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, p. 203-215
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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 : https://doi.org/10.1016/j.anihpc.2013.02.006
Classification:  35D40,  52A41
Keywords: Quasiconvex, Principal curvature, Convex envelope
@article{AIHPC_2014__31_2_203_0,
     author = {Barron, E.N. and Jensen, R.R.},
     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},
     publisher = {Elsevier},
     volume = {31},
     number = {2},
     year = {2014},
     pages = {203-215},
     doi = {10.1016/j.anihpc.2013.02.006},
     zbl = {1302.35104},
     mrnumber = {3181665},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_2_203_0}
}
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/item/AIHPC_2014__31_2_203_0/

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