Quasiconvex functions and nonlinear PDEs
Emmanuel Nicholas Barron, Rafal K. Goebel, Robert R. Jensen · Transactions of the American Mathematical Society · 2013
A second order characterization of functions which have convex level sets (quasiconvex functions) results in the operator L 0 ( D u , D 2 u ) = min { v ⋅ D 2 u v T | | v | = 1 , | v ⋅ D u | = 0 } . L_0(Du,D^2u)= \operatorname {min}\{v\cdot D^2u\,v^T\;|\;|v|=1,|v\cdot Du|=0\}. In two dimensions this is the mean curvature operator, and in any dimension L 0 ( D u , D 2 u ) / | D u | L_0(Du,D^2u)/|Du| is the first principal curvature of the surface S = u − 1 ( c ) . S=u^{-1}(c). Our main results include a comparison principle for L 0 ( D u , D 2 u ) = g L_0(Du,D^2u)=g when g ≥ C g > 0 g \geq C