Problem 1 (yau-contest 2023)
Section: Analysis and Differential Equations

Let $(M_n, \|\cdot\|)$ be the Banach space formed by $n \times n$ real matrices, equipped with Hilbert-Schmidt norm. Prove:

(1) If $\gamma : \mathbb{R} \to M_n$ is a $C^1$-function with $\gamma(0) = I$ (the identity matrix) and $\dot{\gamma}(0) = A \in M_n$, then for $\forall t \in \mathbb{R}$, the sequence $\{\gamma^{n}(t/n)\}_{n=1}^{\infty}$ converges to $\exp(tA)$; in particular, for $\forall A, B \in M_n$, $e^{t(A+B)} = \lim_{n\to\infty} (e^{tA/n} e^{tB/n})^{n}$.

(2) For $\forall A, B \in M_n$, if $e^{t(A+B)} = e^{A} e^{B}$, $\forall t \in \mathbb{R}$, then $[A,B] := AB - BA = 0.$

Problem 2 (yau-contest 2023)
Section: Analysis and Differential Equations

Let $f(t,x,y)$ be a $C^1$ function on $[0,1] \times \mathbb{R}^2$. Let $\varphi$ be a solution of the second-order ordinary differential equation $(*)$
$$\frac{d^2\varphi}{dt^2} = f\big(t, x, \frac{d\varphi}{dt}, \varphi(t)\big), \quad t \in [0,1],$$
such that $\varphi(0)=a$, $\varphi(1)=b$ where $a,b$ are given real numbers. Suppose $\frac{\partial f}{\partial t}(t,x,y) > 0$ for all $(t,x,y) \in [0,1] \times \mathbb{R}^2$. Prove: if $|\beta - b|$, $\beta \in \mathbb{R}$, is sufficiently small, then there exists a solution $\psi$ of $(*)$ such that $\psi(0)=a$, $\psi(1)=\beta$.

Problem 3 (yau-contest 2023)
Section: Analysis and Differential Equations

A function $\varphi : \mathbb{R}^n \to \mathbb{C}$ is called positive definite if for all $k \in \mathbb{N}$, $y_j \in \mathbb{R}^n$, $c_j \in \mathbb{C}$, $j = 1, \ldots, k$, one have $\sum_{i,j=1}^{k} c_i \\overline{c_j} \, \varphi(y_i - y_j) \ge 0$. If $\varphi$ is a measurable positive definite function on $\mathbb{R}^n$. Prove:

(1) $\varphi(-y) = \varphi(y)$ and $|\varphi(y)| \le \varphi(0)$.

(2) For every Lebesgue integrable nonnegative function $f$ on $\mathbb{R}^n$, one have
$$\int_{\mathbb{R}^n} \varphi(x-y) f(x) f(y) \, dx \, dy \ge 0.$$

(3) If the function $f$ is also even, then
$$\int_{\mathbb{R}^n} \varphi(x) \, f * f(x) \, dx \ge 0,$$
where $*$ stands for convolution.

(4) For all $\alpha > 0$, we have
$$\int_{\mathbb{R}^n} \varphi(x) \exp(-\alpha |x|^2) \, dx \ge 0.$$

Problem 4 (yau-contest 2023)
Section: Analysis and Differential Equations

For a $2\pi$-periodic function $x \in L^2([0,2\pi])$, let $x_n(t) = x(nt)$, $n = 1, 2, \ldots$. Prove that $\{x_n\}$ converges weakly in $L^2([0,2\pi])$ and find the limit.

Problem 5 (yau-contest 2023)
Section: Analysis and Differential Equations

Let $f$ be an entire function on $\mathbb{C}$ and there exists a constant $C > 0$ such that $f(z) \le \frac{C}{\sqrt{|z|}} \, |\cos(z)|$ for all $z \in \mathbb{C}$. Prove that $f$ is identically zero.

Problem 6 (yau-contest 2023)
Section: Analysis and Differential Equations

Let $u(t,x)$ satisfy the following equation,
$$u_t + \sum_{i=1}^{n} \psi_i(t, x, u) \, u_{x_i} = \mu \, \Delta u; \quad u(0,x) = u_0,$$
where $u_0 \in C^2(\mathbb{R}^n)$, $\psi_i$, $i=1,\ldots,n$ are of bounded $C^2$-norm, $\Delta = \sum_{i=1}^{n} \partial_i^2$ is the Laplacian in $\mathbb{R}^n$ and $\mu > 0$. Assume $|u_0| \le e^{\Phi/\mu}$ where $\Phi$ is bounded above and has Lipschitz constant $1$. Assume that $(\sum_{i=1}^{n} |\Psi_i(t, x, u)|^2)^{1/2} \le A$, where $A$ is a postive constant.
Prove that there exists a constant $C$ depending only on $n$ that
$$|u(t, x)| \le C \, e^{C \, e^{((A+1)t + \Phi(x) + 2)/\mu}}, \quad \forall t \ge 0.$$