Title:
|
Boundedness of the solution of the third problem for the Laplace equation (English) |
Author:
|
Medková, Dagmar |
Language:
|
English |
Journal:
|
Czechoslovak Mathematical Journal |
ISSN:
|
0011-4642 (print) |
ISSN:
|
1572-9141 (online) |
Volume:
|
55 |
Issue:
|
2 |
Year:
|
2005 |
Pages:
|
317-340 |
Summary lang:
|
English |
. |
Category:
|
math |
. |
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. (English) |
Keyword:
|
third problem |
Keyword:
|
Laplace equation |
MSC:
|
31B10 |
MSC:
|
35B45 |
MSC:
|
35B65 |
MSC:
|
35J05 |
MSC:
|
35J25 |
idZBL:
|
Zbl 1081.35013 |
idMR:
|
MR2137140 |
. |
Date available:
|
2009-09-24T11:23:13Z |
Last updated:
|
2020-07-03 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127980 |
. |
Reference:
|
[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 |
Reference:
|
[2] M. Brelot: Éléments de la théorie classique du potentiel.Centre de documentation universitaire, Paris, 1961. MR 0106366 |
Reference:
|
[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) |
Reference:
|
[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 |
Reference:
|
[5] W. H. Fleming: Functions whose partial derivatives are measures.Illinois J. Math. 4 (1960), 452–478. Zbl 0151.05402, MR 0130338, 10.1215/ijm/1255456061 |
Reference:
|
[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 |
Reference:
|
[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 |
Reference:
|
[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. |
Reference:
|
[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. |
Reference:
|
[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. |
Reference:
|
[11] L. L. Helms: Introduction to Potential Theory. Pure and Applied Mathematics 22.John Wiley & Sons, , 1969. MR 0261018 |
Reference:
|
[12] J. Král: Integral Operators in Potential Theory. Lecture Notes in Mathematics 823.Springer-Verlag, Berlin, 1980. MR 0590244 |
Reference:
|
[13] J. Král: The Fredholm method in potential theory.Trans. Amer. Math. Soc. 125 (1966), 511–547. MR 0209503, 10.2307/1994580 |
Reference:
|
[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 |
Reference:
|
[15] N. L. Landkof: Fundamentals of Modern Potential Theory.Izdat. Nauka, Moscow, 1966. (Russian) MR 0214795 |
Reference:
|
[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. MR 1479311, 10.1023/A:1022818618177 |
Reference:
|
[17] D. Medková: Solution of the Robin problem for the Laplace equation.Appl. Math. 43 (1998), 133–155. MR 1609158, 10.1023/A:1023267018214 |
Reference:
|
[18] D. Medková: Solution of the Neumann problem for the Laplace equation.Czechoslovak Math. J. 48 (1998), 768–784. 10.1023/A:1022447908645 |
Reference:
|
[19] D. Medková: Continuous extendibility of solutions of the Neumann problem for the Laplace equation.Czechoslovak Math. J 53 (2003), 377–395. MR 1983459, 10.1023/A:1026239404667 |
Reference:
|
[20] D. Medková: Continuous extendibility of solutions of the third problem for the Laplace equation.Czechoslovak Math. J 53 (2003), 669–688. MR 2000062, 10.1023/B:CMAJ.0000024512.23001.f3 |
Reference:
|
[21] D. Medková: Solution of the Dirichlet problem for the Laplace equation.Appl. Math. 44 (1999), 143–168. 10.1023/A:1022209421576 |
Reference:
|
[22] J. Nečas: Les méthodes directes en théorie des équations élliptiques.Academia, Prague, 1967. MR 0227584 |
Reference:
|
[23] 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:
|
[24] I. Netuka: Generalized Robin problem in potential theory.Czechoslovak Math. J. 22(97) (1972), 312–324. Zbl 0241.31008, MR 0294673 |
Reference:
|
[25] I. Netuka: An operator connected with the third boundary value problem in potential theory.Czechoslovak Math. J. 22(97) (1972), 462–489. Zbl 0241.31009, MR 0316733 |
Reference:
|
[26] I. Netuka: The third boundary value problem in potential theory.Czechoslovak Math. J. 2(97) (1972), 554–580. Zbl 0242.31007, MR 0313528 |
Reference:
|
[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. MR 1293594, 10.1080/00036819208840093 |
Reference:
|
[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. Zbl 0921.31004, MR 1378015, 10.1080/00036819508840313 |
Reference:
|
[29] M. Schechter: Principles of Functional Analysis.Academic Press, , 1973. MR 0445263 |
Reference:
|
[30] W. P. Ziemer: Weakly Differentiable Functions.Springer-Verlag, , 1989. Zbl 0692.46022, MR 1014685 |
. |