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The propagation of damage on the square Ising lattice with a corner geometry is studied by means of Monte Carlo simulations. By imposing free boundary conditions at which competing boundary magnetic fields ±h are applied, the system undergoes a filling transition at a temperature Tf(h) lower than the Onsager critical temperature TC. The competing fields cause the formation of two magnetic domains with opposite orientation of the magnetization, separated by an interface that for T larger than Tf(h) (but T<TC) runs along the diagonal of the sample that connects the corners where the magnetic fields of different orientation meet. Also, for T<Tf(h) this interface is localized either close to the corner where the magnetic field is positive or close to the opposite one, with the same probability. It is found that, just at T= Tf(h), the damage initially propagates along the interface of the competing domains, according to a power law given by D(t) µth. The value obtained for the dynamic exponent (h*=0.89(1)) is in agreement with that corresponding to the wetting transition in the slit geometry (Abraham Model) given by hWT=0.91(1). However, for later times the propagation crosses to a new regime such as h**=0.40(2), which is due to the propagation of the damage into the bulk of the magnetic domains. This result can be understood due to the constraints imposed to the propagation of damage by the corner geometry of the system that cause healing at the corners where the interface is attached. The critical points for the damage spreading transition (TD(h)) are evaluated by extrapolation to the thermodynamic limit by using a finite-size scaling approach. Considering error bars, an overlap between the filling and the damage spreading transitions is found, such that Tf(h)= TD(h). The probability distribution of the damage average position P(l0D) and that of the interface between magnetic domains of different orientation P(l0) are evaluated and compared.