On homogenization problems with oscillating Dirichlet conditions in space–time domains
Yuming Paul Zhang · Journal of the London Mathematical Society · 2022
We prove the homogenization of fully nonlinear parabolic equations with periodic oscillating Dirichlet boundary conditions on certain general prescribed space–time domains. It was proved in (Feldman, J. Math. Pures Appl. 101 (2014), no. 5, 599–622; Feldman and Kim, Ann. Sci. Éc. Norm. Supér 50 (2017), no. 4, 1017–1064) that for elliptic equations, the homogenized boundary data exist at boundary points with irrational normal directions, and it is generically discontinuous elsewhere. However, for parabolic problems, on a flat moving part of the boundary, we prove the existence of continuous homogenized boundary data g ¯ $\bar{g}$ . We also show that, unlike the elliptic case, g ¯ $\bar{g}$ can be discontinuous even if the operator is rotation/reflection invariant.