Title: Recent results on the rates of convergence in periodic homogenization of elliptic equations in nondivergence form
Abstract: Homogenization theory deals with the analysis of averaging effects in problems with structure at one or many microscopic scales, expressed though singular dependence on a small parameter or, equivalently, through dependence on a rapidly oscillating variable. Natural questions are if solutions of such oscillatory problems converge to those of an averaged, or effective problem, and if some explicit rate can be given for this convergence.
We present a couple of results and ongoing work in the framework of periodic homogenization of nonvariational, or nondivergence form elliptic PDEs, addressing these questions in the case of the principal eigenvalue problem associated to positive-homogeneous, uniformly elliptic, second-order operators, and for Hamilton-Jacobi equations with nonlocal diffusion.