  It has been shown that the shape of the luminosity function of white dwarfs
(WDLF) is a powerful tool to check for the possible existence of DFSZ-axions, a
proposed but not yet detected type of weakly interacting particles. With the
aim of deriving new constraints on the axion mass, we compute in this paper new
theoretical WDLFs on the basis of WD evolving models that incorporate for the
feedback of axions on the thermal structure of the white dwarf. We find that
the impact of the axion emission into the neutrino emission can not be
neglected at high luminosities ($M_{\rm Bol}\lesssim 8$) and that the axion
emission needs to be incorporated self-consistently into the evolution of the
white dwarfs when dealing with axion masses larger than $m_a\cos^2\beta\gtrsim
5$ meV (i.e. axion-electron coupling constant $g_{ae}\gtrsim 1.4\times
10^{-13}$). We went beyond previous works by including 5 different derivations
of the WDLF in our analysis. Then we have performed $\chi^2$-tests to have a
quantitative measure of the assessment between the theoretical WDLFs
---computed under the assumptions of different axion masses and normalization
methods--- and the observed WDLFs of the Galactic disk. While all the WDLF
studied in this work disfavour axion masses in the range suggested by
asteroseismology ($m_a\cos^2\beta\gtrsim 10$ meV; $g_{ae}\gtrsim 2.8\times
10^{-13}$) lower axion masses can not be discarded from our current knowledge
of the WDLF of the Galactic Disk. A larger set of completely independent
derivations of the WDLF of the galactic disk as well as a detailed study of the
uncertainties of the theoretical WDLFs is needed before quantitative
constraints on the axion-electron coupling constant can be made.
