundary extension lemma \ref{boundary_extension} and Federer's dimension reduction argument, we can proceed, similar to \cite{SU2} and \cite{HL3}, to  conclude that
${\rm{dim}}_H(\mathcal{S}(u)\cap\Gamma)\le n-3$ for $n\ge 3$, and $\mathcal{S}(u)$ is discrete when $n=3$. This completes the proof. \qed

\section{On the local existence of regular solutions to heat flow}
In this section, we will consider the  gradient flow associated with the minimization problem \eqref{mini_harmonic}, or, equivalently, the parabolic version
of the harmonic map equation \eqref{HM}.  Under some further assumptions on $M^\pm$ and $\Gamma$, to be specified below,
we will establish the local existence of regular solutions of the heat flow under the initial and corresponding boundary conditions.

Before describing the corresponding heat flow problem, we first need to introduce some notations.
For a given $T>0$, let $\{\Gamma(t): t\in [0,T]\}$ be a smooth family of smooth hypersurfaces,
with $\Gamma(0)=\Gamma$, such that
$$\partial \Gamma(t)=\partial\Gamma=\Sigma, \ \forall 0\le t\le T.$$
For $t\in [0,T]$, decompose $\Omega\setminus\Gamma(t)$ into the disjoint union of
two simply connected components $\Omega^+(t)$ and $\Omega^-(t)$, i.e.,
$$\Omega\setminus\Gamma(t)=\Omega^+(t)\cup \Omega^-(t), \ t\in [0,T].$$
Denote $\Omega^\pm=\Omega^\pm(0)$, and write
$$\Omega\setminus\Gamma=\Omega^+\cup\Omega^-,\ \ \partial\Omega\setminus\Sigma=\Sigma^+ \cup\Sigma^{-}$$
so that $\partial\Omega^\pm=\Gamma\cup\Sigma^\pm.$
Set
$$Q_T=\big\{(x,t): \ x\in\Omega, \ 0<t\le T\big\}, \ \partial_pQ_T=(\Omega\times \{0\})\cup (\partial\Omega\times (0,T]),$$
and
$$\Gamma_T=\big\{(x,t): \ x\in \Gamma(t), \ 0<t\le T\big\}, \ Q_T^\pm=\big\{(x,t): \ x\in \Omega^\pm(t), \ 0<t\le T\big\}.$$
The harmonic heat flow problem corresponding to \eqref{HM} can be formulated as follows. We are looking for
$u^\pm:Q_T^\pm\mapsto N^\pm$, with $u^\pm(x,t)\in M^\pm$ for $(x,t)\in \Gamma_T$,  that solves
\begin{equation}\label{HFHM}
\begin{cases}
\partial_tu^+-\Delta u^+=A^+(u^+)(\nabla u^+, \nabla u^+) & \ {\rm{in}}\ Q_T^+,\\
\partial_tu^--\Delta u^-=A^-(u^-)(\nabla u^-, \nabla u^-) & \ {\rm{in}}\ Q_T^-,\\
u^\pm(x, t)= g^\pm(x) & \ (x,t)\in\Sigma^\pm\times [0, T],\\
u=u_0^\pm & \ {\rm{on}}\ \Omega^\pm\times\{0\},\\
\Phi^+(u^+)=u^-& \ {\rm{on}}\ \Gamma_T,\\
\big(\frac{\partial u^+}{\partial\nu}\big)^T=(D\Phi^+(u^+))^t\big[\big(\frac{\partial u^-}{\partial\nu}\big)^T\big] &  \ {\rm{on}}\ \Gamma_T.
\end{cases}
\end{equation}
Here $u_0^\pm:\Omega^\pm\mapsto N^\pm$, with $u_0^\pm(x)\in M^\pm$ satisfying $u_0^-(x)=\Phi^+(u_0^+(x))$
for $x\in\Gamma$, and $g^\pm=u_0^\pm\big|_{\Sigma^\pm}$ are given initial and boundary values.

In order to establish the short time existence of regular solutions to \eqref{HFHM},
we need to set up the problem appropriately by specifying
the assumptions $({\bf A})$, $({\bf B})$, and $({\bf C})$ on $N^\pm$
and $M^\pm$:

\begin{itemize}
\item[({\bf A})] The target Riemannian manifolds $(N^\pm, h^\pm)$ have the same dimension ${\rm{dim}}(N^\pm)=k+m$.
For, otherwise,
if $k_1={\rm{dim}}(N^+)<k_2={\rm{dim}}(N^-)$, then we can replace $(N^+, h^+)$ by
$$\big(\widehat{N^+}=N^+\times \mathbb S^{k_2-k_1}, \ \widehat{h^+}=h^+\oplus h_{\rm{can}}\big),$$
where $h_{\rm{can}}$ denotes the standard metric on $\mathbb S^{k_2-k_1}$. Notice that ${\rm{dim}}(\widehat{N^+})=k_2$.
Moreover, for any map $u:\Omega^+(t)\times [0,T]\to N^+$,
if we define $\tilde{u}(x,t)=(u(x,t),e): \Omega^+(t)\times [0,T]\to \widehat{N^+}$, where $e\in \mathbb S^{k_2-k_1}$,
then we can show that if $u$ is a solution to the heat flow of harmonic maps to $N^+$, then
$\tilde{u}$ is also a solution to the heat flow of harmonic maps to $\widehat{N^+}$. This follows from
the chain rule and the fact that $(N^+,h^+)$ is a totally geodesic sub-manifold of $(\widehat{N^+}, \widehat{h^+})$.

\item[({\bf B})] The manifolds  $M^\pm\subset N^\pm$ are two $k$-dimensional compact smooth sub-manifolds,
with $\partial M^\pm=\emptyset$, such that
there exists a smooth  diffeomorphism $\Phi^+: M^+\mapsto M^-$, whose inverse is denoted by $\Phi^-:M^-\mapsto M^+$.
Moreover, there exists $r_0=r_0(M^+)>0$ such that for any $p^+\in M^+$, $\Phi^+$ can be extended into a smooth
diffeomorphism, still denoted as itself,
$$\Phi^+: B_{r_0}^{N^+}(p^+)=\{p\in N^+:  {\rm{d}}_{N^+}(p, p^+)<r_0\}\mapsto
B_{r_0}^{N^-}(p^-)=\{p\in N^-:  {\rm{d}}_{N^-}(p, p^-)<r_0\},$$
whose inverse is also denoted by $\Phi^-$.

\item[({\bf C})] There exists a $0<r_1=r_1(N^+)\le r_0(M^+)$ such that for any $p^+\in N^+$,
there exists a local parametrization of $B_{r_1}^{N^+}(p^+)$ by $(B_1^k\times B_1^{m}, \phi^+)$, i.e.,
$$U=(U^1, U^2)=\big((u_1,\cdots, u_k), (u_{k+1},\cdots, u_{k+m})\big)\in B_1^k\times B_1^m$$
provides a local  representation of $B_{r_1}^{N^+}(p^+)$ via the
diffeomorphism $\phi^+:B_1^k\times B_1^m\mapsto B_{r_1}^{N^+}(p^+)$.
We may assume that $U(p^+)=(0,0)$, and if $p^+\in M^+$ then
$$U\big(M^+\cap B_{r_1}^{N^+}(p^+)\big)\equiv\big\{U=(U^1, U^2)\in B_1^k\times B_1^m: \ U^2=0\big\},$$
and the Riemannian metric $h^+$ on $B_{r_1}^{N^+}(p^+)$ can be expressed by
$$h^+(U)=\sum_{i,j=1}^{k+m} h_{ij}^+(U)du_i\otimes du_j,\ \forall U\in B_1^k\times B_1^m,$$
and the induced metric of  $h^+$ on $M^+\cap B_{r_1}^{N^+}(p^+)$ is given by
$$\displaystyle h^+(U^1,0)=\sum_{i,j=1}^{k} h_{ij}^+(U^1,0)du_i\otimes du_j, \ \forall U^1\in B_1^k.$$

