Global regularity for the minimal surface equation in Minkowskian geometry

Abstract

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. In the 3D case, such a proof fails as a consequence of the well-known Strichartz inequality “missing endpoint” and one has instead slightly weaker results, which requires smallness of the data in certain W^s,p, p < 2, spaces. In the 2D case, this fails as well, since the free solution of the 2D wave equation fails to be square integrable, but only belongs to L^2,∞, another failure by an endpoint. Thus, an important question is left open: Can one prove global regularity for the 2D minimal surface equation, assuming smallness in unweighted Sobolev spaces only?