Lipschitz区域上Dirichlet问题的端点非正则性——一个初等证明
End-point non-regularity of the Dirichlet problem on Lipschitz domains -- An elementary proof
AI总结：
本文构造平面Lipschitz区域，证明从$H^{3/2}\cap H^1_0$到$H^{-1/2}$的拉普拉斯算子像非闭，通过初等方法揭示边界振荡使Dirichlet问题解的$H^{3/2}$范数无界。
AI中文摘要：
我们构造了一个平面Lipschitz区域，使得从$H^{3/2}\cap H^1_0$到$H^{-1/2}$的拉普拉斯算子的像不是闭的。我们证明，边界的快速振荡会产生边界层，导致Dirichlet问题解的$H^{3/2}$范数无界。我们的证明是初等的，因为它直接基于齐次性论证，未使用调和分析的深层结果，如面积积分估计或Dahlberg、Jerison与Kenig的调和测度。
英文摘要：
We construct a plane Lipschitz domain for which the range of the Laplace operator from $H^{3/2}\cap H^1_0$ to $H^{-1/2}$ is not closed. We show that rapid oscillations of the boundary create a boundary layer that leads to unbounded $H^{3/2}$ norm of the solution of the Dirichlet problem. Our proof is elementary in the sense that it is directly based on homogeneity arguments and does not use deep results of harmonic analysis such as estimates of area integrals or harmonic measures of Dahlberg or Jerison and Kenig.
