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

TL;DR

This paper proves improved low regularity well-posedness results for the time-like minimal surface equation in Minkowski space, advancing understanding of nonlinear wave equations satisfying the null condition.

Contribution

It establishes the first low regularity well-posedness results for the time-like minimal surface equation, improving regularity thresholds by 3/8 derivatives in 2D and 1/4 in higher dimensions.

Findings

Enhanced regularity thresholds for well-posedness.

First such results for this specific equation.

Significant derivative improvements over previous results.

Abstract

It has long been conjectured that for nonlinear wave equations which 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.