Workshop on Pdes 2022 - Bonheure

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

Prof. Denis Bonheure
Université Libre de Bruxelles 

Date: 15th February 10:00 pm - 10:45 pm (UTC -3) Link: PPGM/UFSCar Youtube channel

Abstract: We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely

 −div (∇u/√(1−|∇u|2 )) = ρ   in RN,

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 RN, provided that ρ ∈ Lq(RN)∩Lm(RN) with q>N and m ∈ [1, 2N/(N+2)].