Problem 1 (yau-contest 2019)
Section: Analysis and Differential Equations

Show that there is no non-zero $f \in C_0^{\infty}(\mathbb{R}^2)$ (compactly supported smooth function) so that its Fourier transform $\hat f(\xi)$ is also compactly supported.

Problem 2 (yau-contest 2019)
Section: Analysis and Differential Equations

Prove the following classical interior Schauder estimates:

There exists a universal constant $C$, for all smooth compactly supported functions $u, f \in C_0^{\infty}(\mathbb{R}^3)$ with $\Delta u = f$, we have
\[\|u\|_{C^{2,\alpha}} \le C\,\|f\|_{C^{0,\alpha}},\]
where $0<\alpha<1$ and $\|\cdot\|_{C^{k,\alpha}}$ are H\"older norms.

Problem 3 (yau-contest 2019)
Section: Analysis and Differential Equations

Let $(X, \mathcal{A}, \mu)$ be a probability space and let $T: X \to X$ be a measurable and measure preserving map, i.e., for all $A \in \mathcal{A}$, we have $\mu(T^{-1}(A)) = \mu(A)$. For $A, B \in \mathcal{A}$, if $\mu(A-B) = \mu(B-A) = 0$, we say that $A = B$ a.e.

Assume $A \in \mathcal{A}$ such that $T^{-1}(A) = A$ a.e. Prove that there exists a set $B \in \mathcal{A}$ so that $T^{-1}B = B$ and $A = B$ a.e.

Problem 4 (yau-contest 2019)
Section: Analysis and Differential Equations

Is there an entire function $f$ with infinitely many zeroes, so that for every $r \in (0,1)$, there exist constants $A_r, B_r < \infty$ such that
\[ |f(z)| \le A_r e^{B_r |z|^r} \]
for every $z \in \mathbb{C}$?

Problem 5 (yau-contest 2019)
Section: Analysis and Differential Equations

Let $u(t,x,y)$ be a smooth real function defined on $\mathbb{R} \times \mathbb{R}^2$ where $t \in \mathbb{R}$ and $(x,y) \in \mathbb{R}^2$. We assume that it solves the following semilinear wave equation:
\[ -\frac{\partial^2}{\partial t^2} u + \Delta u = u^3. \]
If the supports of the initial data $u(0,x)$ and $\frac{\partial u}{\partial t}(0,x)$ are compact, prove that, for all $t_0 \in \mathbb{R}$, the supports of $u(t_0,x)$ and $\frac{\partial u}{\partial t}(t_0,x)$ are compact.