PDE II - May 2024

Qualifying Exam – Partial Differential Equations II – May 2024

INSTRUCTIONS: The exam is closed-book – Notes, resources or communications are strictly prohibited ! You may use the well-known theorems or results without proof, but you must indicate such a result clearly with all necessary steps. You may also use the result of one problem to solve another problem on the exam.

In all the following, unless specified, $\mathrm{\Omega }$ will denote a bounded open set in ${\mathbb{R}}^n$.

#1. (15 points)

Let $p>1$, $n\ge 2$ and $\mathrm{\Omega }=\{x\in {\mathbb{R}}^n:1<|x|<2\}$. Assume $v\in W_0^{1,p}(1,2)$ and let $u(x)=v(|x|)$. Show that $u\in W_0^{1,p}(\mathrm{\Omega })\cap C^{0,\gamma }(\bar{\mathrm{\Omega }})$, where $\gamma =1-\frac{1}{p}$.

#2. (10 points)

Let $B_R(a)$ denote the open ball in ${\mathbb{R}}^n$ with center $a$ and radius $R$. Prove the inequality

where $u_R$ is the average of $u$ over $B_R(a)\setminus {\bar{B}}_{R/2}(a)$ and $C_n$ is a constant independent of $R$ and $a$.

#3.

Let $u\in H_{\mathrm{loc}}^1({\mathbb{R}}^n)$ be a weak solution of equation $-\sum _{i,j=1}^n{D}^i(a_{ij}(x)D^{j}u)=0$ in $B_R(0)$ for all $R>0$, where the equation is assumed to be uniformly elliptic in all of ${\mathbb{R}}^n$.

(a) (10 points) Show that there exists a constant $C$ such that for all $R>0$ and $\lambda \in \mathbb{R}$,

(b) (10 points) Assume $|Du|\in L^2({\mathbb{R}}^n)$. Show that $u$ is constant almost everywhere on ${\mathbb{R}}^n$.

#4. (15 points)

Let $\beta \in C^1({\mathbb{R}}^n)$ and $\alpha \in C^1(\mathbb{R})$ with ${\alpha }^{\prime }\ge 0$. Show that given any $f$ and $g$, the boundary value problem

cannot have more than one solution in $C^2(\mathrm{\Omega })\cap C^1(\bar{\mathrm{\Omega }})$.

#5.

Let $\mathrm{\Phi }\in C^1(\bar{\mathrm{\Omega }};{\mathbb{R}}^n)$ and $c\in C(\bar{\mathrm{\Omega }})$. Consider the operator

(a) (10 points) Find the formal adjoint operator $L^{\ast }$ of $L$ on $H_0^1(\mathrm{\Omega })$ in the sense that the bilinear form $B^{\ast }$ associated with $L^{\ast }$ satisfies $B^{\ast }[u,v]=B[v,u]$ for all $u,v\in H_0^1(\mathrm{\Omega })$, where $B$ is the bilinear form associated with $L$.

(b) (15 points) Show that $Lu=f$ has a unique weak solution $u\in H_0^1(\mathrm{\Omega })$ for each $f\in L^2(\mathrm{\Omega })$ provided that one of the following conditions holds:

(i) $c\ge 0$ in $\mathrm{\Omega }$.　(ii) $c\ge \mathrm{div}\mathrm{\Phi }$ in $\mathrm{\Omega }$.　(iii) $c\ge \frac{1}{2}\mathrm{div}\mathrm{\Phi }$ in $\mathrm{\Omega }$.

#6. (15 points)

Let $n\ge 2$, $1<p<n$, $0<q<\frac{np+p-n}{n-p}$ and $q\ne p-1$. Use constrained minimization to show that the boundary value problem

has a nontrivial weak solution in $W_0^{1,p}(\mathrm{\Omega })$ satisfying $u\ge 0$ almost everywhere in $\mathrm{\Omega }$.