狄利克雷-诺依曼映射、边界拉普拉斯算子和霍曼德重新发现的手稿
The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander's rediscovered manuscript
摘要：
狄利克雷-诺依曼（DtN）映射与相应边界拉普拉斯算子的平方根有多接近？近年来对这一问题进行了积极的调查。有些令人惊讶的是，其中涉及的许多技术可以追溯到20世纪50年代新发现的Hörmander手稿。我们提出了Hörmander的方法及其应用，重点是特征值估计和谱渐近。特别是，我们得到的结果DtN映射的非光滑边界的黎曼设置，DtN运营商的亥姆霍兹方程和DtN运营商的微分形式。
英文摘要：
How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hörmander from the 1950s. We present Hörmander's approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.
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