A uniqueness result for the quasiconvex operator and first order PDEs for convex envelopes
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, pp. 203-215.

The operator involved in quasiconvex functions is L(u)= min |y|=1,y·Du=0 yD 2 uy 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.

DOI: 10.1016/j.anihpc.2013.02.006
Classification: 35D40, 52A41
Keywords: Quasiconvex, Principal curvature, Convex envelope
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     title = {A uniqueness result for the quasiconvex operator and first order {PDEs} for convex envelopes},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Barron, E.N.; Jensen, R.R. A uniqueness result for the quasiconvex operator and first order PDEs for convex envelopes. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, pp. 203-215. doi : 10.1016/j.anihpc.2013.02.006. http://www.numdam.org/articles/10.1016/j.anihpc.2013.02.006/

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