We consider the sharp interface limit of a semilinear wave equation of Ginzburg-Landau type in R^(1+n) with potential W(u), depending on a small parameter h>0, where u takes values in R^k, k= 1, 2, and the potential term W(u) is a double-well potential if k=1 and a mexican hat if k=2. For fixed h>0 we find some special solutions, constructed around minimal surfaces in 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 appearance 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.
Time-like minimal submanifolds as singular limits of nonlinear wave equations
ORLANDI, Giandomenico
2010-01-01
Abstract
We consider the sharp interface limit of a semilinear wave equation of Ginzburg-Landau type in R^(1+n) with potential W(u), depending on a small parameter h>0, where u takes values in R^k, k= 1, 2, and the potential term W(u) is a double-well potential if k=1 and a mexican hat if k=2. For fixed h>0 we find some special solutions, constructed around minimal surfaces in 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 appearance 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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