\end{itemize}

It is readily seen that for $p^+\in M^+$ and $p^-=\Phi^+(p^+)$, through the diffeomorphism
$\Phi^+:B_{r_0}^{N^+}(p^+)\mapsto B_{r_0}^{N^-}(p^-)$, $U=(U^1,U^2)\in B_1^k\times B_1^m$
provides a local parametrization of $B_{r_1}^{N^-}(p^-)$
through the diffeomorphism $\phi^-:=\Phi^+(\phi^+): B_1^k\times B_1^m\mapsto B_{r_1}^{N^-}(p^-)$.
In particular, $U(p^-)=(0,0)$,
$$U\big(M^-\cap B_{r_1}^{N^-}(p^-)\big)\equiv\big\{U=(U^1, U^2)\in B_1^k\times B_1^m: \ U^2=0\big\},$$
and the Riemannian metric $h^-$ on $B_{r_1}^{N^-}(p^-)$ can be expressed by
$$h^-(U)=\sum_{i,j=1}^{k+m} h_{ij}^-(U) du_i\otimes du_j,\ \forall U\in B_1^k\times B^m_1, $$
and the induced metric of $h^-$ on $M^-\cap B_{r_1}^{N^-}(p^-)$ is given by
$$\displaystyle h^-(U^1,0)=\sum_{i,j=1}^{k} h_{ij}^-(U^1,0)du_i\otimes du_j, \ \forall U^1\in B_1^k.$$

We may assume henceforth that $r_1(N^+)=r_0(M^+)$ in the assumptions ({\bf B}) and ({\bf C}).

\begin{remark} Under the assumptions $({\bf A})$, $({\bf B})$, and $({\bf C})$, it is not hard to see that by choosing a sufficiently small $r_0=r_0(M^+)>0$, under the above local parametrization
of  $B_{r_0}^{N^\pm}(p^\pm)$, the local representations of the Riemannian metrics $h^\pm$ enjoy the following properties:
\begin{equation*}
h^\pm(U)=\sum_{i,j=1}^k h_{ij}^\pm(U^1, U^2) du_i\otimes du_j+\sum_{i,j=k+1}^{k+m} h_{ij}^\pm(U^1, U^2) du_i\otimes du_j, \ \forall U=(U^1,U^2)\in B_1^k\times B_1^m,
\end{equation*}
such that
\begin{equation}\label{asym}
\sum_{i,j=k+1}^{k+m}|h^\pm_{ij}(U^1, U^2)|\le C|U^2|, \forall U=(U^1,U^2)\in B_1^k\times B_1^m,
\end{equation}
for some $C>0$ depending only on $M^\pm$ and $N^\pm$.
\end{remark}

Now we are ready to state a theorem on the local existence of regular solutions to \eqref{HFHM}, whose full proof will be given
in another future work.
\begin{theorem}\label{existence} Under the assumptions $({\bf A})$, $({\bf B})$, and $({\bf C})$ on $N^\pm$ and $M^\pm$,
for $0<\alpha<1$, let $u_0^\pm\in C^{1+\alpha}(\overline{\Omega^\pm}, N^\pm)$
and $g^\pm=u_0^\pm\big|_{\overline{\Sigma^\pm}}\in C^{1+\alpha}(\overline{\Sigma^\pm}, N^\pm)$
be given initial and boundary data such that $u_0^\pm(\Gamma)\subset M^\pm$ satisfies
$u_0^-(x)=\Phi^+(u_0^+(x))$ and $(\frac{\partial u_0^-}{\partial\nu}(x))^T
=D\Phi^+(u^+_0(x))(\frac{\partial u_0^+}{\partial\nu}(x))^T$ for $x\in\Gamma$. Then there exist $T_0>0$, depending on
$\|u_0^\pm\|_{C^{1,\alpha}(\Omega^\pm)}$, and a unique solution
$u^\pm\in C^{1+\alpha,\frac{1+\alpha}2}(Q_{T_0}^\pm, N^\pm)$ of the initial and boundary value problem
\eqref{HFHM}.
\end{theorem}

The proof of Theorem \eqref{existence} is more delicate than the usual proofs of short time smooth solutions to the heat flow of harmonic maps
under the Dirichlet boundary condition (cf. \cite{CL1}, \cite{Hamilton} ) or the free boundary condition (cf. \cite{S3}). It involves to
first show the local existence of regular solutions over small balls, and then patch these local solutions by extending the Schwarz
alternating method on linear parabolic equations to the quasilinear harmonic map heat flows into small neighborhoods
of points in $N^\pm$. For this, we have to overcome major difficulties that arise near the interface $\Gamma$. A detailed proof
will be addressed in a forthcoming work.

In this part, we will indicate a proof of Theorem \ref{existence} when the images of $u^\pm$ is contained
in a single coordinate chart of $N^\pm$. Before doing it, we want to rewrite the system \eqref{HFHM} in an intrinsic form
near a small neighborhood of a point $(x_0,t_0)\in \Gamma_T$ and also derive a generalized energy inequality.

\subsection{Local representation of \eqref{HFHM}}
For $t_0\in (0,T)$ and $x_0\in\Gamma(t_0)$, choose a small $\delta_0>0$, depending on
$\|u^\pm\|_{C^0(Q_T^\pm)}$, such that
$$u^\pm\big(Q_T^\pm\cap P_{\delta_0}(x_0,t_0)\big)\subset
B_{r_0}^{N^\pm}(p_0^\pm), \ {\rm{with}}\ \ p_0^\pm=u^\pm(x_0,t_0)\in M^\pm.$$
where  $P_{\delta_0}(x_0,t_0)=B_{\delta_0}(x_0)\times (t_0-\delta_0^2, t_0+\delta_0^2)$.
Then, by employing the local representations given by the assumptions
({\bf B}) and ({\bf C}) on $M^\pm, N^\pm$, we can rewrite the
harmonic heat flow equation \eqref{HFHM} as
\begin{equation}\label{heatflow1}
\begin{cases}
\partial_t U-\Delta U=\Gamma^+(U)(\nabla U, \nabla U) &\ {\rm{in}} \ Q^+_T\cap P_{\delta_0}(x_0,t_0),\\
\partial_t U-\Delta U=\Gamma^-(U)(\nabla U, \nabla U) &\ {\rm{in}} \ Q^{-}_T\cap P_{\delta_0}(x_0,t_0),
\end{cases}
\end{equation}
where
$U=(U^1,U^2): Q^+_T\cap P_{\delta_0}(x_0,t_0)\mapsto B_1^k\times B_1^m$
is the local representation of $u=u^\pm: Q^+_T\cap P_{\delta_0}(x_0,t_0)\mapsto N$,
and $\Gamma^\pm(\cdot)(\cdot,\cdot)$ is the Christoffel symbol of $N^\pm$.

Observe that within this  local coordinate system,
the boundary condition \eqref{HFHM}$_4$
on the free interface $\Gamma_T$ gives rise to
\begin{equation}\label{interface_bdry10}
{U}^2=0,\ {\rm{on}}\ \Gamma_T\cap P_{\delta_0}(x_0,t_0),
\end{equation}
and by \eqref{asym} the boundary condition \eqref{HFHM}$_5$ on the free interface $\Gamma_T$
reduces
to
\begin{equation}\label{interface_bdry20}
\sum_{j=1}^{k}h^+_{ij}(U^1,0)\frac{\partial{(U^1)^j}}{\partial \nu}
=\sum_{j=1}^{k}h_{ij}^-(U^1,0)\frac{\partial{(U^1)^j}}{\partial\nu},  \ 1\le i\le k,
\ {\rm{on}}\ \Gamma_T\cap P_{\delta_0}(x_0,t_0).
\end{equation}

\subsection{Parametrization of domains}

\medskip
\indent Since $\Omega^\pm(t)$ is $t$-dependent over $[0,T]$, in this subsection we will re-parametrize the domains
and rewrite  \eqref{HFHM} so that it can be viewed as the heat flow of harmonic maps over fixed domain
but with time-dependent metrics on the domain.

