partial-differential-equations-by-evans · Problem 9

Nonlinear First-Order PDE

Mathematics — Calculus · harmonic-extension-odd-reflection

Problem

Let U+U^+ denote the open half-ball {x∈Rn∣∣x∣<1, xn>0}\{x \in \mathbb{R}^n \mid |x| < 1,\, x_n > 0\}. Assume u∈C(Uˉ+)u \in C(\bar{U}^+) is harmonic in U+U^+, with u=0u = 0 on ∂U+∩{xn=0}\partial U^+ \cap \{x_n = 0\}. Set v(x):={u(x)if xn≥0−u(x1,…,xn−1,−xn)if xn<0v(x) := \begin{cases} u(x) & \text{if } x_n \geq 0 \\ -u(x_1, \ldots, x_{n-1}, -x_n) & \text{if } x_n < 0 \end{cases} for x∈U=B0(0,1)x \in U = B^0(0,1). Prove vv is harmonic in UU.

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