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Keywords:
Laplace equation; Dirichlet problem; Neumann problem; Robin problem
Laplace equation; Dirichlet problem; Neumann problem; Robin problem
Summary:
Dirichlet, Neumann and Robin problem for the Laplace equation is investigated on the open set with holes and nonsmooth boundary. The solutions are looked for in the form of a double layer potential and a single layer potential. The measure, the potential of which is a solution of the boundary-value problem, is constructed.
References:
[1] 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
[2] Yu. D. Burago V. G. Maz’ ya: Potential theory and function theory for irregular regions. Seminars in mathematics V. A. Steklov Mathematical Institute, Leningrad, 1969. MR 0240284
[3] 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
[4] 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.
[5] 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.
[6] 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.
[7] M. Chlebík: Tricomi Potentials. Thesis, Mathematical Institute of the Czechoslovak Academy of Sciences, Praha 1988 (in Slovak).
[8] J. Král: On double-layer potential in multidimensional space. Dokl. Akad. Nauk SSSR, 159 (1964). MR 0176210
[9] J. Král: Integral Operators in Potential Theory. Lecture Notes in Mathematics 823, Springer-Verlag, Berlin, 1980. MR 0590244
[10] J. Král: The Fredholm method in potential theory. Trans. Amer. Math. Soc., 125 (1966), 511–547. MR 0209503
[11] J. Král I. Netuka: Contractivity of C. Neumann’s operator in potential theory. Journal of the Mathematical Analysis and its Applications, 61 (1977), 607–619. MR 0508010
[12] J. Král W. L. Wendland: Some examples concerning applicability of the Fredholm-Radon method in potential heory. Aplikace matematiky, 31 (1986), 239–308. MR 0854323
[13] 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
[14] V. Maz’ya A. Solov’ev: On the boundary integral equation of the Neumann problem in a domain with a peak. Amer. Math. Soc. Transl., 155 (1993), 101–127.
[15] D. Medková: The third boundary value problem in potential theory for domains with a piecewise smooth boundary. Czech. Math. J., 47 (1997), 651–679. MR 1479311
[16] D. Medková : Solution of the Neumann problem for the Laplace equation. Czech. Math. J. (in print).
[17] D. Medková: Solution of the Robin problem for the Laplace equation. preprint No.120, Academy of Sciences of the Czech republic, 1997. MR 1609158
[18] I. Netuka: Smooth surfaces with infinite cyclic variation. Čas. pěst. mat., 96 (1971). MR 0284553 | Zbl 0204.08002
[19] I. Netuka: The Robin problem in potential theory. Comment. Math. Univ. Carolinae, 12 (1971), 205–211. MR 0287021 | Zbl 0215.42602
[20] I. Netuka: Generalized Robin problem in potential theory. Czech. Math. J., 22 (97) (1972), 312–324. MR 0294673 | Zbl 0241.31008
[21] I. Netuka: An operator connected with the third boundary value problem in potential theory. Czech Math. J., 22 (97) (1972), 462–489. MR 0316733 | Zbl 0241.31009
[22] I. Netuka: The third boundary value problem in potential theory. Czech. Math. J., 2 (97) (1972), 554–580. MR 0313528 | Zbl 0242.31007
[23] I. Netuka: Fredholm radius of a potential theoretic operator for convex sets. Čas. pěst. mat., 100 (1975), 374–383. MR 0419794 | Zbl 0314.31006
[24] C. Neumann: Untersuchungen über das logarithmische und Newtonsche Potential. Teubner Verlag, Leipzig, 1877.
[25] C. Neumann: 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.
[26] C. Neumann: Über die Methode des arithmetischen Mittels. Hirzel, Leipzig, 1887 (erste Abhandlung), 1888 (zweite Abhandlung).
[27] J. Radon: Über Randwertaufgaben beim logarithmischen Potential. Sitzber. Akad. Wiss. Wien, 128, 1919, 1123–1167.
[28] 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
[29] 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
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