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Some examples concerning applicability of the Fredholm-Radon method in potential theory
Applications of Mathematics, Tome 31 (1986) no. 4, pp. 293-308
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Simple examples of bounded domains $D\subset \bold R^3$ are considered for which the presence of peculiar corners and edges in the boundary $\delta D$ causes that the double layer potential operator acting on the space $\Cal S(\delta D)$ of all continuous functions on $\delta D$ can for no value of the parameter $\alpha$ be approximated (in the sub-norm) by means of operators of the form $\alpha I+T$ (where $I$ is the identity operator and $T$ is a compact linear operator) with a deviation less then $|\alpha|$; on the other hand, such approximability turns out to be possible for $\alpha = \frac 12$ if a new norm is introduced in $\Cal S(\delta D)$ with help of a suitable weight function.
Simple examples of bounded domains $D\subset \bold R^3$ are considered for which the presence of peculiar corners and edges in the boundary $\delta D$ causes that the double layer potential operator acting on the space $\Cal S(\delta D)$ of all continuous functions on $\delta D$ can for no value of the parameter $\alpha$ be approximated (in the sub-norm) by means of operators of the form $\alpha I+T$ (where $I$ is the identity operator and $T$ is a compact linear operator) with a deviation less then $|\alpha|$; on the other hand, such approximability turns out to be possible for $\alpha = \frac 12$ if a new norm is introduced in $\Cal S(\delta D)$ with help of a suitable weight function.
DOI : 10.21136/AM.1986.104208
Classification : 31B20, 47A53, 47B38
Keywords: double layer potential; Fredholm-Radom method in potential theory; rectangular; compact boundary; Dirichlet problem; Neumann problem 
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Král, Josef; Wendland, Wolfgang. Some examples concerning applicability of the Fredholm-Radon method in potential theory. Applications of Mathematics, Tome 31 (1986) no. 4, pp. 293-308. doi: 10.21136/AM.1986.104208

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