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

2021/01/21 by Denis Bonheure, Bonheure, Denis, Alessandro Iacopetti +1 · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2101.08594

openalex publication_date 2021/01/21 · openalex created_date 2023/08/18 · openalex updated_date 2026/07/28

Abstract

We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely -div((∇ u)/(√(1-|∇ u|²)))= ρ \hboxin ℝ^N, where 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,(2N)/(N+2)].

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