  (abridged) The ambiguous origin of [CII] 158um in the interstellar medium
complicates its use for diagnostics concerning the star-formation rate and
physical conditions in photodissociation regions (PDRs). We observed the giant
HII region N11 in the Large Magellanic Cloud with SOFIA/GREAT in order to
investigate the origin of [CII] to obtain the total H2 gas content, the
fraction of CO-dark H2 gas, and the influence of environmental effects such as
stellar feedback. We present an innovative spectral decomposition method that
allows statistical trends to be derived. The [CII] line is resolved in velocity
and compared to HI and CO, using a Bayesian approach to decompose the profiles.
A simple model accounting for collisions in the neutral atomic and molecular
gas was used in order to derive the H2 column density traced by C+. The profile
of [CII] most closely resembles that of CO, but the integrated [CII] line width
lies between that of CO and that of HI. Using various methods, we find that
[CII] mostly originates from the neutral gas. We show that [CII] mostly traces
the CO-dark H2 gas but there is evidence of a weak contribution from neutral
atomic gas preferentially in the faintest components. Most of the molecular gas
is CO-dark. The fraction of CO-dark H2 gas decreases with increasing CO column
density, with a slope that seems to depend on the impinging radiation field
from nearby massive stars. Finally we extend previous measurements of the
photoelectric-effect heating efficiency, which we find is constant across
regions probed with Herschel, with [CII] and [OI] being the main coolants in
faint and diffuse, and bright and compact regions, respectively, and with PAH
emission tracing the CO-dark H2 gas heating where [CII] and [OI] emit. Our
study highlights the importance of velocity-resolved PDR diagnostics and higher
spatial resolution for HI observations.
