Problem 1 (yau-contest 2015)
Section: Analysis and Differential Equations

Let $f_n \in L^2(R)$ be a sequence of measurable functions over the line, $f_n \to f$ almost everywhere. Let $\|f_n\|_{L^2} \to \|f\|_{L^2}$, prove that $\|f_n - f\|_{L^2} \to 0$.

Problem 2 (yau-contest 2015)
Section: Analysis and Differential Equations

Let $f$ be a continuous function on $[a,b]$, define $M_n = \int_a^b f(x)x^n\,dx$. Suppose that $M_n = 0$ for all integers $n \ge 0$, show that $f(x) = 0$ for all $x$.

Problem 3 (yau-contest 2015)
Section: Analysis and Differential Equations

Determine all entire functions $f$ that satisfying the inequality
$$|f(z)| \le |z|^2|Im(z)|^2$$
for $z$ sufficiently large.

Problem 4 (yau-contest 2015)
Section: Analysis and Differential Equations

Describe all functions that are holomorphic over the unit disk $D = \{z\mid |z|<1\}$, continuous on $\bar{D}$ and map the boundary of the disk into the boundary of the disk.

Problem 5 (yau-contest 2015)
Section: Analysis and Differential Equations

Let $T : H_1 \to H_2$, $Q : H_2 \to H_1$ be bounded linear operators of Hilbert spaces $H_1, H_2$. Let $QT = Id - S_1$, $TQ = Id - S_2$ where $S_1$ and $S_2$ are compact operators. Prove $Ker\,T = \{v \in H_1, Tv = 0\}$, $Coker\,T = H_2/Im(T)$ are finite dimensional and $Im(T) = \{Tv \in H_2, v \in H_1\}$ is closed in $H_2$.

Note: $S$ is compact means for every bounded sequence $x_n \in H_1$, $Sx_n$ has a converging subsequence.

Problem 6 (yau-contest 2015)
Section: Analysis and Differential Equations

Let $H_1$ be the Sobolev space on the unit interval $[0,1]$, i.e. the Hilbert space consisting of functions $f \in L^2([0,1])$ such that
$$\|f\|_1^2 = \sum_{n=-\infty}^{\infty} (1+n^2)|\hat{f}(n)|^2 < \infty;$$
where
$$\hat{f}(n) = \frac{1}{2\pi}\int_0^1 f(x)e^{-2\pi i n x}\,dx$$
are Fourier coefficients of $f$. Show that there exists constant $C>0$ such that
$$\|f\|_{L^\infty} \le C\|f\|_1$$
for all $f \in H_1$, where $\|\cdot\|_{L^\infty}$ stands for the usual supremum norm. (Hint: Use Fourier series.)