Regarding the article "Sequential Screening," in Review of Economic Studies, 2000 by Courty and Li:
In Lemma 3.2, the last equality states that $$\frac{dU(t)}{dt}=\int_{\underline{v}}^{\overline{v}}\frac{\partial g(v|t)}{\partial t}u(t,v)dv = -\int_{\underline{v}}^{\overline{v}}\frac{\partial G(v|t)}{\partial t}y(t,v)dv.$$
This (the second equality here) is an application of integration by parts (they show in Lemma 3.1 that $\frac{\partial u(t,v)}{\partial v}=y(t,v)$); but why is the antiderivative term equal to zero? That is, why does
$$\Bigl[\frac{\partial G(v|t)}{\partial t}u(t,v)\Bigr]^{\overline{v}}_{\underline{v}}=0 \ \ \ \forall t \quad ?$$
[Answered by Sergio: because of the constant support condition on $g(\cdot | t)$.]
Background: This article analyzes a direct revelation mechanism, where the agent learns and discloses information about his valuation in two stages, and the principal assigns payments and allocations after the second stage.
Notation:
$t$ and $v$ are the ex ante agent type and ex post valuation, respectively, distributed according to pdfs $f(\cdot)$ and $g(\cdot|t)$. The latter is assumed to have constant support $[\underline{v},\overline{v}]$ $\forall t$.
$G(\cdot|t)$ is the cdf corresponding to $g(\cdot|t)$.
$y(t,v)$ is the probability of allocation to agent of type $t$ with ex post valuation $v$.
$u(t,v)$ is the ex post surplus to the agent in the mechanism, assuming honest reporting of $t$ and $v$. $u(t,v) = vy(t,v)-x(t,v)$, where $x$ is the payment from the agent to the principal.
$U(t)$ is the expected surplus $\int_{\underline{v}}^{\overline{v}}u(t,v)g(v|t)dt$ of an agent of type $t$.
Probably because:
But it is hard to say because the question did not define what $G$ is (I am assuming is a CDF with support on $[\underline v, \overline v]$).