A Sharp Gradient Estimate and $$W^{2,q}$$ Regularity for the Prescribed Mean Curvature Equation in the Lorentz-Minkowski Space

Abstract

We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namelywhere N 3. We first prove a new gradient estimate for classical solutions with smooth data ρ. 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,q and strictly spacelike in R N , provided that ρ ∈ L q (R N ) ∩ L m (R N ) with q > N and m ∈ [1, 2NN +2 ]. Contents