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Lemma 2

We may replace V k with another sequence so that all the ϕ k are continuous and differentiable and are 2 π - periodic.

Proof:

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

Let H be the following subset of the set of square integrable functions L 2 :

H = ϕ L 2 : there is a sequence ϕ k C such that | ϕ - ϕ k | 2 0 .

We claim that every function ϕ H has a weak derivative. To do this, suppose that ϕ is a function which is an L 2 limit of a sequence ϕ k C . Since E ( ϕ ) < E + 1 , then we may show that the sequence ( ϕ k ) t converges weakly to a limit ϕ t . To do so, observe that if f k is a sequence of functions in L 2 such that | | f k | | L 2 is bounded, then there is a weakly convergent subsequence.

Lemma 3

The derivatives ϕ k , θ , ϕ k , t are a bounded sequence in L 2 .

Proof:

By Lemma 1, E ( ϕ k ) E ( ϕ ) = E for a finite E . Recall the definition of E = c o s 2 ϕ + ϕ t 2 + ϕ θ 2 d t d θ . 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 ϕ k , θ , ϕ k , t are a bounded sequence in L 2 , they have corresponding subsequences which converge weakly to functions ϕ t , ϕ θ .

Hence, we can define the weak energy E ( ϕ ) for any function ϕ H . Let

E H = inf ϕ H E ( ϕ ) ,

then E H E . We will show that this inf is attained.

Take a sequence ϕ k H so that E ( ϕ k ) E H . Here we prove that a subsequence ϕ k have an L 2 limit ϕ H .

For this, note that by definition of H we can choose for each ϕ k a smooth function ϕ k C so that

ϕ k - ϕ k L 2 < 1 k .

Lemma 4

There is a continuous function ϕ such that ϕ k converge uniformly to ϕ .

Proof:

| ϕ k ( x ) - ϕ k ( y ) | = y x ϕ k ' ( t ) d t y x d t 1 2 y x ϕ ' k 2 ( t ) 1 2
= | x - y | 1 2 ϕ ' k 2 ( t ) 1 2 | x - y | ( E + 1 ) 2 π

Hence, the sequence ϕ k converges uniformly to some function ϕ . By the triangle inequality, ϕ k ϕ in L 2 , as well. Since H is a closed set in L 2 , then ϕ H and accordingly, ϕ has a weak derivative ϕ t .

Lemma 5

The sequence ( ϕ k ) t converges weakly to ϕ t .

Proof:

We take advantage of the fact that for every f L 2 , there is a sequence of test functions f m C c ( ( 0 , 1 ) ) such that f m f in L 2 .

So then, take any f L 2 . We need to show that

( ϕ k ) t f - ϕ t f 0 .

For this, take using the triangle inequality:

( ϕ k ) t f - ϕ t f | ( ϕ k ) t | | f - f m | + ( ( ϕ k ) t - ϕ t ) f m + | ϕ t | | f - f m |

and using Cauchy-Schwartz:

( ϕ k ) t L 2 f - f m L 2 + ( ( ϕ k ) t - ϕ t ) f m + ϕ t L 2 f - f m L 2

using the definition of the sequence ϕ k :

( E H + 1 ) f - f m L 2 + ( ( ϕ k ) t - ϕ t ) f m + ϕ t L 2 f - f m L 2

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

= ( E H + 1 ) f - f m L 2 + ( ϕ k - ϕ ) ( f m ) t + ϕ t L 2 f - f m L 2

and using Cauchy-Schwartz again we have:

( ϕ k ) t f - ϕ t f
( E H + 1 ) f - f m L 2 + ϕ k - ϕ L 2 ( f m ) t L 2 + ϕ t L 2 f - f m L 2 .

Letting k FIRST, and then m , we get the result.

By Lemma 3, since ( ϕ k ) t ϕ t weakly, we get

ϕ t 2 lim k ( ϕ k ) t 2 .

We may show that c o s 2 ϕ k c o s 2 ϕ . Because f ( x ) = c o s 2 ( x ) is continuous, c o s 2 ( x ) c o s ( y ) as | x - y | 0 ; thus, because ϕ k - ϕ 0 by definition of weak convergence, c o s 2 ϕ k c o s 2 ϕ .

We have shown that the components ϕ t 2 and c o s 2 ϕ allow ϕ to attain the minimum E H . Now, note that if η is a test function, then ϕ + η H , and so we automatically get

Questions & Answers

what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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