×

zbMATH — the first resource for mathematics

Continuous extendibility of solutions of the third problem for the Laplace equation. (English) Zbl 1080.35009
Summary: A necessary and sufficient condition for the continuous extendibility of a very weak solution of the third problem for the Laplace equation is given. Here the boundary condition is a real measure on the boundary of the domain. It is shown that the solution is continuously extendible to the closure of the domain if and only if the single layer potential corresponding to the boundary condition is continuously extendible to the closure of the domain. For such boundary condition a general form of a solution is given.

MSC:
35B65 Smoothness and regularity of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] V. Anandam and M. A. Al-Gwaiz: Global representation of harmonic and biharmonic functions. Potential Anal. 6 (1997), 207-214. · Zbl 0883.31002
[2] V. Anandam and M. Damlakhi: Harmonic singularity at infinity in \(R^n\). Real Anal. Exchange 23 (1997/8), 471-476. · Zbl 0938.31003
[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. · Zbl 0697.31005
[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 · Zbl 0172.14903
[5] L. E. Fraenkel: Introduction to Maximum Principles and Symmetry in Elliptic Problems. Cambridge Tracts in Mathematics 128. Cambridge University Press, 2000.
[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. · Zbl 0639.31003
[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, . · Zbl 1381.45016
[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, . · Zbl 1273.35087
[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, . · Zbl 1381.45039
[10] L. L. Helms: Introduction to Potential Theory. Pure and Applied Mathematics 22. John Wiley & Sons, 1969. · Zbl 0188.17203
[11] J. Král: Integral Operators in Potential Theory. Lecture Notes in Mathematics 823. Springer-Verlag, Berlin, 1980.
[12] J. Král: The Fredholm method in potential theory. Trans. Amer. Math. Soc. 125 (1966), 511-547. · Zbl 0149.07906
[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. · Zbl 0615.31005
[14] N. L. Landkof: Fundamentals of Modern Potential Theory. Izdat. Nauka, Moscow, 1966.
[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. · Zbl 0978.31003
[16] D. Medková: Solution of the Robin problem for the Laplace equation. Appl. Math. 43 (1998), 133-155. · Zbl 0938.31005
[17] D. Medková: Solution of the Neumann problem for the Laplace equation. Czechoslovak Math. J. 48(123) (1998), 768-784. · Zbl 0949.31004
[18] D. Medková: Continuous extendibility of solutions of the Neumann problem for the Laplace equation. Czechoslovak Math. J 53(128) (2003), 377-395. · Zbl 1075.35508
[19] J. Nečas: Les méthodes directes en théorie des équations élliptiques. Academia, Prague, 1967. · Zbl 1225.35003
[20] I. Netuka: Fredholm radius of a potential theoretic operator for convex sets. Čas. pěst. mat. 100 (1975), 374-383. · Zbl 0314.31006
[21] I. Netuka: Generalized Robin problem in potential theory. Czechoslovak Math. J. 22(97) (1972), 312-324. · 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. · Zbl 0241.31009
[23] I. Netuka: The third boundary value problem in potential theory. Czechoslovak Math. J. 22(97) (1972), 554-580. · Zbl 0242.31007
[24] I. Netuka: Continuity and maximum principle for potentials of signed measures. Czechoslovak Math. J. 25(100) (1975), 309-316. · 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. · Zbl 0749.31003
[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. · Zbl 0921.31004
[27] G. E. Shilov: Mathematical analysis. Second special course. Nauka, Moskva, 1965. · Zbl 0137.26203
[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. · Zbl 0868.35001
[29] M. Schechter: Principles of Functional Analysis. Academic press, New York-London, 1973. · Zbl 0211.14501
[30] W. P. Ziemer: Weakly Differentiable Functions. Springer Verlag, 1989. · Zbl 0692.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.