# UMD PDE Qualifying Exams/Aug2010PDE

 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 ${\displaystyle U}$.