Solution of the Neumann problem for the Laplace equation

Dagmar Medková

Czechoslovak Mathematical Journal (1998)

  • Volume: 48, Issue: 4, page 763-784
  • ISSN: 0011-4642

Abstract

top
For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.

How to cite

top

Medková, Dagmar. "Solution of the Neumann problem for the Laplace equation." Czechoslovak Mathematical Journal 48.4 (1998): 763-784. <http://eudml.org/doc/30453>.

@article{Medková1998,
abstract = {For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.},
author = {Medková, Dagmar},
journal = {Czechoslovak Mathematical Journal},
keywords = {single layer potential; generalized normal derivative; single layer potential; generalized normal derivative; double layer potential},
language = {eng},
number = {4},
pages = {763-784},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Solution of the Neumann problem for the Laplace equation},
url = {http://eudml.org/doc/30453},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Medková, Dagmar
TI - Solution of the Neumann problem for the Laplace equation
JO - Czechoslovak Mathematical Journal
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 4
SP - 763
EP - 784
AB - For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.
LA - eng
KW - single layer potential; generalized normal derivative; single layer potential; generalized normal derivative; double layer potential
UR - http://eudml.org/doc/30453
ER -

References

top
  1. Layer potentials on boundaries with corners and edges, Čas. pěst. mat. 113 (1988), 387–402. (1988) MR0981880
  2. Potential theory and function theory for irregular regions, Seminars in Mathematics, V. A. Steklov Mathematical Institute, Leningrad, 1969. (1969) MR0240284
  3. Geometric Measure Theory, Springer-Verlag Berlin, Heidelberg, New York, 1969. (1969) Zbl0176.00801MR0257325
  4. Some remarks on topologically equivalent norms, Izvestija Mold. Fil. Akad. Nauk SSSR 10(76) (1960), 91–95. (1960) 
  5. On the Fredholm radius for operators of the double layer potential type on piecewise smooth boundaries, Vest. Leningrad. Univ. 19(4) (1986), 60–64. (1986) MR0880678
  6. Invertibility of boundary integral operators of elasticity on surfaces with conic points, Report LiTH-MAT-R-91-50, Linköping Univ., Sweden. 
  7. Solvability of a boundary integral equation on a polyhedron, Report LiTH-MAT-R-91-50, Linköping Univ., Sweden. 
  8. Funktionalanalysis, Teubner, Stuttgart, 1975. (1975) Zbl0309.47001MR0482021
  9. Integral Operators in Potential Theory. Lecture Notes in Mathematics 823, Springer-Verlag, Berlin, 1980, pp. . (1980) MR0590244
  10. Some examples concerning applicability of the Fredholm-Radon method in potential theory, Aplikace matematiky 31 (1986), 293–308. (1986) MR0854323
  11. Fundamentals of modern potential theory, Izdat. Nauka, Moscow, 1966. (Russian) (1966) MR0214795
  12. Boundary integral equations. Sovremennyje problemy matematiki, fundamental’nyje napravlenija, 27, Viniti, Moskva, 1988. (Russian) (1988) 
  13. On the convergence of Neumann series for noncompact operator, Czechoslovak Math. J. 41(116) (1991), 312–316. (1991) MR1105448
  14. 10.1023/A:1022818618177, Czechoslovak Math. J. 47(122) (1997), 651–680. (1997) MR1479311DOI10.1023/A:1022818618177
  15. The third boundary value problem in potential theory, Czechoslovak Math. J. 22(97) (1972), 554–580. (1972) Zbl0242.31007MR0313528
  16. Smooth surfaces with infinite cyclic variation, Čas. pěst. mat. 96 (1971), 86–101. (Czech) (1971) Zbl0204.08002MR0284553
  17. Untersuchungen über das logarithmische und Newtonsche Potential, Teubner Verlag, Leipzig, 1877. (1877) 
  18. Zur Theorie des logarithmischen und des Newtonschen Potentials, Berichte über die Verhandlungen der Königlich Sachsischen Gesellschaft der Wissenschaften zu Leipzig 22 (1870), 49–56, 264–321. (1870) 
  19. Über die Methode des arithmetischen Mittels, Hirzel, Leipzig, 1887 (erste Abhandlung), 1888 (zweite Abhandlung). (1887 (erste Abhandlung), 1888 (zweite Abhandlung)) 
  20. Potentialtheoretische Untersuchungen, B. G. Teubner, Leipzig, 1911. (1911) 
  21. Über Randwertaufgaben beim logarithmischen Potential, Sitzber. Akad. Wiss. Wien 128 (1919), 1123–1167. (1919) 
  22. ARRAY(0x9436470), Birkhäuser, Vienna, 1987. (1987) 
  23. 10.1080/00036819208840093, The panel method. Applicable Analysis 45 (1992), 1–4, 135–177. (1992) MR1293594DOI10.1080/00036819208840093
  24. 10.1080/00036819508840313, Applicable Analysis 56 (1995), 109–115. (1995) Zbl0921.31004MR1378015DOI10.1080/00036819508840313
  25. Leçons d’analyse fonctionnelles, Budapest, 1952. (1952) 
  26. Principles of Functional Analysis, Academic Press, 1973. (1973) MR0445263
  27. Theorie des distributions, Hermann, Paris, 1950. (1950) Zbl0037.07301MR0209834
  28. Functional Analysis, Springer-Verlag, Berlin, 1965. (1965) Zbl0126.11504

Citations in EuDML Documents

top
  1. Dagmar Medková, The Neumann problem for the Laplace equation on general domains
  2. Dagmar Medková, The boundary-value problems for Laplace equation and domains with nonsmooth boundary
  3. Dagmar Medková, Continuous extendibility of solutions of the third problem for the Laplace equation
  4. Dagmar Medková, Continuous extendibility of solutions of the Neumann problem for the Laplace equation
  5. Dagmar Medková, Boundedness of the solution of the third problem for the Laplace equation
  6. Dagmar Medková, Solution of the Dirichlet problem for the Laplace equation

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.