Obstacle Problems for Integro-Differential Operators with Partially Vanishing Kernels

In this paper, we study the obstacle problem of some convex operators, which is related to normalized p-Laplacian \(\Delta _p^s\) ${\mathrm{\Delta }}_p^s$ . We prove that the graph of the regular free boundary is \(C^{1,\alpha }\) $C^{1,\alpha }$ and the solution is \(C^{1,s}\) $C^{1,s}$ near these points. Moreover, we show that the set of regular free boundary points is relatively open.