Min-max formulas for nonlocal elliptic operators
  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 \emph{linear} (and nonlocal) case, Courr\`ege had characterized these operators in the 1960's, and in the \emph{local, but nonlinear} case-- e.g. local Hamilton-Jacobi-Bellman operators-- this characterization has also been known for quite some time. Our result gives both a nonlinear extension of Courr\`ege's and a nonlocal extension of well known results for local Hamilton-Jacobi-Bellman equations. 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 a finite dimensional space, where a min-max formula is easier, and then the min-max can be appropriately lifted to the original operator on the infinite dimensional space. As one application, we mention some preliminary results about the structure of Dirichlet-to-Neumann mappings for second order elliptic equations, including fully nonlinear equations. This is the Director's cut, and it contains extra details for our own sanity. 
Nestor Guillen and Russell W. Schwab
math.AP math.FA math.OC math.PR