Assume that $\Psi(\cdot,t): \Omega\times [0,T]\mapsto\Omega$ is a smooth family of diffeomorphism such that
\begin{equation}\label{diff}
\Psi(x,t)=x,  \forall (x,t)\in\partial\Omega\times [0,T];
\ \Psi(\Gamma(t),t)=\Gamma\ {\rm{and}}\ \Psi(\Omega^\pm(t), t)=\Omega^\pm,  \forall\ t\in [0,T].
\end{equation}
For $u^\pm:Q_T^\pm\mapsto N^\pm$, define $\widehat{Q}_T^\pm=\Omega^\pm\times [0,T]$
and $\widehat{u}^\pm:\widehat{Q}_T^\pm\mapsto N^\pm$ through
$${u}^\pm(x,t)=\widehat{u}^\pm(\Psi(x,t), t): \widehat{Q}_T^\pm\mapsto N^\pm.$$

Given that $u^\pm:Q_T^\pm\mapsto N^\pm$ satisfies \eqref{HFHM}, we want to derive the equation
for $\widehat{u}^\pm$ now. To do it,  first set
$$a_{ij}(x,t)=\big(\frac{\partial\Psi_i}{\partial x_\alpha}\frac{\partial\Psi_j}{\partial x_\alpha}\big)(x,t):Q_T
\mapsto \mathbb R^{n\times n},$$
and
$$\widehat{a}_{ij}(y,t)=a_{ij}(x,t): Q_T\mapsto \mathbb R^{n\times n}, \ {\rm{where}}\ (x,t)=\Psi^{-1}(y,t).$$
Then direct calculations imply that
\begin{equation}\label{chain1}
\begin{cases}
\partial_t u^\pm(x,t)=\partial_t \widehat{u}^\pm(\Psi(x,t), t)
+\frac{\partial{\widehat{u}^\pm}}{\partial y_j}(\Psi(x,t), t) \partial_t\Psi_j,\\
\frac{\partial u^\pm}{\partial x_\alpha}=\frac{\partial\widehat{u}^\pm}{\partial y_i}(\Psi(x,t),t)
\frac{\partial\Psi_i}{\partial x_\alpha},
\end{cases}
\end{equation}
and
\begin{eqnarray*}
&&\Delta u^\pm(x,t)=\frac{\partial}{\partial x_\alpha}\big(\frac{\partial\widehat{u}^\pm}{\partial y_i}(\Psi(x,t),t)
\frac{\partial\Psi_i}{\partial x_\alpha}\big)\\
&&=\frac{\partial^2\widehat{u}^\pm}{\partial y_i\partial y_j}(\Psi(x,t), t)\frac{\partial\Psi_i}{\partial x_\alpha}
\frac{\partial\Psi_j}{\partial x_\alpha}+\frac{\partial\widehat{u}^\pm}{\partial y_i}(\Psi(x,t),t)\Delta\Psi_i.
\end{eqnarray*}
Hence \eqref{HFHM}$_{1,2}$ becomes
\begin{equation}\label{heatflow2}
\begin{cases}\displaystyle
\partial_t \widehat{u}^+
-\frac{\partial}{\partial y_i}\big(\widehat{a}_{ij} \frac{\partial\widehat{u}^+}{\partial y_j}\big)
= \widehat{a}_{ij}A^+(\widehat{u}^+)\big(\frac{\partial\widehat{u}^+}{\partial y_i},
\frac{\partial\widehat{u}^+}{\partial y_j}\big)
+A_i\frac{\partial\widehat{u}^+}{\partial y_i} & {\rm{in}} \ \widehat{Q}_T^+,\\
\displaystyle\partial_t \widehat{u}^{-}
-\frac{\partial}{\partial y_i}\big(\widehat{a}_{ij} \frac{\partial\widehat{u}^{-}}{\partial y_j}\big)
= \widehat{a}_{ij}A^-(\widehat{u}^-)\big(\frac{\partial\widehat{u}^{-}}{\partial y_i},
\frac{\partial\widehat{u}^{-}}{\partial y_j}\big)
+A_i\frac{\partial\widehat{u}^{-}}{\partial y_i} & {\rm{in}} \ \widehat{Q}_T^-,
\end{cases}
\end{equation}
where
$$A_i(y,t)=\frac{\partial\widehat{a}_{ij}}{\partial y_j}(y,t)
-(\Delta\Psi_i)(\Psi^{-1}(y,t),t)-(\partial_t\Psi_i)(\Psi^{-1}(y,t),t), \ \forall (y,t)\in Q_T.
$$

Observe that
the boundary condition \eqref{HFHM}$_4$
on the free interface $\Gamma_T$ gives rise to
\begin{equation}\label{interface_bdry1}
\widehat{u}^-(y, t)=\Phi^+(\widehat{u}^+)(y,t), \ \ \forall (y,t)\in \Gamma\times [0,T],
\end{equation}
while the boundary condition \eqref{HFHM}$_5$ on the free interface $\Gamma_T$ gives rise
to
\begin{equation}\label{interface_bdry2}
\big(\frac{\partial\widehat{u}^-}{\partial \nu}\big)^T(y, t)
=D\Phi^+(\widehat{u}^+)\big(\frac{\partial\widehat{u}^+}{\partial \nu}\big)^T(y, t),  \ \forall (y,t)\in \Gamma\times [0,T],
\end{equation}
where $\nu (=\nu(t))$ is the unit outer normal of $\Gamma$  with respect to the metric
$\widehat{g}(t)=\widehat{a}_{ij}(y,t)\,dy^idy^j$.

\medskip
First we observe that a sufficiently regular solution of \eqref{HFHM} enjoys a generalized energy inequality.
For $1<p<\infty$, $T>0$, and an open set $E\subset\mathbb R^n$, denote
$$W^{2,1}_p(E\times [0,T])=\Big\{u\in L^p(E\times [0,T]): \ \partial_t u, \ \nabla^2 u\in L^p(E\times [0,T])\Big\}.$$
We have
\begin{lemma}\label{energy_ineq1} For $T>0$, and $g\in C^{1}({\Sigma^\pm}, N^\pm)$, if $u^\pm\in W^{2,1}_2({Q_T^\pm},  N^\pm)$,
with $\nabla \frac{\partial u^\pm}{\partial t}\in L^2(Q_T^\pm)$,
is a strong solution of \eqref{HFHM}, then there exists constant $C>0$ depending on $\Gamma_T$ such that
\begin{equation}\label{energy_ineq2}
E(u(t))+\frac14\int_{s}^te^{C(t-\tau)}\big(\int_{\Omega^+(t)} |\partial_t u^+|^2+\int_{\Omega^-(t)} |\partial_t u^-|^2\big)\,dxd\tau\le e^{C(t-s)}E(u(s)),
\end{equation}
for all $0\le s< t\in [0, T]$.
\end{lemma}

\noindent{\bf Proof.\quad}}\def\endpf{\hfill$\Box$
Let $\Psi(\cdot, t): \Omega\times [0,T]\mapsto\Omega$ be a smooth family of diffeomorphism given by
\eqref{diff}.
Define $\widehat{u}^\pm: \widehat{Q}^\pm_T\mapsto N^\pm$ by
$$u^\pm(x,t)=\widehat{u}^\pm(\Psi(x,t),t), \ \forall (x,t)\in \widehat{Q}^\pm_T.$$
Then $\widehat{u}^\pm$ solves \eqref{heatflow2} in $\widehat{Q}^\pm_T$,
\eqref{interface_bdry1} and \eqref{interface_bdry2} on $\Gamma_T$,
and the Dirichlet boundary condition:
\begin{equation}\label{dirichlet}
\widehat{u}^\pm(y,t)=g^\pm(y), \ (y,t)\in\partial\Omega\times [0,T].
\end{equation}
Within this time dependent parametrization, we can write
$$E(u(t))=\frac12\int_{\Omega^+} \widehat{a}_{\alpha\beta}\langle \frac{\partial \widehat{u}^+}{\partial y_\alpha}, \frac{\partial \widehat{u}^+}{\partial y_\beta}\rangle
\,dv_{\widehat{g}}
+\frac12\int_{\Omega^-} \widehat{a}_{\alpha\beta}\langle \frac{\partial \widehat{u}^-}{\partial y_\alpha}, \frac{\partial \widehat{u}^-}{\partial y_\beta}\rangle\,dv_{\widehat{g}},
$$
where $dv_{\widehat{g}}=\sqrt{\widehat{g}}\,dy$, and $\widehat{g}(y,t)={\rm{det}}(\nabla\Psi)(\Psi^{-1}(y,t),t)$.

