Maximum principle for $\varphi\in C^0(\overline{B},\mathbb{R}^2)\cap C^2(B,\mathbb{R}^2)$ Let $B:= \{w\in \mathbb{R}^2 : \lvert w\rvert < 1\}$ be the unit-circle.
Suppose that $\varphi\in C^0(\overline{B},\mathbb{R}^2)\cap C^2(B,\mathbb{R}^2)$ is harmonic in $B$ (both component functions of $\varphi$ are harmonic) and maps $\partial B$ onto the boundary $\partial \Omega$ of a convex domain $\Omega\subset \mathbb{R}^2$. Then $\varphi(B)$ lies in $\Omega$.
The book I am reading states that this follows from the maximum principle for harmonic functions. However, I wasn't able to proof the statement using the fact that both component functions of $\varphi$ are harmonic functions in the classical sense and satisfy the maximum principle. Does there exist a maximum principle for vector-valued harmonic functions that I am not aware of or how can one proof this fact otherwise?
 A: $\newcommand{\R}{\mathbb{R}}$
$\newcommand{\vp}{\varphi}$
If I understand correctly, this is a consequence of the standard maximum principle.
If $q \colon \R^2 \to \R$ is a linear functional, then the composition $q \circ \vp \colon \overline{B} \to \R$ is a harmonic function, and we can apply the standard maximum principle. More precisely, if $C \subseteq \R^2$ is closed and convex, and $\vp(\partial B) \subseteq C$, then the maximum principle tells us
$$
q \circ \vp \le \sup_{x \in \partial B} q(\vp(x)) \le \sup_{y \in C} q(y) 
\quad \text{on } \overline{B}.
$$
Since the set $C$ is closed and convex, it can be described as an intersection of a family (possibly infinite) of half-planes. More precisely,
$$
C = \bigcap_{q \colon \R^2 \to \R \text{ linear}} \big\{ y \in \R^2 : q(y) \le \sup_{y \in C} q(y) \big\}. 
$$
By the previous reasoning, each point $\vp(x)$ (for $x \in \overline{B}$) lies in every set of the form $\big\{ y \in \R^2 : q(y) \le \sup_{y \in C} q(y) \big\}$, and so we showed that $\vp(x) \in C$ for all $x \in \overline{B}$.
With some care, one can apply the strong maximum principle for $q \circ \vp$ and infer a stronger statement for $\vp$ (that is, $\vp(x) \in \Omega$ for $x \in B$, not just $\vp(x) \in \overline{\Omega}$).
