Assume that $g\in C(\mathbb{R}^{n-1})\cap L^\infty(\mathbb{R}^{n-1})$. Then construct a function $u$ satisfying
- $u\in C^\infty(\mathbb{R}_+^n)\cap L^\infty(\mathbb{R}_+^n)\cap C(\overline{\mathbb{R}_+^n})$
- $\Delta u=0$ in $\mathbb{R}_+^n$
- $u(x',0)=g(x')$ on $\mathbb{R}^{n-1}$
Use the results Liouville's Theorem and
- Let $U^+$ denote the open half-ball $\{x\in\mathbb{R}^n \hspace{0.25cm} | \hspace{0.25cm} |x|<1, x_n>0\}$. Assume $u\in C^2(\overline{U^+})$ is harmonic in $U^+$, with $u=0$ on $\partial U^+ \cap \{x_n=0\}$. Set [ v(x):=\begin{cases} u(x) & \text{ if } x_n\geq 0 \\ -u(x_1,\dots,x_{n-1},-x_n) & \text{ if } x_n < 0 \end{cases} ] for $x\in U=B^0(0,1)$. Prove $v\in C^2(U)$ and thus $v$ is harmonic within $U$.
- Now assume only that $u\in C^2(U^+)\cap C(\overline{U^+})$. Show that $v$ is harmonic within $U$.
to prove that there is only one such solution (that is there is just one bounded solution.)
Note here $\overline{\mathbb{R}_+^n}=\{(x',x_n):x'\in\mathbb{R}^{n-1} \text{ and } x_n\geq 0\}$.
So is my thinking correct? From the result that is in the blockquote we get that $u$ is harmonic. And to apply Liousville's Theorem, I need to show that $u$ is bounded. Once I do that I am done.
How? How would I prove that $u$ is bounded based off the information I have? Also it says "only one" do I have to show unqiueness? If so, how?