From $u^\pm\in W^{2,1}_2(Q_T^\pm, N)$ and  $\nabla\frac{\partial u^\pm}{\partial t}\in L^2(Q_T^\pm)$,
$\widehat{u}^\pm\in W^{2,1}_2(\widehat{Q}^\pm_T, N^\pm)$ and
$\nabla\frac{\partial \widehat{u}^\pm}{\partial t}\in L^2(\widehat{Q}^\pm_T)$.
By direct calculations, we have that
\begin{eqnarray*}
\frac{d}{dt}E(u(t))&=&\int_{\Omega^+}\widehat{a}_{\alpha\beta}\langle \frac{\partial}{\partial y_\alpha} (\partial_t{\widehat{u}^+}),
\frac{\partial \widehat{u}^+}{\partial y_\beta}\rangle\sqrt{\widehat{g}}\,dy+\int_{\Omega^-}\widehat{a}_{\alpha\beta}\langle \frac{\partial}{\partial y_\alpha} (\partial_t{\widehat{u}^-}),
\frac{\partial \widehat{u}^-}{\partial y_\beta}\rangle\sqrt{\widehat{g}}\,dy\\
&&+\frac12\big(\int_{\Omega^+} \langle \frac{\partial \widehat{u}^+}{\partial y_\alpha}, \frac{\partial \widehat{u}^+}{\partial y_\beta}\rangle
\partial_t(\widehat{a}_{\alpha\beta}\sqrt{{\widehat{g}}})\,dy
+\int_{\Omega^-} \langle \frac{\partial \widehat{u}^-}{\partial y_\alpha}, \frac{\partial \widehat{u}^-}{\partial y_\beta}\rangle
\partial_t(\widehat{a}_{\alpha\beta}\sqrt{{\widehat{g}}})\,dy\big)
\\
&=& I(t)+II(t).
\end{eqnarray*}
It is easy to see that
$$|II(t)|\le CE(u(t)).$$
While, applying the integration by parts, \eqref{heatflow2}, the boundary conditions \eqref{interface_bdry1}, \eqref{interface_bdry2},
and \eqref{dirichlet}, and the
fact that $\partial_t \widehat{u}^-(x,t)=D\Phi^+(u^+)(\partial_t \widehat{u}^{+})(x,t)\in T_{u^\pm(x,t)} M^\pm$ for $(x,t)\in \Gamma_T$,
and $\partial_t \widehat{u}^\pm(x,t)=0$ on $\Sigma^\pm\times [0,T]$, we can show that the boundary contributions on both
$\Gamma$ and $\partial\Omega$ are zeroes. Hence
we can estimate $I$ by
\begin{eqnarray*}
I(t)&=&-\int_{\Omega^+}\langle \partial_t\widehat{u}^+, \frac{\partial}{\partial y_\alpha}(\widehat{a}_{\alpha\beta}\frac{\partial\widehat{u}^+}{\partial y_\beta})\rangle
\,dv_{\widehat{g}}-\int_{\Omega^-}\langle \partial_t\widehat{u}^-, \frac{\partial}{\partial y_\alpha}(\widehat{a}_{\alpha\beta}\frac{\partial\widehat{u}^-}{\partial y_\beta})\rangle
\,dv_{\widehat{g}}\\
&&-
\int_{\Omega^+}\langle \partial_t\widehat{u}^+, \widehat{a}_{\alpha\beta}\frac{\partial\widehat{u}^+}{\partial y_\beta}\rangle \frac{\partial\sqrt{\widehat{g}}}{\partial y_\alpha}
\,dy-\int_{\Omega^-}\langle \partial_t\widehat{u}^-, \widehat{a}_{\alpha\beta}\frac{\partial\widehat{u}^-}{\partial y_\beta}\rangle
\frac{\partial\sqrt{\widehat{g}}}{\partial y_\alpha}\,dy\\
&=&-\big(\int_{\Omega^+}|\partial_t \widehat{u}^+|^2\,dv_{\widehat{g}}+\int_{\Omega^-}|\partial_t \widehat{u}^-|^2\,dv_{\widehat{g}}\big)\\
&&+\big(\int_{\Omega^+}\langle\partial_t \widehat{u}^+, A_i\frac{\partial \widehat{u}^+}{\partial y_i}\rangle\,dv_{\widehat{g}}+\int_{\Omega^-}\langle\partial_t \widehat{u}^-, A_i\frac{\partial \widehat{u}^-}{\partial y_i}\rangle\,dv_{\widehat{g}}\big)\\
&&-\big(\int_{\Omega^+}\langle \partial_t\widehat{u}^+, \widehat{a}_{\alpha\beta}\frac{\partial\widehat{u}^+}{\partial y_\beta}\rangle \frac{\partial\sqrt{\widehat{g}}}{\partial y_\alpha}
\,dy+\int_{\Omega^-}\langle \partial_t\widehat{u}^-, \widehat{a}_{\alpha\beta}\frac{\partial\widehat{u}^-}{\partial y_\beta}\rangle
\frac{\partial\sqrt{\widehat{g}}}{\partial y_\alpha}\,dy\big)\\
&=& III(t)+IV(t)+V(t).
\end{eqnarray*}
It is easy to see that
$$
|IV(t)|+|V(t)|\le \frac18\big(\int_{\Omega^+}|\partial_t \widehat{u}^+|^2\,dv_{\widehat{g}}+\int_{\Omega^-}|\partial_t \widehat{u}^-|^2\,dv_{\widehat{g}}\big)
+CE(u(t)).
$$
Hence
$$|I(t)|\le -\frac78 \big(\int_{\Omega^+}|\partial_t \widehat{u}^+|^2\,dv_{\widehat{g}}+\int_{\Omega^-}|\partial_t \widehat{u}^-|^2\,dv_{\widehat{g}}\big).
$$
On the other hand, it follows from the chain rule \eqref{chain1} that
\begin{eqnarray*}
&&\int_{\Omega^+}|\partial_t \widehat{u}^+|^2\,dv_{\widehat{g}}+\int_{\Omega^-}|\partial_t \widehat{u}^-|^2\,dv_{\widehat{g}}\\
&&\ge \frac12\big(\int_{\Omega^+(t)}|\partial_t {u}^+|^2\,dx+\int_{\Omega^-(t)}|\partial_t {u}^-|^2\,dx\big)-CE(u(t)).
\end{eqnarray*}
Putting all these estimate together, we obtain
$$
\frac{d}{dt}E(u(t))\le -\frac14 \big(\int_{\Omega^+(t)}|\partial_t {u}^+|^2\,dx+\int_{\Omega^-(t)}|\partial_t {u}^-|^2\,dx\big)+CE(u(t)),
$$
which, combined with Gronwall's inequality, implies \eqref{energy_ineq2}. \qed

\medskip
We will sketch a proof of Theorem \ref{existence} by employing the fixed point argument, under two extra assumptions that
\begin{enumerate}
\item[(i)] the images of
$u_0^\pm$ is contained in a single coordinate chart, i.e.,
\begin{equation}\label{one-chart}
u_0^\pm(x)\subset B_{r_0}^{N^\pm}(p_0^\pm), \ \forall x\in \overline\Omega,
\end{equation}
for a pair of points $p_0^\pm\in M^\pm$ that satisfies $p_0^-=\Phi^+(p_0^+)$;
and
\item[(ii)]
\begin{equation}\label{isom}
\Phi^+:M^+\mapsto M^- \mbox{\ is an isometry}.
\end{equation}
\end{enumerate}

First we will give some heuristic arguments to indicate that the appropriate function spaces for
the local existence of regular solutions are
\begin{eqnarray*}
{\mathcal{C}}^{1+\alpha, \frac{1+\alpha}2}_{(u_0,g)}(Q_T, B_{r_0}^{N^\pm}(p_0^\pm))
&=&\Big\{ u:Q_T\mapsto B_{r_0}^{N^\pm}(p_0^\pm):\ u^\pm=u\big|_{Q_T^\pm}\in C^{1+\alpha, \frac{1+\alpha}2}(Q_T^\pm),\\
&& \ \ \ u=u_0 \ {\rm{in}}\ \Omega\times\{0\}, \ \ u=g \ {\rm{on}}\ \partial\Omega\times [0,T],\\
&& \ \ \  u^\pm(\Gamma_T)\subset M^\pm,
\ u^-=\Phi^+(u^+),\\
&&\ \ \ (\frac{\partial u^+}{\partial\nu})^T=(D\Phi^+(u^+))^t(\frac{\partial u^-}{\partial\nu})^T \ \ {\rm{on}}\ \ \Gamma_T\Big\},
\end{eqnarray*}
which is equipped with the norm
$$
\big\|u\big\|_{\mathcal{C}^{1+\alpha,\frac{1+\alpha}2}(Q_T)}
=\big\|u^+\big\|_{C^{1+\alpha, \frac{1+\alpha}2}(Q_T^+)}
+\big\|u^-\big\|_{C^{1+\alpha, \frac{1+\alpha}2}(Q_T^-)}.
$$

To see this, assume that $\Gamma(t)\equiv \Gamma$ for $0\le t\le T$. Let
$u^\pm\in {\mathcal{C}}_{(u_0,g)}^{1+\alpha, \frac{1+\alpha}2}(Q_T^\pm, B_{r_0}^{N^\pm}(p_0^\pm))$ be given, and
$U=(U^1, U^2):Q_T\mapsto B_1^k\times B_1^m$ be a local representation of $u^\pm:Q_T^\pm\mapsto B_{r_0}^{N^\pm}(p_0^\pm)$.
Consider $V=(V^1, V^2): Q_T\mapsto B_1^k\times B_1^m$ that is a weak solution of
\begin{equation}\label{heatflow3}
\begin{cases}
\partial_t V-\Delta V=\Gamma^+(U)(\nabla U, \nabla U) &\ {\rm{in}}\ Q_T^+,\\
\partial_t V-\Delta V=\Gamma^-(U)(\nabla U, \nabla U) &\ {\rm{in}} \ Q_T^-,
\end{cases}
\end{equation}
under the initial and boundary condition:
\begin{equation}\label{bdry4}
\begin{cases}
V=U_0  & {\rm{on}}\ \partial_p Q_T,\\
V^2(x^+,t)=V^2(x^-,t)=0,\\
\displaystyle\frac{\partial V^1}{\partial \nu}(x^+,t)
=\frac{\partial V^1}{\partial \nu}(x^-,t), & (x,t)\in \Gamma_T.
\end{cases}
\end{equation}
Here $U_0:\Omega\mapsto B_1^k\times B_1^m$ is a local representation of $u_0$.

