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We study the minimal surface equation in Minkowskian geometry in   , which is a well-known quasilinear wave equation. The classical result of Lindblad, [Proc. Amer. Math. Soc. 132: 1095–1102, 2004], establishes global existence of small and smooth solutions (i.e. global regularity), provided the initial data is small, compactly supported and very smooth. In the present paper, we achieve more precise results. We show that, at least when  n  ≥ 4 (or  n  = 3, but with radial data), it is enough to assume the smallness of some scale invariant quantities, involving (unweighted) Sobolev norms only.

Global regularity for the minimal surface equation in Minkowskian geometry