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

Let $\phi \in C([a,b], \mathbb{R})$. Suppose for every function $h \in C^1([a,b], \mathbb{R})$, $h(a) = h(b) = 0$, we have
$$\int_a^b \phi(x) h(x)\, dx = 0.$$
Prove that $\phi(x) = 0$.

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

Let $f$ be a Lebesgue integrable function over $[a, b+\delta]$, $\delta > 0$, prove that
$$\lim_{h\to 0^+} \int_a^b |f(x+h) - f(x)|\, dx \to 0.$$

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

Let $L(q, q', t)$ be a function of $(q, q', t) \in TU \times \mathbb{R}$, $U$ is an open domain in $\mathbb{R}^n$. Let $\gamma : [a,b] \to U$ be a curve in $U$. Define a functional
$$S(\gamma) = \int_a^b L(\gamma(t), \gamma'(t), t)\, dt.$$
We say that $\gamma$ is an extremal if for every smooth variation of $\gamma$, $\phi(t,s)$, $s \in (-\delta, \delta)$, $\phi(t,0) = \gamma(t)$, $\phi_s = \phi(t,s)$, we have $\left.\frac{dS(\phi_s)}{ds}\right|_{s=0} = 0$. Prove that every extremal $\gamma$ satisfies the Euler-Lagrange equation:
$$\frac{d}{dt}\left(\frac{\partial L}{\partial q'_j}\right) = \frac{\partial L}{\partial q_j}.$$

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

Let $f : U \to U$ be a holomorphic function with $U$ a bounded domain in the complex plane. Assuming $0 \in U$, $f(0) = 0$, $f'(0) = 1$, prove that $f(z) = z$.

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

Let $T : H_1 \to H_2$ be a bounded operator of Hilbert spaces $H_1, H_2$. Let $S : H_1 \to H_2$ be a compact operator, that is, for every bounded sequence $\{v_n\} \in H_1$, $S v_n$ has a converging subsequence. Show that $\mathrm{Coker}(T+S) = H_2 / \mathrm{Im}(T+S)$ is finite dimensional and $\mathrm{Im}(T+S)$ is closed in $H_2$. (Hint: Consider equivalent statements in terms of adjoint operators.)

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

Let $u \in C^2(\overline{\Omega})$, $\Omega \subset \mathbb{R}^d$ is a bounded domain with a smooth boundary.

1) Let $u$ be a solution of the equation $\Delta u = f$, $u|_{\partial\Omega} = 0$, $f \in L^2(\Omega)$. Prove that there is a constant $C$ depends only $\Omega$ such that
$$\int_{\Omega} \left(\sum_{j=1}^n \left(\frac{\partial u}{\partial x_j}\right)^2 + u^2\right) dx \le C \int_{\Omega} f^2(x)\, dx.$$

2) Let $\{u_n\}$ be a sequence of harmonic functions on $\Omega$, such that $\|u_n\|_{L^2(\Omega)} \le M < \infty$, for a constant $M$ independent of $n$. Prove that there is a converging subsequence $\{u_{n_k}\}$ in $L^2(\Omega)$.