*Published Paper*

**Inserted:** 22 nov 2008

**Last Updated:** 16 mar 2010

**Journal:** Physica D

**Volume:** 239

**Pages:** 335-339

**Year:** 2010

**Abstract:**

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.

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