Isoperimetric Inequality : in differentiable $C^1$-case.

Question

I'm studying Fourier series on these days, and I'm going to ask one of them because there are very many blocked parts in the practice questions.

Is is applied isoperimetric ineqality in particular case. I'm very glad if you give me an answer.

The question is like this.

Differential periodic function $f:\mathbb R\to\mathbb C $ satisfies these conditions.

• Period of $f$ is $2\pi$.
• Over the interval $(-\pi,\pi]$, it is injective and $|f'|=1$.
• n-th Fourier coefficient of $f$ is $\hat{f}(n)$.

Then answer following these question.

1. Proove this : Length of the simple closed curve created by the image of $f$ is $2\pi\sum_{-\infty}^{\infty}n^2|\hat f(n)|^2$.
2. Using Green's Theorem, proove this : The area surrounded by this curve is $\frac1{4i} \int_{-\pi}^{\pi}(f'(t)\bar{f(t)}-f(t)\bar{f'(t)})dt$.
3. Proove this : The area surrounded by this curve is $\pi\sum_{-\infty}^{\infty}n|\hat f(n)|^2$.
4. When is the area surrounded by a curve which has constant length the largest?

I think the 4th problem will be easily done by using Cauchy-Schwarz inequality with 1st problem and 3rd problem and it is not clear.

3rd problem is maybe clear, expressing $f'(t)\bar{f(t)}-f(t)\bar{f'(t)}$ as sum of the Fourier series and just integral them.

So my problem is 1st, 2nd. I thought about it for a week, but I didn't get any clues.

And I cannot understand. Why Green's Theorem is used here? Wasn't the Green's Theorem is for the line integral?

Answer 1

Here's a couple hints to help you solve the problem.

(1): The arclength of a curve parametrized by $\mathbf{r}(t)=(u(t),v(t)), t\in(-\pi,\pi)$ or equivalently $f(t)=u(t)+iv(t)$ is given by

$$L=\int_{-\pi}^{\pi} \sqrt{\left(\frac{du}{dt}\right)^2+\left(\frac{dv}{dt}\right)^2}=\int_{-\pi}^{\pi}|f'(t)|dt=2\pi$$

Because of $|f'|=1$ it is also true that $L=\int_0^{2\pi} |f'(t)|^2 dt$. Expand in Fourier modes to obtain the result.

(2): It is a consequence of Green's theorem that the area of a closed curve is given by

$$A= \frac{1}{2}\oint dt \left(u(t)\frac{dv(t)}{dt}- v(t)\frac{du(t)}{dt}\right)$$

Note however that

$$u(t)=\frac{f(t)+\bar{f}(t)}{2}, v(t)=\frac{f(t)-\bar{f}(t)}{2i} $$

(3): As done in (1), expand (2) in Fourier modes.

(4): Note that $n\leq n^2 ~\forall ~n\in \mathbb{Z}$. Then it must be true that

$$\sum_n n|\hat{f}(n)|^2\leq \sum_n n^2|\hat{f}(n)|^2\Rightarrow A\leq \frac{L}{2} $$

When does the equality hold here? Which Fourier modes can be non-zero? Determining this should give a simple curve that satisfies the equality constraint.