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Article

Keywords:
third problem; Laplace equation; continuous extendibility
Summary:
A necessary and sufficient condition for the continuous extendibility of a solution of the third problem for the Laplace equation is given.
References:
[1] V.  Anandam and M. A.  Al-Gwaiz: Global representation of harmonic and biharmonic functions. Potential Anal. 6 (1997), 207–214. DOI 10.1023/A:1017908608650 | MR 1452543
[2] V.  Anandam and M.  Damlakhi: Harmonic singularity at infinity in  $R^n$. Real Anal. Exchange 23 (1997/8), 471–476. MR 1639952
[3] T. S.  Angell, R.  E.  Kleinman and J.  Král: Layer potentials on boundaries with corners and edges. Čas. pěst. mat. 113 (1988), 387–402. MR 0981880
[4] Yu. D.  Burago and V. G.  Maz’ya: Potential theory and function theory for irregular regions. Zapiski Naučnyh Seminarov LOMI 3 (1967), 1–152 (In Russian).
[5] L. E.  Fraenkel: Introduction to Maximum Principles and Symmetry in Elliptic Problems. Cambridge Tracts in Mathematics 128. Cambridge University Press, 2000. MR 1751289
[6] N. V.  Grachev and V. G. Maz’ya: On the Fredholm radius for operators of the double layer potential type on piecewise smooth boundaries. Vest. Leningrad. Univ. 19 (1986), 60–64. MR 0880678
[7] N. V.  Grachev and 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, .
[8] N. V.  Grachev and V. G.  Maz’ya: Solvability of a boundary integral equation on a polyhedron. Report LiTH-MAT-R-91-50, Linköping Univ., Sweden, .
[9] N. V.  Grachev and 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, .
[10] L. L.  Helms: Introduction to Potential Theory. Pure and Applied Mathematics 22. John Wiley & Sons, 1969. MR 0261018
[11] J.  Král: Integral Operators in Potential Theory. Lecture Notes in Mathematics 823. Springer-Verlag, Berlin, 1980. MR 0590244
[12] J. Král: The Fredholm method in potential theory. Trans. Amer. Math. Soc. 125 (1966), 511–547. DOI 10.2307/1994580 | MR 0209503
[13] J. Král and W. L.  Wendland: Some examples concerning applicability of the Fredholm-Radon method in potential theory. Aplikace matematiky 31 (1986), 293–308. MR 0854323
[14] N. L.  Landkof: Fundamentals of Modern Potential Theory. Izdat. Nauka, Moscow, 1966. (Russian) MR 0214795
[15] D.  Medková: The third boundary value problem in potential theory for domains with a piecewise smooth boundary. Czechoslovak Math.  J. 47(122) (1997), 651–679. DOI 10.1023/A:1022818618177 | MR 1479311
[16] D.  Medková: Solution of the Robin problem for the Laplace equation. Appl. Math. 43 (1998), 133–155. DOI 10.1023/A:1023267018214 | MR 1609158
[17] D.  Medková: Solution of the Neumann problem for the Laplace equation. Czechoslovak Math.  J. 48(123) (1998), 768–784. DOI 10.1023/A:1022447908645 | MR 1658269
[18] D.  Medková: Continuous extendibility of solutions of the Neumann problem for the Laplace equation. Czechoslovak Math.  J 53(128) (2003), 377–395. DOI 10.1023/A:1026239404667 | MR 1983459
[19] J.  Nečas: Les méthodes directes en théorie des équations élliptiques. Academia, Prague, 1967. MR 0227584
[20] I.  Netuka: Fredholm radius of a potential theoretic operator for convex sets. Čas. pěst. mat. 100 (1975), 374–383. MR 0419794 | Zbl 0314.31006
[21] I.  Netuka: Generalized Robin problem in potential theory. Czechoslovak Math.  J. 22(97) (1972), 312–324. MR 0294673 | Zbl 0241.31008
[22] I.  Netuka: An operator connected with the third boundary value problem in potential theory. Czechoslovak Math.  J. 22(97) (1972), 462–489. MR 0316733 | Zbl 0241.31009
[23] I.  Netuka: The third boundary value problem in potential theory. Czechoslovak Math.  J. 22(97) (1972), 554–580. MR 0313528 | Zbl 0242.31007
[24] I.  Netuka: Continuity and maximum principle for potentials of signed measures. Czechoslovak Math.  J. 25(100) (1975), 309–316. MR 0382690 | Zbl 0309.31019
[25] A.  Rathsfeld: The invertibility of the double layer potential in the space of continuous functions defined on a polyhedron. The panel method. Appl. Anal. 45 (1992), 1–4, 135–177. DOI 10.1080/00036819208840093 | MR 1293594
[26] A.  Rathsfeld: The invertibility of the double layer potential operator in the space of continuous functions defined over a polyhedron. The panel method. Erratum. Appl. Anal. 56 (1995), 109–115. DOI 10.1080/00036819508840313 | MR 1378015 | Zbl 0921.31004
[27] G. E.  Shilov: Mathematical analysis. Second special course. Nauka, Moskva, 1965. (Russian) MR 0219869
[28] Ch. G.  Simader and H.  Sohr: The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains. Pitman Research Notes in Mathematics Series 360, Addison Wesley Longman Inc., 1996. MR 1454361
[29] M.  Schechter: Principles of Functional Analysis. Academic press, New York-London, 1973. MR 0467221
[30] W. P.  Ziemer: Weakly Differentiable Functions. Springer Verlag, 1989. MR 1014685 | Zbl 0692.46022
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