Time-like lorentzian minimal submanifolds as singular limits of nonlinear wave equations

We consider the sharp interface limit $ϵ\to 0$ of the semilinear wave equation $u_{tt}-\mathrm{\Delta }u+\mathrm{\nabla }W(u)\mathrm{/}{ϵ}^2=0$ in $\mathbf R^{1+n}$,where$u$takesvaluesin$\mathbf R^k$,$k = 1,2$,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 $ϵ>0$ 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.