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

We consider the sharp interface limit ϵ → 0 + of the semilinear wave equation □ u + ∇ W ( u ) / ϵ 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 ϵ > 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.