  Many low-mass white dwarfs are being discovered in the field of our galaxy
and some of them exhibit $g$-mode pulsations, comprising the extremely low-mass
variable (ELMV) stars class. Despite it is generally believed that these stars
are characterized by thick H envelopes, from stellar evolution considerations,
the existence of low-mass WDs with thin H envelopes is also possible. We have
performed detailed asteroseismological fits to all the known ELMVs to search
for a representative model by employing a set of fully evolutionary models that
are representative of low-mass He-core white dwarf stars with a range of
stellar masses $[0.1554-0.4352]\ M_{\odot}$, effective temperatures
$[6000-10000]\ $K, and also with a range of H envelope thicknesses $-5.8
\lesssim \log(M_{\rm H}/M_{\star}) \lesssim -1.7$, hence expanding the space of
parameters. We found that some of the stars under analysis are characterized by
thick H envelopes, but others are better represented by models with thin H
envelope.
