A condenser is a pair $(D,E)$, where $D$ is a domain in the plane and $E$ is a compact subset of $D$. The capacity of the condenser $(D,E)$ is defined by:
$$\text{cap}(D,E) = \inf \int_{D} |\nabla u|^2 dm$$
where the infimum is taken over all Lipschitz functions $u : \mathbb{C} \rightarrow \mathbb{R}$ (defined on $\mathbb{C}$) such that $u \leq 0$ on $\partial D$ and $u \geq 1$ on $E$. Also $dm$ is the usual Lebesgue measure on $\mathbb{C}$.
I believe this definition would be the same if we insisted on taking the infimum over all Lipschitz functions $u$, defined on $\mathbb{C}$, such that $u = 0$ on $\partial D$ and $u = 1$ on $E$, for the following reason:
$\text{trun}(u) \stackrel{\mathrm{def}}{=} \min(\max(u,0),1)$ is still Lipschitz in the whole complex plane, and its Dirichlet integral (i.e.$ \int_{D} |\nabla \text{trun}(u)|^2 dm$) will coincide with $u$'s over $u^{-1}(0,1) \cap D$, and on $u^{-1}(-\infty,0] \cap D$ will vanish, since on this set $\nabla \text{trun}(u) = \nabla \max(u,0) = 0 $ $dm$-almost everywhere.
Similarly its Dirichlet integral over $u^{-1}[1,+\infty) \cap D$ should vanish, and therefore its Dirichlet integral over $D$ should be less than or equal to $u$'s Dirichlet integral.
However, the paper I am reading makes the following remark, which makes me think there has to be a flaw in my reasoning:
"We want to stress here that if $D = \mathbb{D}$, then the infimum can be taken over all admissible functions as above with an additional requirement that $u(z) = 0$ for all $z \in \partial \mathbb{D}$".
So it's unlikely to be true that my reasoning is sound.
Therefore I would like to know where, if in some place, my argument fails.