It follows from the regularity of linear parabolic equations
that $V\in C^{1+\alpha, \frac{1+\alpha}2}(Q_T^\pm)$.
Moreover, since
\begin{align*}
\|\Gamma^+(U)(\nabla U, \nabla U)\|_{L^\infty(Q_T^+)}+
\|\Gamma^-(U)(\nabla U, \nabla U)\|_{L^\infty(Q_T^-)}\le C\|\nabla U\|_{L^\infty(Q_T)}^2,
\end{align*}
it follows from the $W^{2,1}_p$-theory of linear parabolic equations that
$V\in W^{2,1}_p(Q_T^\pm)$ and
\begin{align*}
\big\|V\big\|_{W^{2,1}_p(Q_T^\pm)}\le C(p) \big(\|\nabla U\|_{L^\infty(Q_T)}^2+\|U_0\|_{C^{1+\alpha}(\Omega^\pm)}\big),
\end{align*}
for any $1<p<\infty$.

By the Sobolev's embedding theorem (\cite{LSU} Lemma II.3.3), we conclude that
$V\in C^{1+\alpha, \frac{1+\alpha}2}(Q_T^\pm)$ and
\begin{align*}
\big\|V\big\|_{C^{1+\alpha,\frac{1+\alpha}2}(Q_T^\pm)}\le C(p)\big(\|\nabla U\|_{L^\infty(Q_T)}^2
+\|U_0\|_{C^{1+\alpha}(\Omega^\pm)}\big).
\end{align*}

\medskip
\noindent{\bf Proof of Theorem \ref{existence} under the assumptions \eqref{one-chart} and \eqref{isom}}:

\medskip

For a pair of initial and boundary data $(u_0,g)$ given by Theorem \ref{existence},
let $U_0:\partial_p Q_T\mapsto B_1^k\times B_1^m$ be a local representation of $u_0$.
It follows from the assumptions \eqref{one-chart} and \eqref{isom} that
$u\in \mathcal{C}^{1+\alpha, \frac{1+\alpha}2}_{(u_0,g)}(Q_T, B_{r_0}^{N^\pm}(p_0^\pm))$ if and only if
its local representation $U$ belongs to the space
\begin{eqnarray*}
\mathcal{C}^{1+\alpha, \frac{1+\alpha}2}_{{U_0}}(Q_T, B_1^k\times B_1^m)
&=&\Big\{U=(U^1,U^2)\in C^{1+\alpha, \frac{1+\alpha}2}(Q_T^\pm, B_1^k\times B^m_1):
U=U_0 \ {\rm{on}}\ \partial_p Q_T,\\
&&\ \ U^2(x^+,t)=U^2(x^-,t)=0, \frac{\partial U^1}{\partial\nu}(x^+,t)=\frac{\partial U^1}{\partial\nu}(x^-,t),
\ (x,t)\in\Gamma_T\Big\}.
\end{eqnarray*}
Now we define $\widehat{U_0}=(\widehat{U_0}^1, \widehat{U_0}^2):Q_T\mapsto B_1^k\times B_1^m$
to the solution of the heat equation in $Q_T^\pm$:
\begin{equation}\label{HE}
\begin{cases}
\partial_t\widehat{U_0}-\Delta \widehat{U_0}=0 & \ {\rm{in}}\ Q_T^\pm,\\
\widehat{U_0}= U_0& \ {\rm{on}}\ \partial_p Q_T,\\
\widehat{U_0}^2(x^+,t)=\widehat{U_0}^2(x^-,t)=0& \ {\rm{on}}\ \Gamma_T,\\
\displaystyle\frac{\partial\widehat{U_0}^1}{\partial\nu}(x^+,t)
=\frac{\partial\widehat{U_0}^1}{\partial\nu}(x^-,t)
&  \ {\rm{on}}\ \Gamma_T.
\end{cases}
\end{equation}
From the condition on ${U_0}$, we know that there exists $\epsilon_0>0$ such that
$$|U_0^1|\le 1-4\epsilon_0, \ |U_0^2|\le 1-4\epsilon_0, \ {\rm{in}}\ \ \Omega.$$
Hence by the maximum principle, we have that
$$|\widehat{U_0}^1|\le 1-2\epsilon_0, \ |\widehat{U_0}^2|\le 1-2\epsilon_0, \ {\rm{in}}\ \ Q_T,$$
and hence $\widehat{U_0}\in \mathcal{C}^{1+\alpha, \frac{1+\alpha}2}_{{U_0}}(Q_T, B_{1-2\epsilon_0}^k\times B_{1-2\epsilon_0}^m)$.

As a consequence, for any $0<\epsilon\le\epsilon_0$, we can see that
$$\mathbb{B}(\widehat{U_0}, \epsilon)=\Big\{U\in \mathcal{C}^{1+\alpha, \frac{1+\alpha}2}_{{U_0}}(Q_T, B_1^k\times B_1^m):
\big\|U-\widehat{U_0}\big\|_{\mathcal{C}^{1+\alpha,\frac{1+\alpha}2}(Q_T)}<\epsilon\Big\}
$$
is a ball in $\mathcal{C}^{1+\alpha, \frac{1+\alpha}2}_{U_0}(Q_T,
B_1^k\times B_1^m)$ with center $\widehat{U_0}$ and radius $\epsilon$.

Now we define the solution map $\mathbb{T}: \mathbb{B}(\widehat{U_0},\epsilon)
\mapsto \mathcal{C}^{1+\alpha, \frac{1+\alpha}2}_{\widehat{U_0}}(Q_T, B_1^k\times B_1^m)$ by letting
$V=\mathbb{T}(U), U\in \mathbb{B}(\widehat{U_0},\epsilon)$, be the solution of
\begin{equation}\label{fix1}
\partial_t V-\Delta V=\begin{cases} \Gamma^+(U)(\nabla U,\nabla U) & {\rm{on}}\ Q_T^+\\
                                                        \Gamma^-(U)(\nabla U,\nabla U) & {\rm{on}}\ Q_T^-,
                                                        \end{cases}
\end{equation}
subject to the initial and boundary condition \eqref{bdry4}.

Now we need the following Lemma.

\begin{lemma} \label{fixpt} There exist $\epsilon>0$ and $T>0$ such that
$\mathbb{T}: \mathbb{B}(\widehat{U_0}, \epsilon)\mapsto \mathbb{B}(\widehat{U_0}, \epsilon)$ is a contractive map,
i.e., for any  $\theta\in (0,1)$, we can find $\epsilon>0$ and $T>0$ such that
\begin{equation}
\big\|\mathbb{T}(U_1)-\mathbb{T}(U_2)\big\|_{\mathcal{C}^{1+\alpha,\frac{1+\alpha}2}(Q_{T})}
\le \theta \big\|U_1-U_2\big\|_{\mathcal{C}^{1+\alpha,\frac{1+\alpha}2}(Q_{T})},
\ \forall U_1, U_2\in \mathbb{B}(\widehat{U_0}, \epsilon).
\end{equation}
Therefore there exists a unique $U\in \mathbb{B}(\widehat{U_0}, \epsilon)$ such that $U=\mathbb{T}(U)$.
In particular, if $u^\pm=u\big|_{Q_{T}^\pm}: Q_{T}^\pm\mapsto N^\pm$ has $U$ as its
local representation, then $u$ is a unique regular solution of \eqref{HFHM} in $Q_{T}$.
\end{lemma}

