Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations

Guy Barles (1) , Emmanuel Chasseigne (1) , Cyril Imbert (2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper is concerned with Hölder regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth conditions on the equation. These results are concerned with a large class of integro-differential operators whose singular measures depend on $x$ and also a large class of equations, including Bellman-Isaacs Equations.

Mots clés: Hölder regularity; integro-differential equations; Lévy operators; general non-local operators; viscosity solutions

Domaines: Equations aux dérivées partielles [math.AP]

Guy Barles, Emmanuel Chasseigne, Cyril Imbert. Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. Journal of the European Mathematical Society, 2011, 13, pp.1-26. ⟨10.4171/JEMS/242⟩. ⟨hal-00179690v2⟩