Problem 1 (yau-contest 2021)
Section: Analysis and Differential Equations

Prove that $f(x) \equiv 0$ is the only solution in $L^2(R^n)$ such that
$$\Delta f = 0.$$

Problem 2 (yau-contest 2021)
Section: Analysis and Differential Equations

Let $X \subset C([0,1])$ be a finite dimensional linear subspace of the space of real-valued continuous functions on $[0,1]$. Show that, for a sequence of functions $\{f_k\}_{k\ge 1} \subset X$, if it converges pointwise, it converges uniformly.

Problem 3 (yau-contest 2021)
Section: Analysis and Differential Equations

(a) For $f \in L^1(R^n)$, $g \in L^\infty(R^n)$, show that their convolution $f * g$ is a well-defined continuous function.

(b) Let $E \subset R^n$ be a Lebesgue measurable set with Lebesgue measure $m(E) > 0$. Prove that
$$E - E := \{x - y \mid x \in E, y \in E\}$$
contains an open neighborhood of $0 \in R^n$.

Problem 4 (yau-contest 2021)
Section: Analysis and Differential Equations

Assume that $P$ is a polynomial with complex coefficients. Prove that there exists infinitely many solutions of the following equations on C:
$$e^z = P(z).$$

Problem 5 (yau-contest 2021)
Section: Analysis and Differential Equations

Let $f$ be a bounded holomorphic function defined on $B = \{ z \mid 0 < \operatorname{Re}(z) < 1 \}$ that can be extended as a continuous function on $\overline{B}$. Let
$$A_0 = \sup_{\operatorname{Re}(z)=0} |f(z)| > 0,\quad A_0 = \sup_{\operatorname{Re}(z)=1} |f(z)| > 0.$$
Prove that for all $z \in B$, we have
$$|f(z)| \le (A_0)^{1-\operatorname{Re}(z)}(A_1)^{\operatorname{Re}(z)}.$$

Problem 6 (yau-contest 2021)
Section: Analysis and Differential Equations

Assume that $\Omega \subset R^n$ is a bounded domain with smooth boundary. Prove that there exists a positive constant $\varepsilon_0$ so that for all real numbers $\varepsilon < \varepsilon_0$, for all $f \in L^2(\Omega)$, there exists a unique $u \in H_0^1(\Omega)$ so that
$$-\Delta u + \varepsilon \sin(u) = f$$
in the sense of distributions.