# 4.3 Minimizing the energy of vector fields on surfaces of revolution  (Page 2/4)

 Page 2 / 4

## Lemma 2

We may replace ${V}_{k}$ with another sequence so that all the ${\varphi }_{k}$ are continuous and differentiable and are $2\pi -$ periodic.

## Proof:

Let ${\varphi }_{1}$ be a linear, piecewise approximation of $\varphi$ ( $\varphi$ and ${\varphi }_{1}$ have the same values at the endpoints of the interval). Because $\varphi$ is piecewise continuous, the square integral of the approximation ${\varphi }_{1}$ will also approximate the square integral of $\varphi$ . The “edges” of ${\varphi }_{1}$ may be smoothed out by composing ${\varphi }_{1}$ with functions of the form ${e}^{\frac{1}{x}}$ while retaining its properties of approximation.

Let $\mathcal{H}$ be the following subset of the set of square integrable functions ${L}^{2}$ :

$\mathcal{H}=\left\{\varphi ,\in ,{\mathcal{L}}^{2},:,\text{there},\phantom{\rule{4.pt}{0ex}},\text{is},\phantom{\rule{4.pt}{0ex}},\text{a},\phantom{\rule{4.pt}{0ex}},\text{sequence},\phantom{\rule{4.pt}{0ex}},{\varphi }_{k},\in ,C,\phantom{\rule{4.pt}{0ex}},\text{such},\phantom{\rule{4.pt}{0ex}},\text{that},\int ,{|\varphi -{\varphi }_{k}|}^{2},\to ,0\right\}.$

We claim that every function $\varphi \in H$ has a weak derivative. To do this, suppose that $\varphi$ is a function which is an ${\mathcal{L}}^{2}$ limit of a sequence ${\varphi }_{k}\in C$ . Since $E\left(\varphi \right) , then we may show that the sequence ${\left({\varphi }_{k}\right)}_{t}$ converges weakly to a limit ${\varphi }_{t}$ . To do so, observe that if ${f}_{k}$ is a sequence of functions in ${\mathcal{L}}^{2}$ such that $||{f}_{k}{||}_{{\mathcal{L}}^{2}}$ is bounded, then there is a weakly convergent subsequence.

## Lemma 3

The derivatives ${\varphi }_{k,\theta }$ , ${\varphi }_{k,t}$ are a bounded sequence in ${\mathcal{L}}^{2}$ .

## Proof:

By Lemma 1, $E\left({\varphi }_{k}\right)\to E\left(\varphi \right)=E$ for a finite $E$ . Recall the definition of $E=\int \int co{s}^{2}\varphi +{\varphi }_{t}^{2}+{\varphi }_{\theta }^{2}dtd\theta$ . Because $E$ is the integral finite sum of these positive quantities, these positive quantities must in turn be finite and thus, bounded.

Then, because these derivatives ${\varphi }_{k,\theta }$ , ${\varphi }_{k,t}$ are a bounded sequence in ${L}^{2}$ , they have corresponding subsequences which converge weakly to functions ${\varphi }_{t},\phantom{\rule{0.166667em}{0ex}}{\varphi }_{\theta }$ .

Hence, we can define the weak energy $E\left(\varphi \right)$ for any function $\varphi \in \mathcal{H}$ . Let

${E}_{H}=\underset{\varphi \in \mathcal{H}}{inf}E\left(\varphi \right),$

then $\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{E}_{H}\le E$ . We will show that this inf is attained.

Take a sequence ${\varphi }_{k}\in \mathcal{H}$ so that $E\left({\varphi }_{k}\right)\to {E}_{\mathcal{H}}.$ Here we prove that a subsequence ${\varphi }_{k}$ have an ${\mathcal{L}}^{2}$ limit $\varphi \in \mathcal{H}.$

For this, note that by definition of $\mathcal{H}$ we can choose for each ${\varphi }_{k}$ a smooth function ${\varphi }_{k}^{\infty }\in \mathcal{C}$ so that

$\parallel {\varphi }_{k}-{\varphi }_{k}^{\infty }{\parallel }_{{\mathcal{L}}^{2}}<\frac{1}{k}.$

## Lemma 4

There is a continuous function $\varphi$ such that ${\varphi }_{k}$ converge uniformly to $\varphi$ .

## Proof:

$|{\varphi }_{k}\left(x\right)-{\varphi }_{k}\left(y\right)|=\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left|{\int }_{y}^{x},{\varphi }_{k}^{\text{'}},\left(t\right),d,t\right|\le {\left({\int }_{y}^{x},d,t\right)}^{\frac{1}{2}}{\left({\int }_{y}^{x},\varphi ,{\text{'}}_{k}^{2},\left(t\right)\right)}^{\frac{1}{2}}$
$={|x-y|}^{\frac{1}{2}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\left(\int ,\varphi ,{\text{'}}_{k}^{2},\left(t\right)\right)}^{\frac{1}{2}}\le \sqrt{\frac{|x-y|\left(E+1\right)}{2\pi }}$

Hence, the sequence ${\varphi }_{k}^{\infty }$ converges uniformly to some function $\varphi$ . By the triangle inequality, ${\varphi }_{k}\to \varphi$ in ${\mathcal{L}}^{2}$ , as well. Since $\mathcal{H}$ is a closed set in ${\mathcal{L}}^{2}$ , then $\varphi \in \mathcal{H}$ and accordingly, $\varphi$ has a weak derivative ${\varphi }_{t}$ .

