Let a functional $H[\phi]$ of a map $\phi\in\mathbb{R}^{\mathbb{R}^4}$ be given by: $$ H(x^0) = \int_{\mathbb{R}^3} \left[\frac{1}{2}\sum_{j=0}^{3}\left(\partial_{x^j}\phi\right)^2-\frac{1}{2}\mu^2\phi^2+\frac{\lambda}{4!}\phi^4\right]dx^1dx^2dx^3 $$
where $x^j$ are the arguments of $\phi$.
Find all maps $\phi\in\mathbb{R}^{\mathbb{R}^4}$ which are continuous and compactly supported such that $H(x^0)$ is minimal.
What I have tried: I wrote down the variation of $H$, and arrived at the fact that field configurations which extremize $H$ should obey the equation of motion $$-\mu^2\phi+\frac{\lambda}{3!}\phi^3-\sum_{i=0}^{3}\partial_{x^i}\,^2\phi=0$$
However, in the book I am trying to follow (Peskin & Schroeder QFT page 348) it is claimed that such maps $\phi$ which minimize $H[\phi]$ are such that are constant maps with value $\phi_0$ such that this value $\phi_0$ minimizes the function $V\in\mathbb{R}^{\mathbb{R}}$ where $V$ is given by $V(\alpha)=-\frac{1}{2}\mu^2\alpha^2+\frac{\lambda}{4!}\phi^4$.
While I can heuristically see why this is true (the derivative term in $H$ is always positive, and for constant maps it is zero; after ensuring that this term is zero (its minimal value) we go ahead to minimize the rest of $H$) I would like a more general / clear argument why this is true, if it is indeed true.