partial-differential-equations-by-evans · Problem 4

Nonlinear First-Order PDE

Mathematics — Calculus · subharmonic-functions

Problem

We say

v \in C^2(\bar{U})

is subharmonic if

-\Delta v \leq 0 \quad \text{in } U.

(a) Prove for subharmonic

v

that

v(x) \leq \int_{B(x,r)} v\, dy \quad \text{for all } B(x,r) \subset U.

(b) Prove that therefore

\max_{\bar{U}} v = \max_{\partial U} v

. (c) Let

\phi : \mathbb{R} \to \mathbb{R}

be smooth and convex. Assume

u

is harmonic and

v := \phi(u)

. Prove

v

is subharmonic. (d) Prove

v := |Du|^2

is subharmonic, whenever

u

is harmonic.