partial-differential-equations-by-evans · Problem 5

Nonlinear First-Order PDE

Mathematics — Calculus · maximum-principle-estimate

Problem

Prove that there exists a constant

C

, depending only on

n

, such that

\max_{B(0,1)} |u| \leq C\Big(\max_{\partial B(0,1)} |g| + \max_{B(0,1)} |f|\Big)

whenever

u

is a smooth solution of

\begin{cases} -\Delta u = f & \text{in } B^0(0,1) \\ u = g & \text{on } \partial B(0,1). \end{cases}