Problem 1 (yau-contest 2011)
Section: Analysis and Differential Equations

a) Compute the integral: $\int_{-\infty}^{\infty} \frac{x \cos \alpha x}{(x^2+1)(x^2+2^2)}\, dx$.

b) Show that there is a continuous function $f : [0,+\infty) \to (-\infty,+\infty)$ such that $f \ne 0$ and $f(4x) = f(2x) + f(x)$.

Problem 2 (yau-contest 2011)
Section: Analysis and Differential Equations

Solve the following problem:
$$
\begin{cases}
\dfrac{d^2 u}{dx^2} - u(x) = 4e^{-x}, & x \in (0,1),\\
u(0) = 0, & \dfrac{du}{dx}(0) = 0.
\end{cases}
$$

Problem 3 (yau-contest 2011)
Section: Analysis and Differential Equations

Find an explicit conformal transformation of an open set $U=\{\lvert z\rvert>1\}\setminus(-\infty,-1]$ to the unit disc.

Problem 4 (yau-contest 2011)
Section: Analysis and Differential Equations

Assume $f\in C^2[a,b]$ satisfying $\lvert f(x)\rvert\le A$, $\lvert f'(x)\rvert\le B$ for each $x\in[a,b]$ and there exists $x_0\in[a,b]$ such that $\lvert f''(x_0)\rvert\le D$, then $\lvert f'(x)\rvert\le 2\sqrt{AB}+D$, $\forall x\in[a,b]$.

Problem 5 (yau-contest 2011)
Section: Analysis and Differential Equations

Let $C([0,1])$ denote the Banach space of real valued continuous functions on $[0,1]$ with the sup norm, and suppose that $X\subset C([0,1])$ is a dense linear subspace. Suppose $l:X\to\mathbb{R}$ is a linear map (not assumed to be continuous in any sense) such that $l(f)\ge 0$ if $f\in X$ and $f\ge 0$. Show that there is a unique Borel measure $\mu$ on $[0,1]$ such that $l(f)=\int f\,d\mu$ for all $f\in X$.

Problem 6 (yau-contest 2011)
Section: Analysis and Differential Equations

For $s\ge 0$, let $H^s(T)$ be the space of $L^2$ functions on the circle $T=\mathbb{R}/(2\pi\mathbb{Z})$ whose Fourier coefficients $\hat f_n=\int_0^{2\pi} e^{-inx} f(x)\,dx$ satisfy $\sum (1+n^2)^s\lvert\hat f_n\rvert^2<\infty$, with norm $\lVert f\rVert_s^2=(2\pi)^{-1}\sum (1+n^2)^s\lvert\hat f_n\rvert^2$.

a. Show that for $r>s\ge 0$, the inclusion map $i:H^r(T)\to H^s(T)$ is compact.

b. Show that if $s>1/2$, then $H^s(T)$ includes continuously into $C(T)$, the space of continuous functions on $T$, and the inclusion map is compact.