The operator involved in quasiconvex functions is 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 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 is the minimal principal curvature of a surface, we show by example that does not have a unique solution, even if . Finally, we complete the identification of first order evolution problems giving the convex envelope of a given function.
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}, 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://archive.numdam.org/articles/10.1016/j.anihpc.2013.02.006/} }
TY - JOUR AU - Barron, E.N. AU - Jensen, R.R. TI - A uniqueness result for the quasiconvex operator and first order PDEs for convex envelopes JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 203 EP - 215 VL - 31 IS - 2 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2013.02.006/ DO - 10.1016/j.anihpc.2013.02.006 LA - en ID - AIHPC_2014__31_2_203_0 ER -
%0 Journal Article %A Barron, E.N. %A Jensen, R.R. %T A uniqueness result for the quasiconvex operator and first order PDEs for convex envelopes %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 203-215 %V 31 %N 2 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2013.02.006/ %R 10.1016/j.anihpc.2013.02.006 %G en %F 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://archive.numdam.org/articles/10.1016/j.anihpc.2013.02.006/
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