Journal of the European Mathematical Society


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Volume 13, Issue 1, 2011, pp. 1–26
DOI: 10.4171/JEMS/242

Published online: 2010-10-21

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

Guy Barles[1], Emmanuel Chasseigne[2] and Cyril Imbert[3]

(1) Université de Tours, France
(2) Université François Rabelais, Tours, France
(3) Université Paris-Est Créteil Val de Marne, France

This paper is concerned with the 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.

Keywords: Hölder regularity, integro-differential equations, Lévy operators, general non-local operators, viscosity solutions

Barles Guy, Chasseigne Emmanuel, Imbert Cyril: Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. 13 (2011), 1-26. doi: 10.4171/JEMS/242