Global regularity for the minimal surface equation in Minkowskian geometry
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Atanas Stefanov
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 Ws,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 L2,∞, 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?
© de Gruyter 2011
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- The sh-Lie algebra perturbation lemma
- Gradient estimates in Orlicz spaces for nonlinear elliptic equations with BMO nonlinearity in nonsmooth domains
- Geography of non-formal symplectic and contact manifolds
- Hp → Hp boundedness implies Hp → Lp boundedness
- Global regularity for the minimal surface equation in Minkowskian geometry
- On elementarily κ-homogeneous unary structures
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Artikel in diesem Heft
- The sh-Lie algebra perturbation lemma
- Gradient estimates in Orlicz spaces for nonlinear elliptic equations with BMO nonlinearity in nonsmooth domains
- Geography of non-formal symplectic and contact manifolds
- Hp → Hp boundedness implies Hp → Lp boundedness
- Global regularity for the minimal surface equation in Minkowskian geometry
- On elementarily κ-homogeneous unary structures
- Kernels, regularity and unipotent radicals in linear algebraic monoids
- Mod-Gaussian convergence: new limit theorems in probability and number theory
- Uniform large deviation for pinned hyperbolic Brownian motion
