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

We consider the sharp interface limit epsilon -> 0(+) of the semilinear wave equation square u + del W(u)/epsilon(2) = 0 in R(1+n), where u takes values in 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 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. (C) 2009 Elsevier B.V. All rights reserved