\noindent{\bf Proof.\quad}}\def\endpf{\hfill$\Box$
For $\mathbb{U}\in \mathbb{B}(\widehat{U_0},\epsilon)$, since $V-\widehat{U_0}$ satisfies
\begin{equation}\label{fix2}
\partial_t (V-\widehat{U_0})-\Delta (V-\widehat{U_0})=\begin{cases} \Gamma^+(U)(\nabla U,\nabla U) & {\rm{on}}\ Q_T^+\\
                                                        \Gamma^-(U)(\nabla U,\nabla U) & {\rm{on}}\ Q_T^-,
                                                        \end{cases}
\end{equation}
and
$$\begin{cases}
V-\widehat{U_0}=0 &\ {\rm{on}}\ \partial_p Q_T,\\
(V-\widehat{U_0})^2(x^+,t)=(V-\widehat{U_0})^2(x^-,t)=0 &\ {\rm{on}}\ \Gamma_T,\\
\frac{\partial(V-\widehat{U_0})^1}{\partial\nu}(x^+,t)
=\frac{\partial(V-\widehat{U_0})^1}{\partial\nu}
(x^-,t) &\ {\rm{on}}\ \Gamma_T.
\end{cases}
$$
Hence, similar to the earlier discussion, we have that for some $p=p(\alpha)>n+2$,
\begin{eqnarray*}
\big\|V-\widehat{U_0}\big\|_{\mathcal{C}^{1+\alpha, \frac{1+\alpha}2}(Q_T)}
&\le& C\big\|V-\widehat{U_0}\big\|_{W^{2,1}_p(Q_T^\pm)}\\
&\le& C\big\||\nabla U|^2\big\|_{L^p(Q_T)}\\
&\le& C\big\|\nabla U\big\|_{L^\infty(Q_T)}^2 T^{\frac{1}{p}}\\
&\le & C(\|\widehat{U_0}\|_{\mathcal{C}^{1+\alpha, \frac{1+\alpha}2}(Q_T)}+\epsilon)^2 T^{\frac{1}{p}}\\
&< & \epsilon,
\end{eqnarray*}
provided we choose a sufficiently small $T=T_0>0$, depending only on $U_0$ and $\alpha$.
Hence $V=\mathbb{T}({U})\in \mathbb{B}(\widehat{U_0},\epsilon)$.

For $i=1, 2$, let  $U_i\in \mathbb{B}(\widehat{U_0},\epsilon)$ and $V_i=\mathbb{T}(U_i)$. Then
\begin{equation}\label{fix3}
\partial_t (U_1-U_2)-\Delta (U_1-U_2)=\begin{cases} \Gamma^+(U_1)(\nabla U_1,\nabla U_1)
-\Gamma^+(U_2)(\nabla U_2,\nabla U_2) & {\rm{on}}\ Q_T^+\\
\Gamma^-(U_1)(\nabla U_1,\nabla U_1)-\Gamma^-(U_2)(\nabla U_2,\nabla U_2) & {\rm{on}}\ Q_T^-,
\end{cases}
\end{equation}
and
$$\begin{cases}
U_1-{U_2}=0 &\ {\rm{on}}\ \partial_p Q_T,\\
(U_1-{U_2})^2(x^+,t)=(U_1-{U_2})^2(x^-,t)=0 &\ {\rm{on}}\ \Gamma_T,\\
\frac{\partial(U_1-{U_2})^1}{\partial\nu}(x^+,t)
=\frac{\partial(U_1-{U_2})^1}{\partial\nu}
(x^-,t) &\ {\rm{on}}\ \Gamma_T.
\end{cases}
$$
Hence we can conclude  that  for any $\theta\in (0,1)$ such that for $p=p(\alpha)>n+2$,
 \begin{eqnarray*}
&&\big\|U_1-{U_2}\big\|_{\mathcal{C}^{1+\alpha, \frac{1+\alpha}2}(Q_T)}\\
&\le& C\big\|U_1-{U_2}\big\|_{W^{2,1}_p(Q_T^\pm)}\\
&\le& C\big\|(|\nabla U_1|+|\nabla U_2|)^2(U_1-U_2)+(|\nabla U_1|+|\nabla U_2|)|\nabla(U_1-U_2)|\big\|_{L^p(Q_T)}\\
&\le& C\big\||\nabla U_1|+|\nabla U_2|\big\|_{L^p(Q_T)}^2 \|U_1-U_2\|_{L^\infty(Q_T)}
+C\big\||\nabla U_1|+|\nabla U_2|\big\|_{L^p(Q_T)}\|\nabla(U_1-U_2)\|_{L^\infty(Q_T)}\\
&\le & C(1+\|\widehat{U_0}\|_{\mathcal{C}^{1+\alpha, \frac{1+\alpha}2}(Q_T)}^2) T^{\frac{1}{p}}
\big\|U_1-{U_2}\big\|_{\mathcal{C}^{1+\alpha, \frac{1+\alpha}2}(Q_T)}\\
&< & \theta \big\|U_1-{U_2}\big\|_{\mathcal{C}^{1+\alpha, \frac{1+\alpha}2}(Q_T)},
\end{eqnarray*}
provided $T=T_0>0$ is chosen so that
$$C(1+\|\widehat{U_0}\|_{\mathcal{C}^{1+\alpha, \frac{1+\alpha}2}(Q_T)}^2) T_0^{\frac{1}{p}}\le \theta.$$
This completes the proof of both Lemma \ref{fixpt} and Theorem \ref{existence} under the assumptions
\eqref{one-chart} and \eqref{isom}.
\qed

\section{Boundary monotonicity inequality of \eqref{HFHM}}
\setcounter{equation}{0}
\setcounter{theorem}{0}

In this section, we will derive a boundary monotonicity
inequality on \eqref{HFHM}, analogous to Struwe's monotonicity formula, which may have its own interest.

To simplify the presentation, we assume that
$$\Omega=\mathbb R^n, \ T>0, \ {\rm{and}}\ \Gamma(t)=\Gamma=\partial\mathbb R^n_+ \  {\rm{for}}\ 0\le t\le T.$$

Let $u^\pm:\mathbb R^n_\pm\times [0,+\infty)\to N^\pm$, with $u^\pm(x, t)\in M^\pm$ for $(x,t)\in \partial \mathbb R^n_+\times (0,\infty)$, satisfy
\begin{equation}\label{heatflow5}
\begin{cases}
\partial_t u^+-\Delta u^+=A^+(u^+)(\nabla u^+, \nabla u^+)  & {\rm{in}}\ \mathbb R^n_+\times (0,+\infty)\\
\partial_t u^- -\Delta u^-=A^-(u^-)(\nabla u^-, \nabla u^-)  & {\rm{in}}\ \mathbb R^n_-\times (0,+\infty)\\
\Phi^+(u^+)=u^- & {\rm{in}}\ \partial \mathbb R^n_+\times (0,+\infty)\\
(\frac{\partial u^+}{\partial x_n})^T=\Big(D\Phi^+(u^+)\Big)^t\Big[(\frac{\partial u^-}{\partial x_n})^T\Big]
 & {\rm{on}}\ \partial\mathbb R^n_+\times (0,+\infty).
\end{cases}
\end{equation}

