Russell W. Schwab

PERIODIC HOMOGENIZATION FOR NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

In this note, we prove the periodic homogenization for a family of nonlinear nonlocal “elliptic” equations with oscillatory coefficients. Such equations include but are not limited to Bellman equations for the control of pure jump processes and the Isaacs equations for differential games of pure jump processes. The existence of an effective equation and convergence of the solutions of the family of the original equations is obtained. An inf-sup formula for the effective equation is also provided.

STOCHASTIC HOMOGENIZATION FOR SOME NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

In this note we prove the stochastic homogenization for a large class of fully nonlinear elliptic integro-differential equations in stationary ergodic random environments. Such equations include, but are not limited to, Bellman equations and the Isaacs equations for the control and differential games of some pure jump processes in a random, rapidly varying environment. The translation invariant and non-random effective equation is identified, and the almost everywhere in ω, uniform in x convergence of the family solutions of the original equations is obtained. Even in the linear case of the equations contained herein the results appear to be new.

NEUMANN HOMOGENIZATION VIA INTEGRO-DIFFERENTIAL OPERATORS, PART 2: SINGULAR GRADIENT DEPENDENCE

We continue the program initiated in a previous work, applying integro-differential methods to Neumann homogenization problems. We target the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-conormal oscillatory Neumann conditions. Our analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. Also, we use homogenization results for regular Dirichlet problems to build barriers for the oscillatory Neumann problem with the singular gradient term. We note that our method allows us to recast some existing results for fully nonlinear Neumann homogenization into this same framework.