We define the psh envelope of a (possibly singular) metric $\psi$ on ${L^{\mathrm{an}}}$ by

$$P(\psi):=\sup\{\phi\in\operatorname{PSH}({L^{\mathrm{an}}})\mid\phi\leq\psi\}^{*}.$$

As before, $\phi^{*}$ denotes the usc regularization of a singular metric $\phi$. In all cases we need to consider, $\psi$ will be the sum of a metric in ${\mathcal{E}}^{1}({L^{\mathrm{an}}})$ and a continuous function on ${X^{\mathrm{an}}}$. In particular, $\psi$ is usc, $P(\psi)\in{\mathcal{E}}^{1}({L^{\mathrm{an}}})$ and $P(\psi)\leq\psi$.

This envelope construction was in fact already mentioned at the end of §6 as it plays a key role in the regularization theorem. The psh envelope is an analogue of the convex hull; see Figure 1.

The key fact about the psh envelope is that the composition $E\circ P$ is differentiable and that $(E\circ P)^{\prime}=E^{\prime}\circ P$. More precisely, we have:

Granted this result, let us show how to solve the Monge-Ampère equation. Pick $\phi\in{\mathcal{E}}^{1}({L^{\mathrm{an}}})$ that maximizes $F_{\mu}(\phi)=E(\phi)-\int(\phi-\phi_{0})\mu$ and consider any $f\in C^{0}({X^{\mathrm{an}}})$. For any $t\in{\mathbf{R}}$ we have

	$\displaystyle E(P(\phi+tf))-\int(\phi+tf-\phi_{0})\mu$	$\displaystyle\leq E(P(\phi+tf))-\int(P(\phi+tf)-\phi_{0})\mu$
		$\displaystyle\leq E(\phi)-\int(\phi-\phi_{0})\mu.$

Since the left hand side is differentiable at $t=0$, the derivative must be zero, which amounts to $\int f\operatorname{MA}(\phi)-\int f\mu=0$. Since $f\in C^{0}({X^{\mathrm{an}}})$ was arbitrary, this means that $\operatorname{MA}(\phi)=\mu$, as desired.

The proof of this differentiability results proceeds by first reducing to the case when $\phi$ and $f$ are continuous. A key ingredient is then

In other words, the Monge-Ampère measure $\operatorname{MA}(P(\phi))$ is supported on the locus $P(\phi)=\phi$. A version of this for functions of one variable is illustrated in Figure 1.

To prove this result, we can reduce to the case when $\phi$ is a smooth/model metric. In the complex case, Theorem 8.2 was proved by Berman and the first author in [BB10] using the pluripotential theoretic technique known as “balayage”. In the non-Archimedean setting, Theorem 8.2 is deduced in [BFJ15] from the asymptotic orthogonality of Zariski decompositions in [BDPP13] and is for this reason called the orthogonality property. The assumption in Theorem 4.1 that the variety $X$ be defined over a smooth $k$-curve is used exactly in order to apply the result from [BDPP13].

The solution to the non-Archimedean Monge-Ampère equation $\operatorname{MA}(\phi)=\mu$ can be made slightly more explicit in the case when the support of $\mu$ is a singleton, $\mu=d_{L}\,\delta_{x}$, where $d_{L}:=(L^{n})$ and $x\in{X^{\mathrm{an}}}$ belongs to some dual complex; such points $x$ are known as quasimonomial or Abhyankar points.

The fiber $L^{\mathrm{an}}_{x}$ of ${L^{\mathrm{an}}}$ above $x\in{X^{\mathrm{an}}}$ is isomorphic to the Berkovich affine line over the complete residue field ${\mathcal{H}}(x)$. Fix any nonzero $y\in L^{\mathrm{an}}_{x}$ and set

$$\phi_{x}:=\sup\{\phi\in\operatorname{PSH}({L^{\mathrm{an}}})\mid\|y\|_{\phi}\geq 1\}.$$

By [BFJ15, Prop.8.6], $\operatorname{MA}(\phi_{x})$ is supported on $x$, so $\operatorname{MA}(\phi_{x})=d_{L}\,\delta_{x}$. It would be interesting to find an example of a divisorial point $x\in{X^{\mathrm{an}}}$ such that $\phi_{x}$ is not a model function.