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

  • E.N. Barron

    Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, United States
  • R.R. Jensen

    Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, United States

Abstract

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.

Cite this article

E.N. Barron, R.R. Jensen, A uniqueness result for the quasiconvex operator and first order PDEs for convex envelopes. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 2, pp. 203–215

DOI 10.1016/J.ANIHPC.2013.02.006