Existence and regularity for integro‐differential free transmission problem

Abstract We study an integro‐differential free transmission problem associated with the Bellman–Isaacs‐type operator that is solution‐dependent. The existence of a viscosity solution is proved by constructing solutions of suitable auxiliary problems for such a nonlocal problem. We also identify circumstances under which the gradient of the solution enjoys an interior Hölder regularity whose estimates remain uniform as the degree of the equation approaches 2.