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

• 成果类型: Article
• 署名作者: Ai, Albert; Ifrim, Mihaela; Tataru, Daniel
• 署名单位: University of Wisconsin System; University of Wisconsin Madison; University of California System; University of California Berkeley
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01231-3
• 发表日期: 2024
• 页码: 745-891
• 关键词: 2nd-order hyperbolic operators; linear wave-equations; local well-posedness; nonsmooth coefficients; global regularity; strichartz; counterexamples; SINGULARITIES; EXISTENCE; energy

摘要

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.