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Lie群論・表現論セミナー
過去の記録 ～06/21｜次回の予定｜今後の予定 06/22～
| 開催情報 | 火曜日 16:30～18:00 数理科学研究科棟(駒場) 126号室 |
|---|---|
| 担当者 | 小林俊行 |
| セミナーURL | https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html |
2008年10月14日(火)
16:30-18:00 数理科学研究科棟(駒場) 126号室
Jan Moellers 氏 (Paderborn University)
The Dirichlet-to-Neumann map as a pseudodifferential
operator
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
Jan Moellers 氏 (Paderborn University)
The Dirichlet-to-Neumann map as a pseudodifferential
operator
[ 講演概要 ]
Both Dirichlet and Neumann boundary conditions for the Laplace equation are of fundamental importance in Mathematics and Physics. Given a compact connected Riemannian manifold M with boundary partialM the Dirichlet-to-Neumann operator Lambdag maps Dirichlet boundary data f to the corresponding Neumann boundary data \\Lambda_g f =(\\partial_\\nu u)|_{\\partial M} where u denotes the unique solution to the Dirichlet problem laplacegu=0 in M, u|partialM=f.
The main statement is that this operator is a first order elliptic pseudodifferential operator on the boundary partialM.
We will first give a brief overview of how to define the Dirichlet-to-Neumann operator as a map Lambdag:H1/2(partialM)longrightarrowH−1/2(partialM) between Sobolev spaces. In order to show that it is actually a pseudodifferential operator we introduce tangential pseudodifferential operators. This allows us to derive a
microlocal factorization of the Laplacian near boundary points. Together with a regularity statement for the heat equation this will finally give the main result.
[ 参考URL ]Both Dirichlet and Neumann boundary conditions for the Laplace equation are of fundamental importance in Mathematics and Physics. Given a compact connected Riemannian manifold M with boundary partialM the Dirichlet-to-Neumann operator Lambdag maps Dirichlet boundary data f to the corresponding Neumann boundary data \\Lambda_g f =(\\partial_\\nu u)|_{\\partial M} where u denotes the unique solution to the Dirichlet problem laplacegu=0 in M, u|partialM=f.
The main statement is that this operator is a first order elliptic pseudodifferential operator on the boundary partialM.
We will first give a brief overview of how to define the Dirichlet-to-Neumann operator as a map Lambdag:H1/2(partialM)longrightarrowH−1/2(partialM) between Sobolev spaces. In order to show that it is actually a pseudodifferential operator we introduce tangential pseudodifferential operators. This allows us to derive a
microlocal factorization of the Laplacian near boundary points. Together with a regularity statement for the heat equation this will finally give the main result.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
