UMD PDE Qualifying Exams/Aug2010PDE

Problem 1

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

Solution

(a)

Test