MiS Preprint Repository

Given a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ and two integro-differential operators $L^1,L^2$ of the form $L^{j}u(x)=p.v.{\int }_{\mathrm{\Omega }}(u(x)-u(y))k^j(x,y,x-y)dy$ we study the fully nonlinear Bellman equation $$\begin{array}{rl}\underset{j=1,2}{max}\{L^{j}u(x)+a^j(x)u(x)-f^j(x)\}& =0\text{ in }\mathrm{\Omega }(0.1)\end{array}$$ with Dirichlet boundary conditions. Here, $a^j,f^j:\mathrm{\Omega }\to \mathbb{R}$ are nonnegative functions. We prove the existence of a nonnegative function $u:\mathrm{\Omega }\to \mathbb{R}$ which satisfies $(0.1)$ almost everywhere. The main difficulty arises through the nonlocality of $L^j$ and the absence of regularity near the boundary.