A uniqueness result for the quasiconvex operator and first order PDEs for convex envelopes

The operator involved in quasiconvex functions is and this also arises as the governing operator in a worst case tug-of-war (Kohn and Serfaty (2006) [7], ) and principal curvature of a surface. In Barron et al. (2012) [4], a comparison principle for L(u)=g>0 was proved. A new and much simpler proof is presented in this paper based on Barles and Busca (2001) [3], and Lu and Wang (2008) [8]. Since L(u)/|Du| is the minimal principal curvature of a surface, we show by example that L(u)−g|Du|=0 does not have a unique solution, even if g>0. Finally, we complete the identification of first order evolution problems giving the convex envelope of a given function.