  Global weighted $L^{p}$-estimates are obtained for the gradient of solutions
to a class of linear singular, degenerate elliptic Dirichlet boundary value
problems over a bounded non-smooth domain. The coefficient matrix is symmetric,
nonnegative definite, and both its smallest and largest eigenvalues are
proportion to a weight in a Muckenhoupt class. Under a smallness condition on
the mean oscillation of the coefficients with the weight and a Reifenberg
flatness condition on the boundary of the domain, we establish a weighted
gradient estimate for weak solutions of the equation. A class of degenerate
coefficients satisfying the smallness condition is characterized. A counter
example to demonstrate the necessity of the smallness condition on the
coefficients is given. Our $W^{1,p}$-regularity estimates can be viewed as the
Sobolev's counterpart of the H\"{o}lder's regularity estimates established by
B. Fabes, C. E. Kenig, and R. P. Serapioni in 1982.
