Problem 1 (yau-contest 2017)
Section: Analysis

Suppose $f$ is integrable on $[-\pi,\pi]$, prove that $\sum_{n=-\infty}^{\infty} a_n r^{|n|} e^{inx}$ tends to $f(x)$ for a.e. $x$, as $r\to1$, $r<1$. Here $$a_n=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)e^{-inx}\,dx.$$

Problem 2 (yau-contest 2017)
Section: Analysis

Let $H$ be a Hilbert space equipped with an inner product $(\cdot,\cdot)$ and a norm $\|\cdot\|=(\cdot,\cdot)^{1/2}$. A sequence $\{f_k\}$ is converge to $f\in H$ if $\|f_k-f\|\to0$. A sequence $\{f_k\}\subset H$ is said converge weakly to $f\in H$ if $(f_k,g)\to(f,g)$ for any $g\in H$. Prove the following statements:

a) $\{f_k\}$ converges to $f$ if and only if $\|f_k\|\to\|f\|$ and $\{f_k\}$ converges weakly to $f$.

b) If $H$ is a finite dimensional Hilbert space, then the weak convergence implies convergence. Give a counter example to show that weak convergence does not necessarily imply convergence in an infinite dimensional Hilbert space.

Problem 3 (yau-contest 2017)
Section: Analysis

Let $f:\mathbb{C}/\{0\}\to\mathbb{C}$ be a holomorphic function and
$$|f(z)|\le |z|^{2}+\frac{1}{|z|^{1/2}},$$
for $z$ near $0$. Determine all such functions.

Problem 4 (yau-contest 2017)
Section: Analysis

Find a conformal mapping which maps the region $\{z\mid |z-i|<\sqrt{2},\,|z+i|<\sqrt{2}\}$ onto the unit disk.

Problem 5 (yau-contest 2017)
Section: Analysis

If $E$ is a compact set in a region $\Omega$, prove that there exists a constant $M>0$, depending only on $E$ and $\Omega$, such that every positive harmonic function $u(z)$ in $\Omega$ satisfies $$u(z_2)\le M\,u(z_1)$$ for any two points $z_1,z_2\in E.$

Problem 6 (yau-contest 2017)
Section: Analysis

1) For any bounded domain $\Omega\subset\mathbb{R}^n$, there exists a smallest constant $C(\Omega)$, such that
$$\int_{\Omega}|u|^2\,dx\le C(\Omega)\int_{\Omega}\sum_{i=1}^{n}\left|\frac{\partial u}{\partial x_i}\right|^2\,dx$$
for every function $u\in H_0^1(\Omega)=\overline{C_0^\infty(\Omega)}\subset H^1(\Omega)$, where $C_0^\infty(\Omega)$ is the space of smooth functions over $\Omega$ and vanishing on boundary of $\Omega$ and $H^1(\Omega)$ is the Banach space of functions $u\in L^2(\Omega)$, $\nabla u\in L^2(\Omega)^n$ with the norm:
$$\|u\|_{H^1(\Omega)}=\|u\|_{L^2(\Omega)}+\|\nabla u\|_{L^2(\Omega)^n}
=\int_{\Omega}\left(|u|^2+\sum_{i=1}^{n}\left|\frac{\partial u}{\partial x_i}\right|^2\right)\,dx.$$

2) Let $\Pi=\{(x,y)\mid 0<x<a,\,0<y<b\}$, show that $$C(\Pi)\ge \frac{a^2 b^2}{\pi^2(a^2+b^2)}.$$