| Title:
|
The third boundary value problem in potential theory for domains with a piecewise smooth boundary (English) |
| Author:
|
Medková, Dagmar |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
47 |
| Issue:
|
4 |
| Year:
|
1997 |
| Pages:
|
651-679 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper investigates the third boundary value problem $\frac{\partial u}{\partial n}+\lambda u=\mu $ for the Laplace equation by the means of the potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where $\nu $ is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure ${T}\nu $. Denote by ${T}\:\nu \rightarrow {T}\nu $ the corresponding operator on the space of signed measures on the boundary of the investigated domain $G$. If there is $\alpha \ne 0$ such that the essential spectral radius of $(\alpha I-{T})$ is smaller than $|\alpha |$ (for example, if $G\subset R^3$ is a domain “with a piecewise smooth boundary” and the restriction of the Newtonian potential ${\mathcal U}\lambda $ on $\partial G$ is a finite continuous functions) then the third problem is uniquely solvable in the form of a single layer potential (1) with the only exception which occurs if we study the Neumann problem for a bounded domain. In this case the problem is solvable for the boundary condition $\mu \in $ for which $\mu (\partial G)=0$. (English) |
| MSC:
|
31B20 |
| MSC:
|
35J05 |
| MSC:
|
35J25 |
| idZBL:
|
Zbl 0978.31003 |
| idMR:
|
MR1479311 |
| . |
| Date available:
|
2009-09-24T10:09:19Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127385 |
| . |
| Reference:
|
[1] T.S. Angell, R.E. Kleinman, J. Král: Layer potentials on boundaries with corners and edges.Čas. pěst. mat. 113 (1988), 387–402. MR 0981880 |
| Reference:
|
[2] K. Arbenz: Integralgleichungen für einige Randwertprobleme für Gebiete mit Ecken.Promotionsarbeit Nr. 2777, Eidgenössische Technishe Hochschule in Zürich 1958, 1–41. Zbl 0084.09603, MR 0101416 |
| Reference:
|
[3] M. Brelot: Élements de la théorie classique du potential.Les cours de Sorbone, Paris, 1959. MR 0106366 |
| Reference:
|
[4] Ju. D. Burago, V.G. Maz’ya: Some questions in potential theory and function theory for regions with irregular boundaries (Russian).Zapiski nauč. sem. Leningrad. otd. MIAN 3 (1967). |
| Reference:
|
[5] Ju. D. Burago, V.G. Maz’ya and V.D. Sapožnikova: On the theory of potentials of a double and a simple layer for regions with irregular boundaries (Russian).Problems Math. Anal. Boundary Value Problems Integr. Equations. (Russian), 3–34, Izdat. Leningrad. Univ., Leningrad, 1966. MR 0213596 |
| Reference:
|
[6] T. Carleman: Über das Neumann-Poincarésche Problem für ein Gebeit mit Ecken, Inaugural-Dissertation.Uppsala, 1916. |
| Reference:
|
[7] I.I. Daniljuk: Nonregular Boundary Value Problems in the Plane.Nauka, Moskva, 1975. (Russian) MR 0486546 |
| Reference:
|
[8] E. De Giorgi: Su una teoria generale della misura $(r-1)$-dimensionale in uno spazi ad $r$ dimensioni.Annali di Mat. Pura ed Appl. Ser. 4, 36 (1954), 191–213. MR 0062214, 10.1007/BF02412838 |
| Reference:
|
[9] E. De Giorgi: Nuovi teoremi relativi alle misure $(r-1)$-dimensionali in uno spazi ad $r$ dimensioni.Ricerche Mat. 4 (1955), 95–113. MR 0074499 |
| Reference:
|
[10] M. Dont, E. Dontová: Invariance of the Fredholm radius of an operator in potential theory.Čas. pěst. mat. 112 (1987), no. 3, 269–283. MR 0905974 |
| Reference:
|
[11] N. Dunford, J.T. Schwarz: Linear Operators.Interscience, New York, 1963. |
| Reference:
|
[12] J. Elschner: The double-layer potential operator over polyhedral domains I: Solvability in weighted Sobolev spaces.Applicable Analysis 45 (1992), 117–134. Zbl 0749.31002, MR 1293593, 10.1080/00036819208840092 |