## Lemma 5

The sequence ${\left({\varphi }_{k}\right)}_{t}$ converges weakly to ${\varphi }_{t}$ .

## Proof:

We take advantage of the fact that for every $f\in {\mathcal{L}}^{2}$ , there is a sequence of test functions ${f}_{m}\in {C}_{c}^{\infty }\left(\left(0,1\right)\right)$ such that ${f}_{m}\to f$ in ${\mathcal{L}}^{2}$ .

So then, take any $f\in {\mathcal{L}}^{2}.$ We need to show that

$\left|\int ,{\left({\varphi }_{k}\right)}_{t},f,-,\int ,{\varphi }_{t},f\right|\to 0.$

For this, take using the triangle inequality:

$\left|\int ,{\left({\varphi }_{k}\right)}_{t},f,-,\int ,{\varphi }_{t},f\right|\le \int |{\left({\varphi }_{k}\right)}_{t}||f-{f}_{m}|+\left|\int ,\left({\left({\varphi }_{k}\right)}_{t}-{\varphi }_{t}\right),{f}_{m}\right|+\int |{\varphi }_{t}||f-{f}_{m}|$

and using Cauchy-Schwartz:

$\le \parallel {\left({\varphi }_{k}\right)}_{t}{\parallel }_{{\mathcal{L}}^{2}}\parallel f-{f}_{m}{\parallel }_{{\mathcal{L}}^{2}}+\left|\int ,\left({\left({\varphi }_{k}\right)}_{t}-{\varphi }_{t}\right),{f}_{m}\right|+\parallel {\varphi }_{t}{\parallel }_{{\mathcal{L}}^{2}}{\parallel f-{f}_{m}\parallel }_{{\mathcal{L}}^{2}}$

using the definition of the sequence ${\varphi }_{k}:$

$\le \left({E}_{H}+1\right)\parallel f-{f}_{m}{\parallel }_{{\mathcal{L}}^{2}}+\left|\int ,\left({\left({\varphi }_{k}\right)}_{t}-{\varphi }_{t}\right),{f}_{m}\right|+\parallel {\varphi }_{t}{\parallel }_{{\mathcal{L}}^{2}}{\parallel f-{f}_{m}\parallel }_{{\mathcal{L}}^{2}}$

and using the definition of the weak derivative, since ${f}_{m}$ is a smooth test function:

$=\left({E}_{H}+1\right)\parallel f-{f}_{m}{\parallel }_{{\mathcal{L}}^{2}}+\left|\int ,\left({\varphi }_{k}-\varphi \right),{\left({f}_{m}\right)}_{t}\right|+\parallel {\varphi }_{t}{\parallel }_{{\mathcal{L}}^{2}}{\parallel f-{f}_{m}\parallel }_{{\mathcal{L}}^{2}}$

and using Cauchy-Schwartz again we have:

$\left|\int ,{\left({\varphi }_{k}\right)}_{t},f,-,\int ,{\varphi }_{t},f\right|\le$
$\left({E}_{H}+1\right)\parallel f-{f}_{m}{\parallel }_{{\mathcal{L}}^{2}}+\parallel {\varphi }_{k}{-\varphi \parallel }_{{\mathcal{L}}^{2}}\parallel {\left({f}_{m}\right)}_{t}{\parallel }_{{\mathcal{L}}^{2}}+\parallel {\varphi }_{t}{\parallel }_{{\mathcal{L}}^{2}}{\parallel f-{f}_{m}\parallel }_{{\mathcal{L}}^{2}}.$

Letting $k\to \infty$ FIRST, and then $m\to \infty ,$ we get the result.

By Lemma 3, since ${\left({\varphi }_{k}\right)}_{t}\to {\varphi }_{t}$ weakly, we get

$\int {\varphi }_{t}^{2}\le \underset{k\to \infty }{lim}\int {\left({\varphi }_{k}\right)}_{t}^{2}.$

We may show that $\int co{s}^{2}{\varphi }_{k}\to \int co{s}^{2}\varphi$ . Because $f\left(x\right)=co{s}^{2}\left(x\right)$ is continuous, $co{s}^{2}\left(x\right)\to cos\left(y\right)$ as $|x-y|\to 0$ ; thus, because $\parallel {\varphi }_{k}-\varphi \parallel \to 0$ by definition of weak convergence, $co{s}^{2}{\varphi }_{k}\to co{s}^{2}\varphi$ .

We have shown that the components $\int {\varphi }_{t}^{2}$ and $\int co{s}^{2}\varphi$ allow $\varphi$ to attain the minimum ${E}_{H}$ . Now, note that if $\eta$ is a test function, then $\varphi +\eta \in \mathcal{H},$ and so we automatically get

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers! By Rhodes By Rhodes By Rohini Ajay By Yacoub Jayoghli By Saylor Foundation By Savannah Parrish By Brooke Delaney By Kevin Moquin By Saylor Foundation By Nicole Duquette