Neumann Homogenization via Integro-Differential Operators. Part 2: Singular Gradient Dependence
Nestor Guillen, Russell W. Schwab · SIAM Journal on Mathematical Analysis · 2018
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.