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), p. 203-215 / Harvested from Numdam

Résumé

The operator involved in quasiconvex functions is $ L(u)={\mathrm{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.

Détail

Publié le : 2014-01-01

MR 3181665 | Zbl 1302.35104

DOI : https://doi.org/10.1016/j.anihpc.2013.02.006
Classification:  35D40,  52A41

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