Parabolic and elliptic equations with singular or degenerate coefficients: The Dirichlet problem

We consider the Dirichlet problem for a class of elliptic and parabolic equations in upper-half space $\mathbb{R}^d_+$, where coefficients are product $x_d^\alpha, \alpha \in (-\infty, 1),$ bounded uniformly matrix coefficients. Thus, singular or degenerate near boundary $\{x_d =0\}$ they may not locally integrable. The novelty work is that we find proper weights under which existence, uniqueness, regularity solutions Sobolev spaces established. These results appear to be first their kind new even if constant. They also readily extended systems equations.