| Title:
|
Solution of the Dirichlet problem for the Laplace equation (English) |
| Author:
|
Medková, Dagmar |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
44 |
| Issue:
|
2 |
| Year:
|
1999 |
| Pages:
|
143-168 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For open sets with a piecewise smooth boundary it is shown that a solution of the Dirichlet problem for the Laplace equation can be expressed in the form of the sum of the single layer potential and the double layer potential with the same density, where this density is given by a concrete series. (English) |
| Keyword:
|
Laplace equation |
| Keyword:
|
Dirichlet problem |
| Keyword:
|
single layer potential |
| Keyword:
|
double layer potential |
| MSC:
|
31B10 |
| MSC:
|
35C15 |
| MSC:
|
35J05 |
| MSC:
|
35J25 |
| idZBL:
|
Zbl 1060.35041 |
| idMR:
|
MR1667634 |
| DOI:
|
10.1023/A:1022209421576 |
| . |
| Date available:
|
2009-09-22T18:00:23Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134409 |
| . |
| Reference:
|
[1] F. Ahner: The exterior Dirichlet problem for the Helmholtz equation.Journal of mathematical analysis and application 52 (1975), 415–429. Zbl 0326.45001, MR 0418658, 10.1016/0022-247X(75)90069-4 |
| Reference:
|
[2] R. S. Angell, R. E. Kleinman, J. Král: Layer potentials on boundaries with corners and edges.Čas. pěst. mat. 113 (1988), 387–402. MR 0981880 |
| Reference:
|
[3] H. Bauer: Harmonische Räume und ihre Potentialtheorie.Springer Verlag, Berlin, 1966. Zbl 0142.38402, MR 0210916 |
| Reference:
|
[4] A. S. Besicovitch: A general form of the covering principle and relative differentiation of additive functions I.Proc. Cambridge Phil. Soc. 41 (1945), 103–110. MR 0012325 |
| Reference:
|
[5] A. S. Besicovitch: A general form of the covering principle and relative differentiation of additive functions II.Proc. Cambridge Phil. Soc. 42 (1946), 1–10. Zbl 0063.00353, MR 0014414 |
| Reference:
|
[6] N. Boboc, C. Constantinescu, A. Cornea: On the Dirichlet problem in the axiomatic theory of harmonic functions.Nagoya Math. J. 23 (1963), 73–96. MR 0162957 |
| Reference:
|
[7] Yu. D. Burago, V. G. Maz’ ya: Potential theory and function theory for irregular regions.Leningrad, 1969. (Russian Seminars in mathematics V. A. Steklov Mathematical Institute) MR 0240284 |
| Reference:
|
[8] M. Chlebík: Tricomi potentials.Thesis, Mathematical Institute of the Czechoslovak Academy of Sciences, Praha, 1988. (Slovak) |
| Reference:
|
[9] D. Colton, R. Kress: Iterative methods for solving the exterior Dirichlet problem for the Helmholtz equation with applications to the inverse scattering problem for low frequency acoustic waves.Journal of mathematical analysis and applications 77 (1980), 60–72. MR 0591262, 10.1016/0022-247X(80)90261-9 |
| Reference:
|
[10] M. Dont: Non-tangential limits of the double layer potentials.Čas. pěst. mat. 97 (1972), 231–258. Zbl 0237.31012, MR 0444975 |
| Reference:
|
[11] H. Federer: Geometric Measure Theory.Springer-Verlag, 1969. Zbl 0176.00801, MR 0257325 |
| Reference:
|
[12] I. C. Gohberg, M. S. Krein: The basic propositions on defect numbers, root numbers and indices of linear operators.Amer. Math. Soc. Transl. 13 (1960), 185–264. MR 0113146, 10.1090/trans2/013/08 |
| Reference:
|
[13] N. V. Grachev, V. G. Maz’ya: On the Fredholm radius for operators of the double layer potential type on piecewise smooth boundaries.Vest. Leningrad. Univ. 19(4) (1986), 60–64. MR 0880678 |
| Reference:
|
[14] N. V. Grachev, V. G. Maz’ya: Estimates for kernels of the inverse operators of the integral equations of elasticity on surfaces with conic points.Report LiTH-MAT-R-91-06, Linköping Univ., Sweden. |
| Reference:
|
[15] N. V. Grachev, V. G. Maz’ya: Invertibility of boundary integral operators of elasticity on surfaces with conic points.Report LiTH-MAT-R-91-07, Linköping Univ., Sweden. |
| Reference:
|
[16] N. V. Grachev, V. G. Maz’ya: Solvability of a boundary integral equation on a polyhedron.Report LiTH-MAT-R-91-50, Linköping Univ., Sweden. |
| Reference:
|
[17] H. Heuser: Funktionalanalysis.Teubner, Stuttgart, 1975. Zbl 0309.47001, MR 0482021 |
| Reference:
|
[18] J. Köhn, M. Sieveking: Zum Cauchyschen und Dirichletschen Problem.Math. Ann. 177 (1968), 133–142. MR 0227445, 10.1007/BF01350789 |
| Reference:
|
