We consider elliptic equations and systems in divergence form with the conormal or the Robin boundary conditions, with small BMO (bounded mean oscillation) or variably partially small BMO coefficients. We propose a new class of domains which are locally close to a half space (or convex domains) with respect to the Lebesgue measure in the system (or scalar, respectively) case, and obtain the $W_p^1$ estimate for the conormal problem with the homogeneous boundary condition. Such condition is weaker than the Reifenberg flatness condition, for which the closeness is measured in terms of the Hausdorff distance, and the semi-convexity condition. For the conormal problem with inhomogeneous boundary conditions, we also assume that the domain is Lipschitz. By using these results, we obtain the $W_p^1$ and weighted $W_p^1$ estimates for the Robin problem in these domains.

我们考虑具有共正规或罗宾边界条件的发散形式的椭圆方程和系统，具有小 BMO（有界平均振荡）或可变部分小 BMO 系数。我们提出了一类新的域，它们相对于系统（或分别为标量）情况下的 Lebesgue 测度局部接近半空间（或凸域），并获得${宽}_{磷}^1$估计具有齐次边界条件的共正规问题。这种条件比 Reifenberg 平坦度条件弱，后者的接近度是根据 Hausdorff 距离和半凸条件来衡量的。对于非齐次边界条件的共正规问题，我们还假设域是 Lipschitz。通过使用这些结果，我们得到${宽}_{磷}^1$ 和加权 ${宽}_{磷}^1$ 这些领域中罗宾问题的估计。