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

TL;DR

This paper studies the limit of semilinear wave equations as a parameter approaches zero, showing convergence to time-like minimal submanifolds in Minkowski space, thus linking wave equations to geometric minimal surface theory.

Contribution

It establishes the convergence of solutions of nonlinear wave equations to time-like minimal submanifolds, extending the understanding of their singular limits and geometric interpretations.

Findings

Solutions converge to a Radon measure on minimal submanifolds

Results hold even after singularity formation

Links semilinear wave equations to Born-Infeld and minimal surface theories

Abstract

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.