4.3 Minimizing the energy of vector fields on surfaces of revolution  (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 $L^2$ :

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.

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 ${\varphi }_t,{\varphi }_{\theta }$ .

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

then $E_H\le E$ . 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

Lemma 4

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

Proof:

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 $f_m\in C_c^{\infty }\left((0,1)\right)$ 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