partial-differential-equations-by-evans · Problem 8

Nonlinear First-Order PDE

Mathematics — Calculus · poisson-formula-half-space-regularity

Problem

Let

u

be the solution of

\begin{cases} \Delta u = 0 & \text{in } \mathbb{R}^n_+ \\ u = g & \text{on } \partial \mathbb{R}^n_+ \end{cases}

given by Poisson's formula for the half-space. Assume

g

is bounded and

g(x) = |x|

for

x \in \partial \mathbb{R}^n_+

|x| \leq 1

. Show

Du

is not bounded near

x = 0

. (Hint: Estimate

\frac{u(\lambda e_n) - u(0)}{\lambda}