For $(x_0,t_0)\in\mathbb R^n\times (0,+\infty)$ and $0<R\le\sqrt{t_0}$,  let
$$G_{(x_0,t_0)}(x,t)=\frac{1}{(4\pi(t_0-t))^{\frac{n}2}}e^{-\frac{|x-x_0|^2}{4(t_0-t)}},
\ (x,t)\in\mathbb R^n\times (0,t_0)
$$
denoet the backward heat kernel on $\mathbb R^n$.
Set
$$E(u^\pm; (x_0,t_0), R)=R^2\Big[\int_{\mathbb R^n_+\times \{t_0-R^2\}}|\nabla u^+|^2G_{(x_0,t_0)}(x,t)\,dx
+\int_{\mathbb R^n_{-}\times \{t_0-R^2\}}|\nabla u^-|^2G_{(x_0,t_0)}(x,t)\,dx\Big].
$$
\begin{lemma} \label{bdry_mono}
Suppose that $(x_0,t_0)=(0,0)\in\partial\mathbb{R}^n_+\times (-\infty, 0]$ and $u^\pm \in C^2(\overline{\mathbb R^n_\pm}\times (-\infty, 0], N^\pm)$ is a solution to the system (\ref{heatflow5}). Then
\begin{equation}
\frac{d}{dR}E(R)\ge 0. \label{bdry_mono1}
\end{equation}
\end{lemma}
\noindent{\bf Proof.\quad}}\def\endpf{\hfill$\Box$ Write $G(x,t)$ for $G_{(0,0)}(x,t)$ and define
$$u_R^\pm(x,t)=u^\pm(Rx,R^2 t), \ (x,t)\in\mathbb R^n\times (-\infty,0].$$
It is easy to see that
$$E(u^\pm; R)=E(u^\pm_R; 1).$$
For simplicity, we only verify (\ref{bdry_mono1}) at $R=1$. Since
$$\frac{d}{dR}\big|_{R=1} u_R^\pm=x\cdot\nabla u^\pm-2\partial_t u^\pm,$$
we have
\begin{eqnarray*}
&&\frac{d}{dR}\Big|_{R=1}E(u^\pm; R)=\frac{d}{dR}\Big|_{R=1}E(u^\pm_R; 1)\\
&=&2\int_{\mathbb R^n_+\times \{-1\}} \nabla u^+\cdot\nabla(x\cdot\nabla u^+-2\partial_t u^+)
e^{-\frac{|x|^2}4}\\
&&+2\int_{\mathbb R^n_{-}\times \{-1\}} \nabla u^-\cdot\nabla(x\cdot\nabla u^--2\partial_t u^-)
e^{-\frac{|x|^2}4}\\
&=&-2\Big[\int_{\mathbb R^n_+\times \{-1\}} \nabla\cdot(\nabla u^+ e^{-\frac{|x|^2}4})\cdot (x\cdot\nabla u^+-2\partial_t u^+)\\
&&+\int_{\mathbb R^n_{-}\times \{-1\}} \nabla\cdot(\nabla u^-e^{-\frac{|x|^2}4})\cdot (x\cdot\nabla u^--2\partial_t u^-)\Big]\\
&&-2\Big[\int_{\partial \mathbb R^n_+\times \{-1\}} \frac{\partial u^+}{\partial x_n}
\cdot (x\cdot\nabla u^+-2\partial_t u^+)e^{-\frac{|x|^2}4}\\
&&-\int_{\partial \mathbb R^n_{-}\times \{-1\}} \frac{\partial u^-}{\partial x_n}
\cdot (x\cdot\nabla u^--2\partial_t u^-)e^{-\frac{|x|^2}4}\Big].\\
\end{eqnarray*}
Since
$$\nabla\cdot(\nabla u^\pm e^{-\frac{|x|^2}4})
=\Delta u^\pm-\frac{1}{2}x\cdot\nabla u^\pm
=\partial_t u^\pm-A^\pm(u^\pm)(\nabla u^\pm, u^\pm)-\frac{1}{2}x\cdot\nabla u^\pm
$$
and
$$A^\pm(u^\pm)(\nabla u^\pm, u^\pm)\cdot
(x\cdot\nabla u^\pm-2\partial_t u^\pm)=0,$$
we have
\begin{eqnarray*}
&&-2\Big[\int_{\mathbb R^n_+\times \{-1\}} \nabla\cdot(\nabla u^+ e^{-\frac{|x|^2}4})\cdot (x\cdot\nabla u^+-2\partial_t u^+)\\
&&+\int_{\mathbb R^n_{-}\times \{-1\}} \nabla\cdot(\nabla u^-e^{-\frac{|x|^2}4})\cdot (x\cdot\nabla u^--2\partial_t u^-)\Big]\\
&=&\big[\int_{\mathbb R^n_+\times \{-1\}} |x\cdot\nabla u^+-2\partial_t u^+|^2
e^{-\frac{|x|^2}4}+\int_{\mathbb R^n_{-}\times \{-1\}} |x\cdot\nabla u^--2\partial_t u^-|^2
 e^{-\frac{|x|^2}4}\big].
\end{eqnarray*}
Since $x=(x',0)$ for $x\in\partial \mathbb R^n_\pm$, and
$$u^\pm\big(\partial\mathbb R^n_\pm\times (-\infty,0)\big)\subset M^\pm,$$
we have
$$x\cdot\nabla u^\pm-2\partial_t u^\pm\big|_{\partial\mathbb R^n_\pm\times (-\infty, 0)}
\in T_{u^\pm(x,t)}M^\pm$$
so that
$$\frac{\partial u^\pm}{\partial x_n}
\cdot (x\cdot\nabla u^\pm-2\partial_t u^\pm)
=\Big(\frac{\partial u^\pm}{\partial x_n}\Big)^T
\cdot (x\cdot\nabla u^\pm-2\partial_t u^\pm)
\ \ {\rm{on}}\ \ {\partial\mathbb R^n_\pm\times (-\infty, 0)}.
$$
Since
$$u^-(x,t)=\Phi^+(u^+(x,t)) \ \ {\rm{on}}\ \ {\partial\mathbb R^n_\pm\times (-\infty, 0)},$$
we have
$$\nabla_{\rm{tan}} u^-(x,t)=D\Phi^+(u^+(x,t))\nabla_{\rm{tan}} u^+(x,t),
\ \partial_t u^-(x,t)=D\Phi^+(u^+(x,t))\partial_t u^+(x,t)$$
$ {\rm{on}}\ \ {\partial\mathbb R^n_\pm\times (-\infty, 0)}$ and hence
$$x\cdot\nabla u^-(x,t)-2\partial_t u^-(x,t)
=D\Phi^+(u^+(x,t))\big(x\cdot\nabla u^+(x,t)-2\partial_t u^+(x,t)\big)
\ {\rm{on}}\ \ {\partial\mathbb R^n_\pm\times (-\infty, 0)}.$$
Therefore we have
\begin{eqnarray*}
&&\big[\int_{\partial \mathbb R^n_+\times \{-1\}} \frac{\partial u^+}{\partial x_n}
\cdot (x\cdot\nabla u^+-2\partial_t u^+)e^{-\frac{|x|^2}4}\\
&&-\int_{\partial \mathbb R^n_{-}\times \{-1\}} \frac{\partial u^-}{\partial x_n}
\cdot (x\cdot\nabla u^--2\partial_t u^-)e^{-\frac{|x|^2}4}\Big]\\
&=&\int_{\partial\mathbb R^n_+\times \{-1\}}
\big[(\frac{\partial u^+}{\partial x_n})^T-(D\Phi^+(u^+(x,t)))^t
(\frac{\partial u^-}{\partial x_n})^T\big]
\cdot (x\cdot\nabla u^+-2\partial_t u^+)e^{-\frac{|x|^2}4}=0,
\end{eqnarray*}
where we have used the boundary condition (\ref{heatflow5})$_5$ in the last step. Putting all
these calculations together, we obtain
\begin{eqnarray}\label{mono_id2}
&&\frac{d}{dR}\Big|_{R=1}E(u^\pm; R)\nonumber\\
&=&\big[\int_{\mathbb R^n_+\times \{-1\}} |x\cdot\nabla u^+-2\partial_t u^+|^2
e^{-\frac{|x|^2}4}+\int_{\mathbb R^n_{-1}\times \{-1\}} |x\cdot\nabla u^--2\partial_t u^-|^2
 e^{-\frac{|x|^2}4}\big]\ge 0.
\end{eqnarray}
This completes the proof. \qed

redpajama_set_name: RedPajamaArXiv


Too much anxiety about a test is commonly referred to as test anxiety. It is perfectly natural to feel some anxiety when preparing for and taking a test. In fact, a little anxiety can jump start your studying and keep you motivated. However, too much anxiety can interfere with your studying. You may have difficulty learning and remembering what you need to know for the test. Further, too much anxiety may block your performance during the test. You may have difficulty demonstrating what you know during the test.
HOW DO I KNOW IF I HAVE TEST ANXIETY?
I have a hard time getting started studying for a test.
When studying for a test, I find many things that distract me.
I expect to do poorly on a test no matter how much or how hard I study.
When taking a test, I experience physical discomfort such as sweaty palms, an upset stomach, a headache, difficulty breathing, and tension in my muscles.
When taking a test, I find it difficult to understand the directions and questions.
When taking a test, I have difficulty organising my thoughts.
When taking a test, I find my mind wandering to other things.
I usually score lower on a test than I do on assignments and papers.
After a test, I remember information I couldn't recall during the test.
WHAT CAN I DO ABOUT TEST ANXIETY?
Here are some things you can do before, during, and after a test to reduce your test anxiety.
Use good study techniques to gain cognitive mastery of the material that will be covered on the test. This mastery will help you to approach the test with confidence rather than have excessive anxiety. Employ the tips we provide at Preparing to Study.
Follow a plan for taking the test such as the DETER strategy we describe at A Strategy for Taking Tests. Don't panic even if you find the test difficult. Stay with your plan!
Don't worry about other students finishing the test before you do. Take the time that you need to do your best.
Once you finish the test and hand it in, forget about it temporarily. There is nothing more you can do until the graded test is returned to you. Turn your attention and effort to new assignments and tests.
When the graded test is returned to you, analyse it to see how you could have done better. Learn from your mistakes and from what you did well. Apply this knowledge when you take the next test.
You have to know the material to do well on a test. You have to control test anxiety to show what you know.


