PDE II - May 2025

PDE II: Qualifying Exam

Friday May 9, 2025

There are five equally weighted problems. Start with problems that are most familiar.

Question 1

Fix $p\in[1,\infty)$ and $\Omega\subset\mathbb R^d$ open, connected, and bounded. Prove that if $u\in L^p(\Omega)$ satisfies

$$Du=0,\qquad \text{a.e. in }\Omega,$$

then $u\in W^{1,p}(\Omega)$ and there exists $c\in\mathbb R$ such that $u=c$ a.e. in $\Omega$.

Question 2

Fix $p\in[1,\infty)$ and $x\ne y\in\mathbb R$. Define the linear operator $T_{x,y}:C^0(\mathbb R)\to\mathbb R$ by

$$T_{x,y}u=\frac{u(x)-u(y)}{|x-y|^{1/q}},$$

where $q$ is the Sobolev conjugate to $p$. Show that $T_{x,y}$ has an extension to $W^{1,p}(\mathbb R)$ that satisfies

$$|T_{x,y}u|\le\|u\|_{W^{1,p}(\mathbb R)},$$

for all $u\in W^{1,p}(\mathbb R)$.

Question 3

Assume $F:\mathbb R\mapsto\mathbb R$ is $C^1$ with $F'$ bounded. If $\Omega\subset\mathbb R^d$ is bounded and open, then show that $u\in W^{1,p}(\Omega)$ for some $p\in[1,\infty)$ implies that $F(u)\in W^{1,p}(\Omega)$ and

$$\partial_{x_i}F(u)=F'(u)u_{x_i},$$

for $i=1,\ldots,d$.

Question 4

Let $\Omega\subset\mathbb R^d$ be bounded, open, with a smooth boundary. Let $c(x)\ge0$ be a smooth function. Show that there is a solution $u\in C^2(\Omega)$ to

$$\begin{aligned}-\Delta u&=\frac{1}{1+e^{-u}}-c(x)u, &&\text{in }L^2(\Omega),\\u&=0,&&\text{in }L^2(\partial\Omega),\end{aligned}$$

that satisfies $u>0$ on $\Omega$.

Hint: $f(u):=(1+e^{-u})^{-1}$ is an increasing, bounded function and $f(0)>0$.

Question 5

Consider the operator

$$Lu:=-\Delta u+b\cdot Du,$$

where $b\in\mathbb R^d$ is a constant vector of norm 1 and $\Omega\subset\mathbb R^d$ is an open, bounded set with a $C^1$ boundary and exterior unit normal $\nu$. Let $\partial\Omega$ be divided into two disjoint, connected sets $\Gamma_1$ and $\Gamma_2$ each of which has positive surface measure.

1. For $f\in L^2(\Omega)$ and $\alpha\in\mathbb R$, find a weak formulation for the problem

   $$\begin{aligned}Lu&=f,&&\text{in }L^2(\Omega),\\u&=0,&&\text{in }L^2(\Gamma_1),\\\nu\cdot\nabla u+\alpha u&=0,&&\text{in }L^2(\Gamma_2).\end{aligned}$$

   Show for your weak formulation that if $u\in H^2(\Omega)$ and $u$ is a weak solution then $u$ is a strong solution.

2. Show that if $\alpha\ge\frac12$ then for each $f\in L^2(\Omega)$ your weak formulation in part (a) has a unique solution $u\in H^1(\Omega)$, with zero trace on $\Gamma_1$.