<< Chapter < Page | Chapter >> Page > |
We may replace ${V}_{k}$ with another sequence so that all the ${\varphi}_{k}$ are continuous and differentiable and are $2\pi -$ periodic.
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 ${L}^{2}$ :
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 $E\left(\varphi \right)<E+1$ , 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 $\left|\right|{f}_{k}{\left|\right|}_{{\mathcal{L}}^{2}}$ is bounded, then there is a weakly convergent subsequence.
The derivatives ${\varphi}_{k,\theta}$ , ${\varphi}_{k,t}$ are a bounded sequence in ${\mathcal{L}}^{2}$ .
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
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
There is a continuous function $\varphi $ such that ${\varphi}_{k}$ converge uniformly to $\varphi $ .
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 ${\varphi}_{t}$ .
The sequence ${\left({\varphi}_{k}\right)}_{t}$ converges weakly to ${\varphi}_{t}$ .
We take advantage of the fact that for every $f\in {\mathcal{L}}^{2}$ , there is a sequence of test functions ${f}_{m}\in {C}_{c}^{\infty}\left((0,1)\right)$ such that ${f}_{m}\to f$ in ${\mathcal{L}}^{2}$ .
So then, take any $f\in {\mathcal{L}}^{2}.$ We need to show that
For this, take using the triangle inequality:
and using Cauchy-Schwartz:
using the definition of the sequence ${\varphi}_{k}:$
and using the definition of the weak derivative, since ${f}_{m}$ is a smooth test function:
and using Cauchy-Schwartz again we have:
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
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
Notification Switch
Would you like to follow the 'The art of the pfug' conversation and receive update notifications?