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

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

doi:10.1016/j.anihpc.2013.02.006

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

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

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Résumé

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

MR Zbl

DOI : 10.1016/j.anihpc.2013.02.006

Classification : 35D40, 52A41
Keywords: Quasiconvex, Principal curvature, Convex envelope

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     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},
     pages = {203--215},
     publisher = {Elsevier},
     volume = {31},
     number = {2},
     year = {2014},
     doi = {10.1016/j.anihpc.2013.02.006},
     mrnumber = {3181665},
     zbl = {1302.35104},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2013.02.006/}
}

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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, Tome 31 (2014) no. 2, pp. 203-215. doi: 10.1016/j.anihpc.2013.02.006

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