Problem 1 (yau-contest 2018)
Section: Analysis

Let the sequence of functions $\{f_n\}_{n=1}^\infty$ in $L^2(\mathbb{R}^d)$ satisfy that $\|f_n\|_{L^2} = 1$.

(1) Show that there exists a subsequence of functions $\{f_{n_j}\}_{j=1}^\infty$ such that $f_{n_j}$ converges weakly to some function $f$ in $L^2(\mathbb{R}^d)$, i.e.
$$ (f_{n_j}, g) \to (f, g) $$
for all $g \in L^2(\mathbb{R}^d)$.

(2) If $f_n \to f$ weakly in $L^2(\mathbb{R}^d)$, and $\|f_n\|_{L^2} \to \|f\|_{L^2}$ as $n \to \infty$. Show that $\|f_n - f\|_{L^2} \to 0$ as $n \to \infty$.

Problem 2 (yau-contest 2018)
Section: Analysis

Let $f : U \to \mathbb{C}$ be a non-constant holomorphic function where $U \subset \mathbb{C}$ is the open set containing the closure $\overline{D}$ of the unit disk $D = \{ z \in \mathbb{C} \mid |z| < 1 \}$. If $|f(z)| = 1$, for all $z \in \partial D$, Prove that $D \subset f(\overline{D})$.

Problem 3 (yau-contest 2018)
Section: Analysis

Prove that if a sequence of harmonic function on the open disk converges uniformly on compact subset of the disk, then the limit is harmonic.

Problem 4 (yau-contest 2018)
Section: Analysis

Let $\mu$ be a Borel measure on $\mathbb{R}^n$. Let $\rho > 0$, a fixed positive number, and $B_{\rho}(x) = \{ y \in \mathbb{R}^n \mid d(x,y) < \rho \}$. For $x \in \mathbb{R}^n$, define a function:
$$\theta(x) : x \mapsto \mu\big(B_{\rho}(x)\big)$$
1) Show that $\theta$ is upper semi-continuous, i.e. for every $x \in \mathbb{R}^n$, $\theta(x) \ge \limsup_{y \to x} \theta(y)$.
2) Give an example of a Borel measure $\mu$, such that the function $\theta$ is not continuous.

Problem 5 (yau-contest 2018)
Section: Analysis

Let $g$ denote a smooth function on $\mathbb{R}^n$ with compact support. Let $f$ denote the function given by the formula
$$ f(x) = \frac{1}{n(n-1)\alpha(n)} \int_{\mathbb{R}^n} \frac{1}{|x-y|^{n-2}}\, g(y)\, dy. $$
Here $\alpha(n)$ is volume of the unit ball in $\mathbb{R}^n$.
(a) Prove that the integral that defines $f$ converges for each $x \in \mathbb{R}^n$.
(b) Prove that $f$ is differentiable and that the gradient of $f$ is given by the formula
$$ \nabla f\big|_{x} = \frac{1}{n(n-1)\alpha(n)} \int_{\mathbb{R}^n} \frac{1}{|x-y|^{n-2}}\, (\nabla g)\big|_{y} \, dy. $$
(c) Prove the $f$ obeys $-\Delta f = g$ with $\Delta$ denoting $\displaystyle \frac{\partial^2}{\partial x_1^2} + \cdots + \frac{\partial^2}{\partial x_n^2}$.

Problem 6 (yau-contest 2018)
Section: Analysis

If $u$ is a positive harmonic function on $\mathbb{R}^n \setminus \{0\}$ ($n \ge 2$), then exist constants $a \ge 0, b \ge 0$ such that
$$ u(x) = a + b|x|^{2-n} $$
for all $x \in \mathbb{R}^n \setminus \{0\}$.