Global regularity for the minimal surface equation in Minkowskian geometry

Abstract

Abstract. We study the minimal surface equation in Minkowskian geometry in R n R 1 C , which is a well-known quasilinear wave equation. The classical result of Lindblad, [10], 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 D 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;1 , 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?

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

“…On the mathematical side, the question of global existence for small, smooth and localized initial data was considered in work of Lindblad [31], Brendle [8], Stefanov [40] and Wong [48]. The stability of a nonflat steady solution, called the catenoid, was studied in [10,29].…”

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

“…On the mathematical side, the question of global existence for small, smooth and localized initial data was considered in work of Lindblad [29], Brendle [7] and Stefanov [38]. The stability of a nonflat steady solution, called the catenoid, was studied in [27,9].…”