text:
O.V.S. Bulleid
OVSB - Legacy
OVSB - Bristol Display
Loco Classes
Loco - Classes
Loco - Survivors
West Country - 75th
'Merchant Navy'
MN - Intro
MN - History
MN - David's - 21C1
MN - John L's - 21C2
MN - Roger's - 21C3
MN - John C's - 21C6
MN - Russell's
MN - Derby
MN - Charles'
MN - Len's
'CCl' Electric Loco
'Q1' Class
'Q1' - Full size
'Q1' - Valve Gear
'Q1' - Peter
'Q1' - Nick
'Q1' - John
'Q1' - Len
'Q1' - Harry
'Q1' - Rob
'Q1' - Apprentice
'Light' Pacifics
WC - Bernard
WC - Stefan
WC - Martyn
BB - Keith's
WC - Doug
BB - Clive
WC - Richard
WC - Mike's
WC - Alec
WC - Francis
BB - Andrew
BB - Alan
WC - Rob
'36001 - Leader'
'Leader' - Intro
'Leader' - Detail
'Leader' - Kevan's
'Leader' - Kevan's, Pt 2
'Class 12' Shunter
'10201' Diesel
'MN' rebuilds
MN - Mike
MN - Kevin
MN - Bob
MN - Ian
MN - Ross
MN - Bernard
MN - Cliff
MN - John B
MN - Tony Hu
MN - James
MN - Tony Ha
MN - Keith
'11001' Shunter
'4-DD' EMU
'CC1' Turf Burner
Bulleid Books
Published Drawings
Ariel Article Index
Ariel Errors List
Greenly MN Drwgs
Site Background
A web-site by Rob Speare
O.V.S. Bulleid - His Legacy
The period when O.V.S. Bulleid was C.M.E. of the Southern Railway is relatively recent, and much of the infrastructure from the time is still in use.
Being late in the development of steam, many of his designs were not superceeded, and as these locomotives were relatively new, many of the Pacifics were working until the end of B.R. steam, when steam engines were withdrawn en masse.
Many of the Bulleid pacifics that survive came from Dai Woodham's scrapyard. These escaped the cutting torch when individuals and groups began purchasing the locomotives; years later ALL these discarded engines had been saved.
There are no original air-smoothed 'Merchant Navy' class locos now, as many years after their construction and well into the British Railways era, the entire class was re-built, mainly due to problems associated with the enclosed motion oil baths. Their appearance was much changed, stripped of their air-smoothed casing and receiving a new cylindrical smoke box, more resembling the large B.R. standard classes, and possibly the best looking pacific locomotives running in Britain, of their period.
A similar exercise was commenced on the 'West Country' and 'Battle of Britain' class locomotives, but with a change of National Policy now pointing towards diesels, the re-builds stopped with half the 110 members still in the original air-smoothed form. Of the 20 survivors of the smaller Pacifics, 10 are of the as-built air-smoothed design.
35027 'Port Line' was the first ex-Barry 'Merchant Navy' to be restored to working order, having languishing in the open sea air and becoming a rusting hulk at Barry, from March 1967 until leaving for restoration in December 1982.
Having initially been restored to steam in 1974, a major overhaul costing £350,000 returned 35028 'Clan Line' to the main-line in November 2006.
The N.R.M. at York has 35029 'Ellerman Lines' sliced open on permanent display, with a motorised drive rotating the wheels and motion. Battle of Britain class 'Winston Churchill' is also part of the York based national collection.
In preservation, ownership is often in the hands of a group of individuals, often formalised into a Company. Southern Locomotives Ltd. which started as the group that originally restored 'Port Line', now owns five Bulleid pacifics in various states of repair.
Other groups have become closely associated with a particular railway; for example the Bulleid Society owns two Bulleid light pacifics which are based on the Bluebell Railway.
Similarly whilst 34007 'Wadebridge' was originally owned by many small shareholders, the 'MHRPS' now appears to be the major shareholder.
The quirky looking Q1 class worked into the 1960s, but were starting to suffer from rusting of their relatively lightweight construction. C1 is preserved as part on the National collection, and for over 25 years it was kept on the Bluebell Railway. Although not in continuous service there, it did have a major overhaul, and many enthusiasts were able to see it in action. In 2005 it was recalled to the N.R.M at York and has been on display in the exhibition's great hall.
Nothing from Bulleid's final S.R. design, the unconventional Leader, nor the later Irish turf burning version was retained.
Although the three Bulleid 'booster' class locos were scrapped, one member of the later class which used the same 'booster' principle has been preserved. The NRM owned Class 71, E5001 has been restored to green livery.
One of the Bulleid 350hp shunters was bought by the NCB when withdrawn in 1971. It now runs on the Spa Valley line.
And as for the fruition of more recent projects :
34007 'Wadebridge' completed restoration in December 2006, and has run regularly since, although with a borrowed tender.
34070 'Manston', the last locomotive to be built by the Southern (Nov 1947), spent 18 years in Barry scrapyard. After an estimated £350,000 rebuild 'Manston' entered service on the Swanage Railway in 2008.
34059 'Sir Archibald Sinclair', is restored and running on the Bluebell Railway (2009).
34053 'Sir Keith Park', has completed restoration, now running on the Severn Valley Railway (2012).
34081 '92 Squadron', is in an advanced state of restoration at Wandsworth by a dedicated group.
34010 'Sidmouth', has finally commenced restoration at Swanage, from ex Barry condition.
Sadly, it's not all good news. Given the enormous cost of restoration, and on-going funding required to run these large locomotives, there are still a number that have been rescued, but remain rusty carcasses with no prospect of restoration.
But importantly, we do have access to a body of work from a visionary designer of fairly recent history represented in many examples; a number of these are running on preserved railways, allowing us to examine and enjoy.
These will continue to inspire model engineers to build interesting live steam models, as hopefully the associated series of articles can reveal.


Search engines can be a great place to start, but usually your search will send you to a blog or website that is using an image you like, but 9 times out of 10 the site does not source that image. It is important to get your graphics from the source so you know if it is copyright protected or open for use.
To ensure that the images you use on your website and/or blog are properly used you can narrower your search by including forms of licensing in your search.
Like anything on the internet there are hundreds, if not thousands of sources to get images from. Here are a few good articles that list more photo tools and sources.
Where do you go to get your images?


text:
Listings >> Arts & Entertainment >>Event >>Comedy Club
Pretty Funny Women
Comedy Club and Class
www.prettyfunnywomen.com
Longest running, all female stand-up comedy show in the world! PWF has been running strong in several venues for over fifteen years. It features a different line up each show with a roster of over 400 comics, actors and comedy writers. Stand-up has never looked so good!
About the Owner- Lisa Sundstedt is an accomplished stand up comic, writer, singer, actress and producer. Her first professional gig was singing and dancing at Tokyo Disneyland in Japan. She spent another few months singing on an Italian cruise ship, followed by four years on the road with the infamous 50's rock group "Sha Na Na". In 1995 her then-boyfriend said, "pretty women aren't funny!" so Lisa became a stand-up comic and created "Pretty, Funny Women"…the longest running all female comedy show in the country. Their relationship broke up, but her career took off and she immediately landed herself a big time Hollywood agent, a holding deal with Tri Star Television, and the Montreal Comedy Festival where she shared the stage with Ray Romano. She then booked the Chicago Comedy Festival, where she shared a room with Doug Stanhope. She has written on shows like "Girls Behaving Badly", "The Chelsea Handler Show" and "Chelsea Lately", "Coach Charming" for TLC and was a producer on "Parental Discretion with Stefanie Wilder-Taylor" for Nick Mom. She has done hundreds of national commercials and voice-overs, including campaigns for Vons/Safeway, Mattel, Ford, Kraft, and more.
Lisa produces live "Pretty Funny Women" shows at The Comedy Store in Hollywood, Flappers in Burbank & currently teaches stand up comedy classes for women in Los Angeles. Her hobbies include gambling and, well…that's it.
Los Angeles 91423 CA US
Artist Christiana Lewis
Enerji Barre


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