A sharp gradient estimate and \(W^{2,q}\) 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 \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...

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Bibliographic Details
Published in:arXiv.org 2023-08
Main Authors: Bonheure, Denis, Iacopetti, Alessandro
Format: Article
Language:English
Subjects:
Boundary conditions
Curvature
Hyperspaces
Minkowski space
Online Access:Get full text
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Summary: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 \(m\in[1,\frac{2N}{N+2}]\).
ISSN:2331-8422