Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate fully non-linear elliptic equations

Cyril Imbert (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate when ``the gradient is small''. Typical examples are either equations involving the $m$-Laplace operator or Bellman-Isaacs equations from stochastic control problems. We establish an Alexandroff-Bakelman-Pucci estimate and we prove a Harnack inequality for viscosity solutions of such degenerate elliptic equations.

Mots clés: Degenerate fully nonlinear elliptic equation; singular elliptic equation; non-divergence form; Alexandrov-Bakelman-Pucci estimate; weak Harnack inequality; local maximum principle; Harnack inequality; Hölderregularity; viscosity solutions

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert. Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate fully non-linear elliptic equations. 2009. ⟨hal-00366901v2⟩