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 $L\left(u\right)={\mathrm{min}}_{|y|=1,y·Du=0}\phantom{\rule{0.166667em}{0ex}}y{D}^{2}u{y}^{T}$ 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 $L\left(u\right)/|Du|$ is the minimal principal curvature of a surface, we show by example that $L\left(u\right)-g|Du|=0$ 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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