A Sharp Gradient Estimate and W2,q Regularity for the Prescribed Mean Curvature Equation in the Lorentz-Minkowski Space

We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely -div(del u/root 1-|del u|(2)) =rho in R-N , where N (sic) 3. We first prove a new gradient estimate for classical solutions with smooth data rho. As a consequence, we obtain that the unique weak solution of the equation satisfying a homogeneous boundary condition at infinity is locally of class W-2,W-q and strictly spacelike in R-N, provided that rho is an element of L-q (R-N) boolean AND L-m(R-N) with q > N and m is an element of [1, 2N/N+2].