On essential norm of the Neumann operator

Dagmar Medková

Mathematica Bohemica (1992)

  • Volume: 117, Issue: 4, page 393-408
  • ISSN: 0862-7959

Abstract

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One of the classical methods of solving the Dirichlet problem and the Neumann problem in 𝐑 m is the method of integral equations. If we wish to use the Fredholm-Radon theory to solve the problem, it is useful to estimate the essential norm of the Neumann operator with respect to a norm on the space of continuous functions on the boundary of the domain investigated, where this norm is equivalent to the maximum norm. It is shown in the paper that under a deformation of the domain investigated by a diffeomorphism, which is conformal (i.e. preserves angles) on a precisely specified part of boundary, for the given norm there exists a norm on the space of continuous functions on the boundary of the deformated domain such that this norm is equivalent to the maximum norm and the essential norms of the corresponding Neumann operators with respect to these norms are the same.

How to cite

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Medková, Dagmar. "On essential norm of the Neumann operator." Mathematica Bohemica 117.4 (1992): 393-408. <http://eudml.org/doc/29218>.

@article{Medková1992,
abstract = {One of the classical methods of solving the Dirichlet problem and the Neumann problem in $\mathbf \{R\}^m$ is the method of integral equations. If we wish to use the Fredholm-Radon theory to solve the problem, it is useful to estimate the essential norm of the Neumann operator with respect to a norm on the space of continuous functions on the boundary of the domain investigated, where this norm is equivalent to the maximum norm. It is shown in the paper that under a deformation of the domain investigated by a diffeomorphism, which is conformal (i.e. preserves angles) on a precisely specified part of boundary, for the given norm there exists a norm on the space of continuous functions on the boundary of the deformated domain such that this norm is equivalent to the maximum norm and the essential norms of the corresponding Neumann operators with respect to these norms are the same.},
author = {Medková, Dagmar},
journal = {Mathematica Bohemica},
keywords = {reduced boundary; interior normal in Federer’s sense; Neumann operator; compact operator; Hausdorff measure; reduced boundary; interior normal in Federer's sense},
language = {eng},
number = {4},
pages = {393-408},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On essential norm of the Neumann operator},
url = {http://eudml.org/doc/29218},
volume = {117},
year = {1992},
}

TY - JOUR
AU - Medková, Dagmar
TI - On essential norm of the Neumann operator
JO - Mathematica Bohemica
PY - 1992
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 117
IS - 4
SP - 393
EP - 408
AB - One of the classical methods of solving the Dirichlet problem and the Neumann problem in $\mathbf {R}^m$ is the method of integral equations. If we wish to use the Fredholm-Radon theory to solve the problem, it is useful to estimate the essential norm of the Neumann operator with respect to a norm on the space of continuous functions on the boundary of the domain investigated, where this norm is equivalent to the maximum norm. It is shown in the paper that under a deformation of the domain investigated by a diffeomorphism, which is conformal (i.e. preserves angles) on a precisely specified part of boundary, for the given norm there exists a norm on the space of continuous functions on the boundary of the deformated domain such that this norm is equivalent to the maximum norm and the essential norms of the corresponding Neumann operators with respect to these norms are the same.
LA - eng
KW - reduced boundary; interior normal in Federer’s sense; Neumann operator; compact operator; Hausdorff measure; reduced boundary; interior normal in Federer's sense
UR - http://eudml.org/doc/29218
ER -

References

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