Calculus of Variations and Geometric Measure Theory
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G. Bellettini - M. Novaga - G. Orlandi

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

created by novaga on 22 Nov 2008
modified on 10 Nov 2018


Published Paper

Inserted: 22 nov 2008
Last Updated: 10 nov 2018

Journal: Physica D
Volume: 239
Pages: 335-339
Year: 2010

ArXiv: 0811.3741 PDF


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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