On parabolic and elliptic equations with singular or degenerate coefficients
Hongjie Dong, Tuoc Phan
Abstract
We study parabolic and elliptic equations of both divergence and non-divergence form in the half space {x d > 0} whose coefficients are the product of x α d , and uniformly nondegenerate bounded measurable matrix-valued functions, where α ∈ (-1, ∞).As such, the coefficients are singular or degenerate near the boundary of the half space.For equations with the conormal or Neumann boundary condition, we prove the existence, uniqueness, and regularity of solutions in weighted Sobolev spaces and mixed-norm weighted Sobolev spaces when the coefficients are only measurable in the x d direction and have small mean oscillation in the other directions in small cylinders.Our results are new even in the special case when the coefficients are constants.