[19] V. Kordula, V. Müller, V. Rakočević: On the semi-Browder spectrum.Studia Math. 123 (1997), 1–13. MR 1438302 |
| Reference:
|
[20] J. Král: Integral Operators in Potential Theory. Lecture Notes in Mathematics 823.Springer-Verlag, Berlin, 1980. MR 0590244 |
| Reference:
|
[21] J. Král: The Fredholm method in potential theory.Trans. Amer. Math. Soc. 125 (1966), 511–547. MR 0209503, 10.2307/1994580 |
| Reference:
|
[22] J. Král, W. L. Wendland: Some examples concerning applicability of the Fredholm-Radon method in potential theory.Aplikace matematiky 31 (1986), 239–308. MR 0854323 |
| Reference:
|
[23] R. Kress, G. F. Roach: On the convergence of successive approximations for an integral equation in a Green’s function approach to the Dirichlet problem.Journal of mathematical analysis and applications 55 (1976), 102–111. MR 0411214, 10.1016/0022-247X(76)90280-8 |
| Reference:
|
[24] V. G. Maz’ya: Boundary Integral Equations.Sovremennyje problemy matematiki, fundamental’nyje napravlenija, 27, Viniti, Moskva (Russian), 1988. |
| Reference:
|
[25] D. Medková: The third boundary value problem in potential theory for domains with a piecewise smooth boundary.Czechoslov. Math. J. 47 (1997), 651–679. MR 1479311, 10.1023/A:1022818618177 |
| Reference:
|
[26] D. Medková: Solution of the Neumann problem for the Laplace equation.Czechoslov. Math. J. 48 (1998), 768–784. MR 1658269, 10.1023/A:1022447908645 |
| Reference:
|
[27] D. Medková: Solution of the Robin problem for the Laplace equation.Appl. of Math. 43 (1998), 133–155. MR 1609158, 10.1023/A:1023267018214 |
| Reference:
|
[28] M. Miranda: Distribuzioni aventi derivate misure, Insiemi di perimetro localmente finito.Ann. Scuola norm. Sup. Pisa Cl. Sci. 18 (1964), 27–56. Zbl 0131.11802, MR 0165073 |
| Reference:
|
[29] D. Mitrea: The method of layer potentials for non-smooth domains with arbitrary topology.Integr. equ. oper. theory 29 (1997), 320–338. Zbl 0915.35032, MR 1477324, 10.1007/BF01320705 |
| Reference:
|
[30] A. P. Morse: Perfect blancets.Trans. Am. Math. Soc. 61 (1947), 418–442. MR 0020618, 10.1090/S0002-9947-1947-0020618-0 |
| Reference:
|
[31] I. Netuka: Fredholm radius of a potential theoretic operator for convex sets.Čas. pěst. mat. 100 (1975), 374–383. Zbl 0314.31006, MR 0419794 |
| Reference:
|
[32] I. Netuka: An operator connected with the third boundary value problem in potential theory.Czechoslov. Math. J. 22(97) (1972), 462–489. Zbl 0241.31009, MR 0316733 |
| Reference:
|
[33] I. Netuka: The third boundary value problem in potential theory.Czechoslov. Math. J. 2(97) (1972), 554–580. Zbl 0242.31007, MR 0313528 |
| Reference:
|
[34] I. Netuka: Smooth surfaces with infinite cyclic variation (Czech).Čas. pěst. mat. 96 (1971), 86–101. MR 0284553 |
| Reference:
|
[35] I. Netuka: Generalized Robin problem in potential theory.Czechoslov. Math. J. 22(97) (1972), 312–324. Zbl 0241.31008, MR 0294673 |
| Reference:
|
[36] I. Netuka: Double layer potential representation of the solution of the Dirichlet problem.Comment. Math. Un. Carolinae 14 (1973), 183–186. Zbl 0255.31009, MR 0316725 |
| Reference:
|
[37] I. Netuka: Double layer potentials and Dirichlet problem.Czechoslov. Math. J. 24(99) (1974), 59–73. MR 0348127 |
| Reference:
|
[38] A. Rathsfeld: The invertibility of the double layer potential in the space of continuous functions defined on a polyhedron. The panel method.Applicable Analysis 45 (1992), 1–4, 135–177. MR 1293594, 10.1080/00036819208840093 |
| Reference:
|
[39] A. Rathsfeld: The invertibility of the double layer potential in the space of continuous functions defined on a polyhedron. The panel method. Erratum.Applicable Analysis 56 (1995), 109–115. MR 1378015, 10.1080/00036819508840313 |
| Reference:
|
[40] M. Schechter: Principles of Funtional Analysis.Academic Press, London, 1971. MR 0445263 |
| Reference:
|
[41] H. Watanabe: Double layer potentials for a bounded domain with fractal boundary.in Potential Theory. ICPT 94, ed. J. Král, J. Lukeš, I. Netuka, J. Veselý, Walter de Gruyter, Berlin, New York, 1996, pp. 463–471. Zbl 0854.31001, MR 1404730 |
| Reference:
|
[42] K. Yosida: Functional Analysis.Springer Verlag, 1965. Zbl 0126.11504 |
| Reference:
|
[43] W. P. Ziemer: Weakly Differentiable Functions.Springer Verlag, 1989. Zbl 0692.46022, MR 1014685 |
| . |
