Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
third problem; Laplace equation
third problem; Laplace equation
Summary:
A necessary and sufficient condition for the boundedness of a solution of the third problem for the Laplace equation is given. As an application a similar result is given for the third problem for the Poisson equation on domains with Lipschitz boundary.
References:
[1] 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
[2] M. Brelot: Éléments de la théorie classique du potentiel. Centre de documentation universitaire, Paris, 1961. MR 0106366
[3] Yu. D. Burago and V. G. Maz’ya: Potential theory and function theory for irregular regions. Zapiski Naučnyh Seminarov LOMI 3 (1967), 1–152. (Russian)
[4] H. Federer and W. P. Ziemer: The Lebesgue set of a function whose partial derivatives are $p$-th power summable. Indiana Univ. Math. J. 22 (1972), 139–158. MR 0435361
[5] W. H. Fleming: Functions whose partial derivatives are measures. Illinois J. Math. 4 (1960), 452–478. DOI 10.1215/ijm/1255456061 | MR 0130338 | Zbl 0151.05402
[6] L. E. Fraenkel: Introduction to Maximum Principles and Symmetry in Elliptic Problems. Cambridge Tracts in Mathematics 128. Cambridge University Press, Cambridge, 2000. MR 1751289
[7] 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
[8] 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., Linköping.
[9] 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., Linköping.
[10] 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., Linköping.
[11] L. L. Helms: Introduction to Potential Theory. Pure and Applied Mathematics 22. John Wiley & Sons, , 1969. MR 0261018
[12] J. Král: Integral Operators in Potential Theory. Lecture Notes in Mathematics 823. Springer-Verlag, Berlin, 1980. MR 0590244
[13] J. Král: The Fredholm method in potential theory. Trans. Amer. Math. Soc. 125 (1966), 511–547. DOI 10.2307/1994580 | MR 0209503
[14] 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
[15] N. L. Landkof: Fundamentals of Modern Potential Theory. Izdat. Nauka, Moscow, 1966. (Russian) MR 0214795
[16] D. Medková: The third boundary value problem in potential theory for domains with a piecewise smooth boundary. Czechoslovak Math. J. 47 (1997), 651–679. DOI 10.1023/A:1022818618177 | MR 1479311
[17] D. Medková: Solution of the Robin problem for the Laplace equation. Appl. Math. 43 (1998), 133–155. DOI 10.1023/A:1023267018214 | MR 1609158
[18] D. Medková: Solution of the Neumann problem for the Laplace equation. Czechoslovak Math. J. 48 (1998), 768–784. DOI 10.1023/A:1022447908645
[19] D. Medková: Continuous extendibility of solutions of the Neumann problem for the Laplace equation. Czechoslovak Math. J 53 (2003), 377–395. DOI 10.1023/A:1026239404667 | MR 1983459
[20] D. Medková: Continuous extendibility of solutions of the third problem for the Laplace equation. Czechoslovak Math. J 53 (2003), 669–688. DOI 10.1023/B:CMAJ.0000024512.23001.f3 | MR 2000062
[21] D. Medková: Solution of the Dirichlet problem for the Laplace equation. Appl. Math. 44 (1999), 143–168. DOI 10.1023/A:1022209421576
[22] J. Nečas: Les méthodes directes en théorie des équations élliptiques. Academia, Prague, 1967. MR 0227584
[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] I. Netuka: Generalized Robin problem in potential theory. Czechoslovak Math. J. 22(97) (1972), 312–324. MR 0294673 | Zbl 0241.31008
[25] 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
[26] I. Netuka: The third boundary value problem in potential theory. Czechoslovak Math. J. 2(97) (1972), 554–580. MR 0313528 | Zbl 0242.31007
[27] 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), 135–177. DOI 10.1080/00036819208840093 | MR 1293594
[28] 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
[29] M. Schechter: Principles of Functional Analysis. Academic Press, , 1973. MR 0445263
[30] W. P. Ziemer: Weakly Differentiable Functions. Springer-Verlag, , 1989. MR 1014685 | Zbl 0692.46022
Search
Partner of
