UMD PDE Qualifying Exams/Aug2010PDE

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Problem 1[edit]

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.