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

Abstract

We consider the sharp interface limit → 0 of the semilinear wave equation 2u + ∇W (u)/ 2 = 0 in R, where u takes values in R, 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 > 0 we find some special solutions, constructed around minimal surfaces in R. 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 BornInfeld equation.

Cite this paper

@inproceedings{Bellettini2008TimelikeLM, title={Time-like lorentzian minimal submanifolds as singular limits of nonlinear wave equations}, author={Giovanni Bellettini and Matteo Novaga}, year={2008} }