  In this paper, thick branes generated by the mimetic scalar field with
Lagrange multiplier formulation are investigated. We give three typical thick
brane background solutions with different asymptotic behaviors and show that
all the solutions are stable under tensor perturbations. The effective
potentials of the tensor perturbations exhibit as volcano potential,
Po\"{o}schl-Teller potential, and harmonic oscillator potential for the three
background solutions, respectively. All the tensor zero modes (massless
gravitons) of the three cases can be localized on the brane. We also calculate
the corrections to the Newtonian potential. On a large scale, the corrections
to the Newtonian potential can be ignored. While on a small scale, the
correction from the volcano-like potential is more pronounced than the other
two cases. Combining the specific corrections to the Newtonian potential of
these three cases and the latest results of short-range gravity experiments, we
get the constraint on the scale parameter as $k\gtrsim 10^{-4}$eV, and
constraint on the corresponding five-dimensional fundamental scale as $M_\ast
\gtrsim10^5$TeV.
