The time-like minimal surface equation in Minkowski space: low regularity solutions

Abstract

It has long been conjectured that for nonlinear wave equations that satisfy a nonlinear form of the null condition, the low regularity well-posedness theory can be significantly improved compared to the sharp results of Smith-Tataru for the generic case. The aim of this article is to prove the first result in this direction, namely for the time-like minimal surface equation in the Minkowski space-time. Further, our improvement is substantial, namely by \(3/8\) derivatives in two space dimensions and by \(1/4\) derivatives in higher dimensions.