Geodesic
On essential norm of the Neumann operator
Mathematica Bohemica, Tome 117 (1992) no. 4, pp. 393-408

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One of the classical methods of solving the Dirichlet problem and the Neumann problem in $\bold R^m$ 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.
One of the classical methods of solving the Dirichlet problem and the Neumann problem in $\bold R^m$ 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.
DOI : 10.21136/MB.1992.126064
Classification : 31B20, 47B38, 47G10
Keywords: reduced boundary; interior normal in Federer’s sense; Neumann operator; compact operator; Hausdorff measure 
Medková, Dagmar. On essential norm of the Neumann operator. Mathematica Bohemica, Tome 117 (1992) no. 4, pp. 393-408. doi: 10.21136/MB.1992.126064
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