Min–max formulas for nonlocal elliptic operators

Abstract

In this work, we give a characterization of Lipschitz operators on spaces of $$C^2(M)$$ functions (also $$C^{1,1}$$, $$C^{1,\gamma }$$, $$C^1$$, $$C^\gamma $$) that obey the global comparison property—i.e. those that preserve the global ordering of input functions at any points where their graphs may touch, often called “elliptic” operators. Here M is a complete Riemannian manifold. In particular, we show that all such operators can be written as a min–max over linear operators that are a combination of drift–diffusion and integro-differential parts. In the linear (and nonlocal) case, these operators had been characterized in the 1960s, and in the local, but nonlinear case—e.g. local Hamilton–Jacobi–Bellman operators—this characterization has also been known and used since approximately since 1960s or 1970s. Our main theorem contains both of these results as special cases. It also shows any nonlinear scalar elliptic equation can be represented as an Isaacs equation for an appropriate differential game. Our approach is to “project” the operator to one acting on functions on large finite graphs that approximate the manifold, use non-smooth analysis to derive a min–max formula on this finite dimensional level, and then pass to the limit in order to lift the formula to the original operator.