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

Barron, E.N.; Jensen, R.R.

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 203-215.

The operator involved in quasiconvex functions is $L\left(u\right)={\min }_{\left|y\right|=1,y·Du=0}y{D}^{2}u{y}^T$ 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 $L\left(u\right)=g>0$ 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 $L\left(u\right)/\left|Du\right|$ is the minimal principal curvature of a surface, we show by example that $L\left(u\right)-g\left|Du\right|=0$ 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.

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