On nonlocal quasilinear equations and their local limits

We introduce a new class of quasilinear nonlocal operators and study equations involving these operators. The operators are degenerate elliptic and may have arbitrary growth in the gradient. Included are new nonlocal versions of p-Laplace, ∞-Laplace, mean curvature of graph, and even strongly degenerate operators. Our main results are non-trivial comparison, uniqueness, and existence results for viscosity solutions of linear and fully-linear equations involving these operators. We also identify the conditions under which the nonlocal operators converge to local quasilinear operators, and show that the solutions of the corresponding nonlocal equations converge to the solutions of the local limit equations. Finally, we give a (formal) stochastic representation formula for the solutions and provide many examples. Loosely speaking our results imply that for “any” quasi-linear 2nd order local operator, “any” well-posed local equation, and “any” non-local Lévy type operator , there is a corresponding Lévy type quasilinear operator and a well-posed non-local equation. Moreover, the solution of any such local equation can be approximated by the solutions of a multitude of different non-local equations.