PDE II - August 2024

#1. (15 points)

Let $n\ge 2,1<p<n$ and let $\Omega$ be an open set containing $0$ in ${ℝ}^n$. Prove that

$${\int }_{\Omega }\frac{{\left|u\left(x\right)\right|}^p}{{\left|x\right|}^p}\mathrm{dx}\le {\left(\frac{p}{n-p}\right)}^p{\int }_{\Omega }{\left|Du\left(x\right)\right|}^p\mathrm{dx}\forall u\in W_0^{1,p}\left(\Omega \right)$$

#2. (10 points)

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

$${\int }_{B_R\left(a\right)\setminus \overline{B_{\frac{R}{3}}}\left(a\right)}{\left(u-u_R\right)}^2\mathrm{dx}\le C_n{R}^2{\int }_{B_R\left(a\right)\setminus \overline{B_{\frac{R}{3}}}\left(a\right)}{\left|Du\right|}^2\mathrm{dx}\forall u\in H^1\left(B_R\left(a\right)\right)$$

where $u_R$ is the average of $u$ over $B_R\left(a\right)\setminus \overline{B_{\frac{R}{3}}}\left(a\right)$ and $C_n$ is a constant independent of $R$ and $a$.

#3.

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

1. (10 points) Show that there exists a constant $C$ such that for all $R>0$ and $\lambda \in ℝ$,

   $${\int }_{B_{\frac{R}{3}}\left(0\right)}{\left|Du\right|}^2\mathrm{dx}\le \frac{C}{R^2}{\int }_{B_R\left(0\right)\setminus \overline{B_{\frac{R}{3}}}\left(0\right)}{\left(u-\lambda \right)}^2\mathrm{dx}$$

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

#4. (15 points)

Let $\beta \in C^1\left({ℝ}^n\right)$. Show that given any $f$ and $g$, the boundary value problem

$$\{\begin{array}{c}-\Delta u+\beta \left(Du\right)+u^3=f\text{in}\Omega \\ u=g\text{on}\partial \Omega \end{array}$$

can have at most one solution $u\in C^2\left(\Omega \right)\cap C^1\left(\overline{\Omega }\right)$.

#5.

Let $\Phi \in C^1\left(\overline{\Omega };{ℝ}^n\right)$ and $c\in C\left(\overline{\Omega }\right)$. Consider the operator

$$Lu=-\Delta u+\Phi ·Du+cu.$$

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

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

   1. $c\ge 0\text{in}\Omega$.
   2. $c\ge \mathrm{div}\Phi \text{in}\Omega$.
   3. $c\ge \frac{1}{2}\mathrm{div}\Phi \text{in}\Omega$.

#6. (15 points)

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

$$\{\begin{array}{c}-\mathrm{div}\left({\left|Du\right|}^{p-2}Du\right)={\left|u\right|}^{q-1}u\text{in}\Omega \\ u=0\text{on}\partial \Omega \end{array}$$

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