# UMD PDE Qualifying Exams/Aug2010PDE

## Problem 1

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

Test