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

作者：Barron E N^*; Jensen R R

来源：ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, 2014, 31(2): 203-215.

DOI：10.1016/j.anihpc.2013.02.006

• Quasiconvex
• Principal curvature
• Convex envelope

The operator involved in quasiconvex functions is L(u)= min(vertical bar y vertical bar=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 %26gt; 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)/vertical bar Du vertical bar is the minimal principal curvature of a surface, we show by example that L(u) - g vertical bar Du vertical bar= 0 does not have a unique solution, even if g %26gt; 0. Finally, we complete the identification of first order evolution problems giving the convex envelope of a given function.

• 出版日期2014-4