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

Abstract

The operator involved in quasiconvex functions is $L(u)={\min }_{|y|=1,y\cdot Du=0}y{D}^{2}u{y}^T$ 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.