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single layer potential; generalized normal derivative
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.
[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] H. Federer: Geometric Measure Theory. Springer-Verlag Berlin, Heidelberg, New York, 1969. MR 0257325 | Zbl 0176.00801
[4] I. Gohberg, A. Marcus: Some remarks on topologically equivalent norms. Izvestija Mold. Fil. Akad. Nauk SSSR 10(76) (1960), 91–95.
[5] 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
[6] N. V. Grachev, V. G. Maz’ya: Invertibility of boundary integral operators of elasticity on surfaces with conic points. Report LiTH-MAT-R-91-50, Linköping Univ., Sweden.
[7] 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.
[8] H. Heuser: Funktionalanalysis. Teubner, Stuttgart, 1975. MR 0482021 | Zbl 0309.47001
[9] J. Král: Integral Operators in Potential Theory. Lecture Notes in Mathematics 823. Springer-Verlag, Berlin, 1980, pp. . MR 0590244
[10] J. Král, W. L. Wendland: Some examples concerning applicability of the Fredholm-Radon method in potential theory. Aplikace matematiky 31 (1986), 293–308. MR 0854323
[11] N. L. Landkof: Fundamentals of modern potential theory. Izdat. Nauka, Moscow, 1966. (Russian) MR 0214795
[12] V. G. Maz’ya: Boundary integral equations. Sovremennyje problemy matematiki, fundamental’nyje napravlenija, 27. Viniti, Moskva, 1988. (Russian)
[13] D. Medková: On the convergence of Neumann series for noncompact operator. Czechoslovak Math. J. 41(116) (1991), 312–316. MR 1105448
[14] D. Medková: The third boundary value problem in potential theory for domains with a piecewise smooth boundary. Czechoslovak Math. J. 47(122) (1997), 651–680. DOI 10.1023/A:1022818618177 | MR 1479311
[15] I. Netuka: The third boundary value problem in potential theory. Czechoslovak Math. J. 22(97) (1972), 554–580. MR 0313528 | Zbl 0242.31007
[16] I. Netuka: Smooth surfaces with infinite cyclic variation. Čas. pěst. mat. 96 (1971), 86–101. (Czech) MR 0284553 | Zbl 0204.08002
[17] C. Neumann: Untersuchungen über das logarithmische und Newtonsche Potential. Teubner Verlag, Leipzig, 1877.
[18] 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.
[19] C. Neumann: Über die Methode des arithmetischen Mittels. Hirzel, Leipzig, 1887 (erste Abhandlung), 1888 (zweite Abhandlung).
[20] J. Plemelj: Potentialtheoretische Untersuchungen. B. G. Teubner, Leipzig, 1911.
[21] J. Radon: Über Randwertaufgaben beim logarithmischen Potential. Sitzber. Akad. Wiss. Wien 128 (1919), 1123–1167.
[22] J. Radon: ARRAY(0x9436470). Birkhäuser, Vienna, 1987.
[23] 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. DOI 10.1080/00036819208840093 | MR 1293594
[24] A. Rathsfeld: The invertibility of the double layer potential operator in the space of continuous functions defined over a polyhedron. The panel method. Erratum. Applicable Analysis 56 (1995), 109–115. DOI 10.1080/00036819508840313 | MR 1378015 | Zbl 0921.31004
[25] F. Riesz, B. Sz. Nagy: Leçons d’analyse fonctionnelles. Budapest, 1952.
[26] M. Schechter: Principles of Functional Analysis. Academic Press, 1973. MR 0445263
[27] L. Schwartz: Theorie des distributions. Hermann, Paris, 1950. MR 0209834 | Zbl 0037.07301
[28] K. Yosida: Functional Analysis. Springer-Verlag, Berlin, 1965. Zbl 0126.11504
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