Problem 1 (qiuzhen-qualify 2024-fall)
Section: Analysis

Find all the conformal maps from $\mathbb{C} \setminus \{0\}$ onto $\mathbb{C} \setminus \{0\}$. Verify your answer.

Problem 2 (qiuzhen-qualify 2024-fall)
Section: Analysis

Let $\Omega \subset \mathbb{C}$ be a simply connected domain and $u$ a real-valued harmonic function on $\Omega$. Prove that up to a constant, there exists a unique holomorphic function $f$ on $\Omega$ such that $\operatorname{Re}(f) = u$.

Problem 3 (qiuzhen-qualify 2024-fall)
Section: Analysis

Find all $a \in \mathbb{R}$ such that the following boundary value problem has no solution $u = u(x)$:

$$\frac{d^2 u}{dx^2} + a u = 1, \quad u(0) = u(1) = 0.$$

Problem 4 (qiuzhen-qualify 2024-fall)
Section: Analysis

Let $u \in \mathscr{S}'(\mathbb{R}^3)$ be the standard surface measure on the unit sphere $$S^2 = \{ x = (x_1, x_2, x_3) : |x|^2 = x_1^2 + x_2^2 + x_3^2 = 1 \} \subset \mathbb{R}^3.$$ Compute its Fourier transform $\widehat{u}(\xi)$ for $\xi \in \mathbb{R}^3$. (Here we use the following definition for Fourier transform of $L^1$ functions $$\widehat{u}(\xi) = \int_{\mathbb{R}^3} e^{-i x \cdot \xi} \, u(x) \, dx$$ and extend to tempered distributions.)

Problem 5 (qiuzhen-qualify 2024-fall)
Section: Analysis

Consider the following operator

$$T f(x) = \int_0^\infty \frac{f(y)}{x+y} \, dy.$$

(i) Show that for $p \in (1, \infty)$, and any $f \in C_c([0, \infty))$, $$\|T f\|_{L^p((0,\infty))} \le C_p \|f\|_{L^p((0,\infty))},$$ where $$C_p := \int_0^\infty \frac{dx}{x^{1/p}(1+x)}.$$

(ii) Compute $C_p$.

(iii) Show that the result in (i) does not hold for any $C_p > 0$ when $p = 1$ or $p = \infty$.

Problem 6 (qiuzhen-qualify 2024-fall)
Section: Analysis

Find all $C^1$ solutions $u = u(x,y) : \mathbb{R}^2 \to \mathbb{R}$ of the equation

$$|\nabla u| := \sqrt{\left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2} = 1.$$

Problem 7 (qiuzhen-qualify 2024-fall)
Section: Analysis

Let $X$ be a Banach space, and $V \subset X$ be a subspace with finite codimension.

(i) Show that $V$ is not necessarily closed by providing a counter example.

(ii) Suppose there exist a Banach space $Y$ and a bounded linear operator $A$ from $Y$ to $X$, such that $V = R(A)$. Show that $V$ is closed.

Problem 8 (qiuzhen-qualify 2024-fall)
Section: Analysis

(i) Let $\theta: \mathbb{R} \to \mathbb{R}^+$ be a $C^1$ function such that $\|\theta'/\theta\|_{L^\infty(\mathbb{R})} \le 1$. Show that for all $u \in C_c^\infty(\mathbb{R})$, there exists some absolute constant $C$ s.t.,

$$\int_{\mathbb{R}} u^6 \, \theta \le C \|u\|_{L^2(\mathbb{R})}^2 \int_{\mathbb{R}} (|u'|^2 + |u|^2) \, \theta.$$

(ii) Let $\omega: \mathbb{R}^2 \to \mathbb{R}^+$ be a $C^1$ function such that $\|\nabla \omega / \omega\|_{L^\infty(\mathbb{R}^2)} \le 1$. Show that for all $u \in C_c^\infty(\mathbb{R}^2)$, there exists some absolute constant $C$ s.t.,

$$\int_{\mathbb{R}^2} u^4 \, \omega \le C \|u\|_{L^2(\mathbb{R}^2)}^2 \int_{\mathbb{R}^2} (|\nabla u|^2 + |u|^2) \, \omega.$$