A Remark on Global Existence for Small Initial Data of the Minimal Surface Equation in Minkowskian Space Time

t=0 = εg f ∈ C 0 and g ∈ C 0 , i.e. (1.1)-(1.2) has for fixed f and g a solution for all t ≥ 0 if ε > 0 is sufficiently small. This is an interesting model in Lorentzian geometry proposed to me by Hamilton[Ha1]. It is also the equation for a membrane, in field theory, see Hoppe[Ho1]. Also Huisken and Struwe[HS1] have some recent results related to local existence for (1.1). What makes the proof go through also in the physically important case of two space dimensions (φ itself corresponds to the third space dimension) is that the nonlinear terms satisfies the so called ”null condition” of Christodoulou[C1] and Klainerman[K2,K4]. The purpose of this note is to present two simple proofs making use of the extra symmetries of the equation. The first proof uses a version of the method of [K2], that works also in two space dimensions. For equations in divergence form we can get a good L estimate for the solution itself, see [L1], that replaces the conformal energy estimate used in [K2]. The second proof uses a simplified version [C2] of the method of [C1] and it works also in one space dimension due to that the equation satisfies a ”double null condition”, see (1.5). The first proof does not work in the case of one space dimension but it has the advantage that it does not require compact support of initial data but merely some decay at infinity. Let (1.3) = ∂ t − 3