Existence of \(C^\alpha \) solutions to integro-PDEs (Q2314550)

The author proves the existence of a \(C^{\alpha}\) viscosity solution of the following second order integro-PDE: \[ \sup_{\alpha\in J_1}\inf_{\beta\in J_2} \left\{-\operatorname{tr} A_{\alpha \beta}D^2u(x) -I_{\alpha\beta}[x,u]+b_{\alpha\beta}(x) Du(x)+c_{\alpha\beta}(x)u(x)+f_{\alpha\beta}(x)\right\}=0, \] where \(J_1, J_2\) are two index sets. The main tool is an application of the weak Harnack inequality.