  We consider the sharp interface limit $\epsilon \to 0$ of the semilinear wave
equation $u_{tt} - \Delta u + \nabla W(u)/ \epsilon^2 = 0$ in $\mathbf
R^{1+n}$, where $u$ takes values in $\mathbf R^k$, $k = 1,2$, and $W$ is a
double-well potential if $k = 1$ and vanishes on the unit circle and is
positive elsewhere if $k = 2$. For fixed $\epsilon > 0$ we find some special
solutions, constructed around minimal surfaces in $\mathbf R^n$. In the general
case, under some additional assumptions, we show that the solutions converge to
a Radon measure supported on a time-like $k$-codimensional minimal submanifold
of the Minkowski space-time. This result holds also after the appearence of
singularities, and enforces the observation made by J. Neu that this semilinear
equation can be regarded as an approximation of the Born-Infeld equation.
