论文标题
在不满意的衡量度量条件下的非平滑域中的共态和罗宾边界价值问题
The conormal and Robin boundary value problems in nonsmooth domains satisfying a measure condition
论文作者
论文摘要
我们以圆锥形或罗宾边界条件为单位的椭圆形方程和系统，具有较小的BMO（有界平均振荡）或可变的部分小BMO系数。我们提出了一类新的域，该域在系统（或标量）情况下的Lebesgue度量方面接近一半空间（或凸形域），并获得均质边界条件的综合问题的$ W^1_p $估计。这种条件比Reifenberg平坦度条件弱，该条件是根据Hausdorff距离和半跨性别条件来衡量的。对于不均匀边界条件的共同问题，我们还假设该域是Lipschitz。通过使用这些结果，我们获得了这些域中罗宾问题的$ W^1_p $和加权$ W^1_p $估计。
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 $W^1_p$ 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 $W^1_p$ and weighted $W^1_p$ estimates for the Robin problem in these domains.
