On the Martin boundary of Lipschitz strips
Hiroaki Aikawa · Journal of the Mathematical Society of Japan · 1986
\S 1. Introduction.Let $n\geqq 1$ and $m\geqq 1$ .We denote by $P=(X, Y)$ a point in $R^{n+m}=R^{n}\cross R^{m}$ , where $X=(x_{1}, \cdots , x_{n})\in R^{n}$ and $Y=(y_{1}, \cdots , y_{m})\in R^{m}$ .We write $|P|,$ $|X|$ and $|Y|$ for $(\Sigma_{j\Leftarrow 1}^{n}x_{j}^{2}+\Sigma_{j=1}^{m}y_{j}^{2})^{1/2},$ $(\Sigma_{j\approx 1}^{n}x_{j}^{2})^{1/2}$ and $(\Sigma_{j=1}^{m}y_{j}^{2})^{1/2}$ , respectively.We identify $R^{n}$ and $R^{m}$ with $\{(X, Y);Y=0\}$ and $\{(X, Y);X=0\}$ , respectively.We denote by $S^{n-1}$ the unit sphere $\{\alpha\in R^{n} ; |\alpha|=1\}$ with center at the origin in $R^{n}$ .Let $D$ be a bounded domain in $R^{m}$ .We call $L=R^{n}\cross D=\{(X, Y);Y\in D\}$ a strip.If $D$ is a Lipschitz domain, then $L$ is said to be a Lipschitz strip.In this note we consider the Martin compactification of a Lipschitz strip.We denote by $\overline{L}=R^{n}\cross\overline{D}$ the Euclidean closure of $L$ in $R^{n+m}$ .Let $M_{\alpha},$ $\alpha\in$ $S^{n-1}$ , be a point at infinity and let $\hat{L}=\overline{L}\cup\{M_{a} ; \alpha\in S^{n-1}\}$ be a compact topo- logical space with open base $O_{1}\cup O_{2}$ , where $0_{1}=\{U\cap\overline{L};U$ is an open set of $R^{n+m}\}$ and $0_{2}=$ { $U(\alpha,$ $\epsilon,$ $R);\alpha\in S^{n-1},00$ } with $U(\alpha, \epsilon, R)=$ $\{M_{\beta} ; \beta\in S^{n-1}, \Sigma_{i\Rightarrow 1}^{n}\alpha_{i}\beta_{i}>1-\epsilon\}\cup\{(X, Y)\in\overline{L};(1-\epsilon)^{-1}\Sigma_{i=1}^{n}x_{i}\alpha_{i}>|X|>R\}$ .We note that $P_{j}=(X_{j}, Y_{j})\in\overline{L}$ converges to $M_{\alpha}$ if and only if lim $jarrow\infty|X_{j}|=+\infty$ and $\lim jarrow\infty X_{j}/|X_{j}|=\alpha$ .We shall prove THEOREM 1.The Martin compactification of $L$ is homeomorphic to $L$ .H. AIKAWA \S 2. Proof of Theorem 1.We shall use the following notation: Let $X_{0}=(0, \cdots , 0)\in R^{n},$ $Y_{0}=(0, \cdots , 0)$ $\in R^{m}$ and $P_{0}=(X_{0}, Y_{0})\in R^{n+m}$ .Without loss of generality we may assume that $Y_{0}\in D$ , and hence that $P_{0}\in L$ .We let $L_{0}=R^{n}\cross\{Y_{0}\}$ .Denote by $B^{n}(X, r)$ , $B^{m}(Y, r)$ and $B(P, r)$ the n-dimensional open ball with center at $X$ and radius $r$ , the m-dimensional open ball with center at $Y$ and radius $r$ and the $(n+m)-$ dimensional open ball with center at $P$ and radius $r$ , respectively.We may assume that $D\supset B^{m}(Y_{0},5)$ .Let $\pi:R^{n+m}arrow R^{n}$ be the projection defined by $\pi((X, Y))=X$ and let $\pi_{0}(P)=(\pi(P), Y_{0})$ .We put $U_{+}(t)=\{P\in L;(\pi(P))_{n}>t\}$ , $U_{-}(t)=\{P\in L;(\pi(P))_{n} b$ }.Let $Q_{0}=(0, \cdots , 0, b)\in R^{n+m}$ and $\Gamma=\{P=$ $(p_{1}, \cdots , p_{n+m})\in R^{n+m}$ ; $p_{n+m}-b>-2^{-1}|P-Q_{0}|$ } be a cone with vertex at $Q_{0}$ .Let $v$ be a positive harmonic function on $\Gamma$ vanishing on $\partial\Gamma$ .We observe that $v(P)=|P-Q_{0}|^{\delta}v((P-Q_{0})/|P-Q_{0}|+Q_{0})$ with $\delta>0$ .From the Harnack inequality