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

Let $F:\mathbb{R}\to\mathbb{R}$ be a strictly convex function. Let $u:[0,1]\to\mathbb{R}$ be a continuous function, with $\int_0^1 u(x)\,dx=0$.

Show that
$$\int_0^1 F(u(x))\,dx \le \frac{F(\|u\|_\infty)+F(-\|u\|_\infty)}{2},$$
where $\|u\|_\infty := \sup_{x\in[0,1]} |u(x)|$. Also determine when equality occurs.

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

Prove that there exists a universal constant $K$, for all $C^1$ function $f:\mathbb{R}^2\to\mathbb{R}$, if $f\in L^1(\mathbb{R}^2)\cap L^2(\mathbb{R}^2)$ and $\nabla f \in L^2(\mathbb{R}^2)$, we have the following inequality:
$$\|f\|_{L^2(\mathbb{R}^2)}^2 \le K\, \|f\|_{L^1(\mathbb{R}^2)} \,\|\nabla f\|_{L^2(\mathbb{R}^2)}.$$
Can you provide constant $K$ so that $K<10$? In the problem, all the $L^p$-spaces are defined with respect to the Lebesgue measure.

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

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a harmonic function. Suppose
$$\lim_{|x|\to\infty} \frac{|f(x)|}{\ln|x|} = 0.$$
Prove or disprove that $f$ is a constant.

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

(a) Show that there does not exist a holomorphic function $f$ on $\mathbb{C}\setminus\{1,-1\}$ so that
$$f'(z)=\frac{1}{(z^2-1)^{2019}} \quad \text{for all } z\in \mathbb{C}\setminus\{1,-1\}.$$

(b) Show that there exist a set $L\subset\mathbb{C}$ and a holomorphic function $F$ on $\mathbb{C}\setminus L$ so that $L$ has Hausdorff dimension $1$, and
$$F'(z)=\frac{1}{(z^2-1)^{2019}} \quad \text{for all } z\in \mathbb{C}\setminus L.$$

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

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with smooth boundary. Prove that, for all $p>1$ and $1\le q<\infty$, for all $f\in L^p(\Omega)$, there exists a unique $u\in H_0^1(\Omega)$, such that
$$\Delta u = |u|^{q-1}u + f \quad \text{in } \Omega.$$