UMD PDE Qualifying Exams/Aug2010PDE

Problem 1

A superharmonic $u\in C^2(\overline{U})$ satisfies $-\mathrm{\Delta }u\ge 0$ in $U$, where here $U\subset {ℝ}^{𝕟}$ is open, bounded. (a) Show that if $u$ is superharmonic, then $u(x)\ge \frac{1}{\alpha (n)r^n}\underset{B(x,r)}{\int }udy\text{ for all }B(x,r)\subset U$. (b) Prove that if $u$ is superharmonic, then $\underset{\overline{U}}{\min }u=\underset{\partial U}{\min }u.$ (c) Suppose $U$ is connected. Show that if there exists $x_0\in U$ such that $u(x_0)=\underset{\overline{U}}{\min }u$ then $u$ is constant in $U$.

Solution

(a)

Test