A sharp gradient estimate and $W^{2,q}$ regularity for the prescribed mean curvature equation in the Lorentz-Minkowski space

TL;DR

This paper establishes a new gradient estimate and proves $W^{2,q}$ regularity for solutions to the prescribed mean curvature equation in Lorentz-Minkowski space, advancing understanding of spacelike hypersurfaces with prescribed curvature.

Contribution

It introduces a novel gradient estimate for classical solutions and demonstrates $W^{2,q}$ regularity for weak solutions under certain integrability conditions on the data.

Findings

Established a new gradient estimate for solutions.

Proved local $W^{2,q}$ regularity of solutions.

Showed solutions are strictly spacelike under specified conditions.

Abstract

We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely \begin{equation*} -\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= \rho \quad \hbox{in }\mathbb{R}^N, \end{equation*} where $N\geq 3$. 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 $W^{2,q}$ and strictly spacelike in $\mathbb{R}^N$, provided that $\rho\in L^q(\mathbb{R}^N) \cap L^m(\mathbb{R}^N)$ with $q>N$ and $m\in[1,\frac{2N}{N+2}]$.