  We prove existence and uniqueness of a solution for the stochastic Allen-Cahn
equation with logarithmic potential and multiplicative Wiener noise, under
homogeneous Neumann boundary condition. The existence of a solution is obtained
in the variational sense by means of an approximated equation involving a
Yosida regularization of the nonlinearity. The noise is assumed to vanish at
the extremal points of the physical relevant domain and to satisfy a suitable
Lipschitz-continuity property, allowing to prove uniform estimates of the
approximated solution. The passage to the limit is then carried out and
continuous dependence on the initial datum of the solution is verified. Under
an additional assumption, the existence of an analytically strong solution is
proved. Finally, estimates for the derivatives of the logarithmic potential are
derived, in view of the study of an optimal control problem associated to the
stochastic Allen-Cahn equation.
