Some examples concerning applicability of the Fredholm-Radon method in potential theory

Josef Král; Wolfgang L. Wendland

Aplikace matematiky (1986)

  • Volume: 31, Issue: 4, page 293-308
  • ISSN: 0862-7940

Abstract

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Simple examples of bounded domains D ⊂ 𝐑 3 are considered for which the presence of peculiar corners and edges in the boundary δ D causes that the double layer potential operator acting on the space 𝒮 ( δ D ) of all continuous functions on δ D can for no value of the parameter α be approximated (in the sub-norm) by means of operators of the form α I + T (where I is the identity operator and T is a compact linear operator) with a deviation less then | α | ; on the other hand, such approximability turns out to be possible for α = 1 2 if a new norm is introduced in 𝒮 ( δ D ) with help of a suitable weight function.

How to cite

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Král, Josef, and Wendland, Wolfgang L.. "Some examples concerning applicability of the Fredholm-Radon method in potential theory." Aplikace matematiky 31.4 (1986): 293-308. <http://eudml.org/doc/15456>.

@article{Král1986,
abstract = {Simple examples of bounded domains $D\subset \mathbf \{R\}^3$ 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 $\mathcal \{S\}(\delta D)$ 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 $\alpha = \frac\{1\}\{2\}$ if a new norm is introduced in $\mathcal \{S\}(\delta D)$ with help of a suitable weight function.},
author = {Král, Josef, Wendland, Wolfgang L.},
journal = {Aplikace matematiky},
keywords = {double layer potential; Fredholm-Radom method in potential theory; rectangular; compact boundary; Dirichlet problem; Neumann problem; rectangular; compact boundary; Fredholm-Radon method; Dirichlet problem; double layer potential; Neumann problem},
language = {eng},
number = {4},
pages = {293-308},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some examples concerning applicability of the Fredholm-Radon method in potential theory},
url = {http://eudml.org/doc/15456},
volume = {31},
year = {1986},
}

TY - JOUR
AU - Král, Josef
AU - Wendland, Wolfgang L.
TI - Some examples concerning applicability of the Fredholm-Radon method in potential theory
JO - Aplikace matematiky
PY - 1986
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 31
IS - 4
SP - 293
EP - 308
AB - Simple examples of bounded domains $D\subset \mathbf {R}^3$ 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 $\mathcal {S}(\delta D)$ 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 $\alpha = \frac{1}{2}$ if a new norm is introduced in $\mathcal {S}(\delta D)$ with help of a suitable weight function.
LA - eng
KW - double layer potential; Fredholm-Radom method in potential theory; rectangular; compact boundary; Dirichlet problem; Neumann problem; rectangular; compact boundary; Fredholm-Radon method; Dirichlet problem; double layer potential; Neumann problem
UR - http://eudml.org/doc/15456
ER -

References

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  8. J. Král, 10.2307/1994580, Trans. Amer. Math. Soc. 125 (1966), 511-547. (1966) MR0209503DOI10.2307/1994580
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  10. J. Radon, Über lineare Funktionaltransformationen und Funktionalgleichungen, Sitzber. Akad. Wiss. Wien, Math.-Nat. Kl. IIa, 128 (1919), 1083-1121. (1919) 
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  12. F. Riesz B. Sz.-Nagy, Leçons d'analyse fonctionnelle, Akadémiai Kiadó, Budapest 1972. (1972) 
  13. G. Verchota, 10.1016/0022-1236(84)90066-1, J. of Functional Analysis 59 (1984), 572-611. (1984) Zbl0589.31005MR0769382DOI10.1016/0022-1236(84)90066-1
  14. W. Wendland, Lösung der ersten und zweiten Randwertaufgaben des Innen- und Aussengebietes für Potentialgleichung im R 3 durch Randbelegungen, Bericht des Hahn-Meitner Institute für Kernforschung Berlin, HMI-B 41, BM 19 (1965), 1-99. (1965) MR0232010
  15. W. Wendland, 10.1007/BF02161886, Numerische Mathematik 11 (1968), 380-404. (1968) MR0231550DOI10.1007/BF02161886
  16. W. Wendland, Boundary element methods and their asymptotic convergence, Preprint Nr. 690, TH Darmstadt (1982), 1-82. (1982) MR0762829
  17. T. S. Angell R. E. Kleinman J. Král, Double layer potentials on boundaries with corners and edges, Comment. Math. Univ. Carolinae 27 (1986). (1986) 

Citations in EuDML Documents

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  1. Thomas S. Angell, Ralph Ellis Kleinman, Josef Král, Layer potentials on boundaries with corners and edges
  2. Marius Mitrea, Victor Nistor, Boundary value problems and layer potentials on manifolds with cylindrical ends
  3. Dagmar Medková, On the convergence of Neumann series for noncompact operators
  4. Josef Král, Dagmar Medková, Essential norms of the Neumann operator of the arithmetical mean
  5. Dagmar Medková, Solution of the Neumann problem for the Laplace equation
  6. Josef Král, Dagmar Medková, Essential norms of a potential theoretic boundary integral operator in L 1
  7. Dagmar Medková, Solution of the Robin problem for the Laplace equation
  8. Dagmar Medková, The boundary-value problems for Laplace equation and domains with nonsmooth boundary
  9. Dagmar Medková, Invariance of the Fredholm radius of the Neumann operator
  10. Dagmar Medková, On essential norm of the Neumann operator

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