Strong maximum principle

Question

Let $S^{n-1}$ denote sphere in $\mathbb{R}^n$ and let $D$ denote open unit disk in $\mathbb{R}^n$. Let $f$ be homeomorphism of $S^{n-1}$ onto itself. Let $F$ be its harmonic extension given by Poisson integral. Then the result it to prove that $F$ is also an onto map. In the first part of it the result says to assume WLOG, that for $x\in{D}$ $F_1(x)=|F(x)|,F_2(x)=0,\ldots,F_n(x)=0$. Clearly this can be done by postrotating both $F$ and $f$. But then it says that by strong maximum principle, $F_1(x)<\max f_1=1$, so we get that image is contained in $D$. Can anyone tell me what this strong maximum priciple is?\ And secondly if we take $y\in{D\setminus F(D)}$, then the map $\psi$ defined on $S^{n-1}$ as $$\psi(x)=\frac{f(x)-y}{|f(x)-y|}.$$ How is this map one-one? Any hint is appreciated.

Answer 1

The terminology of the "strong maximum principle" is not universal, but from the context, it is the form of the maximum principle that asserts that a non-constant harmonic function $h$ on a bounded domain $U$ that has a continuous extension to $\overline{U}$ satisfies the strict inequality

$$\lvert h(x)\rvert < \max \left\{\lvert h(y) \rvert : y \in \partial U\right\}$$

for every $x\in U$, where the non-strong maximum principle only asserts that

$$\max \left\{\lvert h(x)\rvert : x \in \overline{U}\right\} = \max \left\{\lvert h(y) \rvert : y \in \partial U\right\}$$

for any $h$ that is harmonic in $U$ and continuous on $\overline{U}$. Or maybe the strong maximum principle is stated in a different but equivalent formulation, like that a harmonic function that has a local maximum in $U$ is constant. Both possible versions of the "strong maximum principle" are often just called the "maximum principle" and no "strong maximum principle" is posited.

Regarding the second point, the injectivity of $\psi$ follows from the injectivity of $f$ and the fact that every ray emanating from $y\in D$ intersects the unit sphere $S^{n-1}$ in exactly one point (due to the strict convexity of $D$).