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

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.