Quasiconvex Functions and Nonlinear Pdes
نویسنده
چکیده
A second order characterization of functions which have convex level sets (quasiconvex functions) results in the operator L0(Du,Du) = min{v ·D2u vT | |v| = 1, |v ·Du| = 0}. In two dimensions this is the mean curvature operator, and in any dimension L0(Du,Du)/|Du| is the first principal curvature of the surface S = u−1(c). Our main results include a comparison principle for L0(Du,Du) = g when g ≥ Cg > 0 and a comparison principle for quasiconvex solutions of L0(Du,Du) = 0. A more regular version of L0 is introduced, namely Lα(Du,Du) = min{v ·D2u vT | |v| = 1, |v ·Du| ≤ α}, which characterizes functions which remain quasiconvex under small linear perturbations. A comparison principle is proved for Lα. A representation result using stochastic control is also given, and we consider the obstacle problems for L0 and Lα.
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