| Reference:
|
[13] E. Fabes: Layer potential methods for boundary value problems on Lipschitz domains, in J. Král, J. Lukeš, I. Netuka and J. Veselý (eds.): Potential Theory. Surveys and Problems. Proceedings, Prague 1987. Lecture Notes in Mathematics 1344..Springer-Verlag, Berlin-Heidelberg-New York, 1988. MR 0973881 |
| Reference:
|
[14] E.B. Fabes, M. Jodeit Jr., J.E. Lewis: Double layer potentials for domain with corners and edges.Indiana Univ. Math. J. 26 (1977), 95–114. MR 0432899, 10.1512/iumj.1977.26.26007 |
| Reference:
|
[15] E. Fabes, M. Sand, J.K. Seo: The spectral radius of the classical layer potentials on convex domains.Partial Differential Equations With Minimal Smoothness and Applications, 129–137, IMA Vol. Math. Appl. 42, Springer, New York, 1992. MR 1155859, 10.1007/978-1-4612-2898-1_12 |
| Reference:
|
[16] H. Federer: A note on the Gauss-Green theorem.Proc. Amer. Math. Soc. 9 (1958), 447–451. Zbl 0087.27302, MR 0095245, 10.1090/S0002-9939-1958-0095245-2 |
| Reference:
|
[17] H. Federer: Geometric Measure Theory.Springer-Verlag, 1969. Zbl 0176.00801, MR 0257325 |
| Reference:
|
[18] H. Federer: The Gauss-Green theorem.Trans. Amer. Math. Soc. 58 (1945), 44–76. Zbl 0060.14102, MR 0013786, 10.1090/S0002-9947-1945-0013786-6 |
| Reference:
|
[19] N.V. Grachev: Representations and estimates for inverse operators of the potential theory integral equations in a polyhedron. Potential Theory (Nagoya, 1990), 201–206.de Gruyter, Berlin, 1992. MR 1167235 |
| Reference:
|
[20] N.V. Grachev, V.G. Maz’ya: On the Fredholm radius for operators of double-layer type on the piece-wise smooth boundary (Russian)..Vest. Leningr. un. mat. mech. (1986), no. 4, 60–64. MR 0880678 |
| Reference:
|
[21] N.V. Grachev, V.G. Maz’ya: Representations and estimates for inverse operators of the potential theory integral equations on surfaces with conic points.Soobstch. Akad. Nauk Gruzin SSR 132 (1988), 21–23. (Russian) MR 1020233 |
| Reference:
|
[22] N.V. Grachev, V.G. Maz’ya: Estimates for kernels of the inverse operators of the integral equations of elasticity on surfaces with conic points. Preprint N26, Akad. Nauk SSSR. Inst. of Engin. Studies, Leningrad, 1989..(Russian) |
| Reference:
|
[23] N.V. Grachev, V.G. Maz’ya: On invertibility of the boundary integral operators of elasticity on surfaces with conic points in the spaces generated by norms $C$, $C^\alpha $, $L_p$, Preprint N30, Akad. Nauk SSSR, Inst. of Engin. Studies, Leningrad, 1990..(Russian) |
| Reference:
|
[24] P.R. Halmos: Finite-Dimensional Vector Spaces.D. van Nostrand, Princeton-Toronto--London-New York, 1963. Zbl 0107.01501, MR 0089819 |
| Reference:
|
[25] D.S. Jerison and C.E. Kenig: The Dirichlet problem in non smooth domains.Annals of Mathematics 113 (1981), 367–382. MR 0607897, 10.2307/2006988 |
| Reference:
|
[26] D.S. Jerison and C.E. Kenig: The Neumann problem in Lipschitz domains.Bulletin of AMS 4 (1981), 203–207. MR 0598688, 10.1090/S0273-0979-1981-14884-9 |
| Reference:
|
[27] K. Jörgens: Lineare Integraloperatoren.B.G. Teubner, Stuttgart, 1970. MR 0461049 |
| Reference:
|
[28] J. Král: Integral operators in potential theory..Lecture Notes in Mathematics 823. Springer-Verlag, Berlin-Heidelberg-New York, 1980. MR 0590244 |
| Reference:
|
[29] J. Král: The Fredholm radius of an operator in potential theory.Czechoslovak Math. J. 15(90) (1965), 454–473, 565–588. MR 0190363 |
| Reference:
|
[30] J. Král: Flows of heat and the Fourier problem.Czechoslovak Math. J. 20(95) (1970), 556–597. MR 0271554 |
| Reference:
|
[31] J. Král: Note on sets whose characteristic functions have singed measure for their partial derivatives.Čas. pěst. mat. 86 (1961), 178–194. (Czech) MR 0136697 |
| Reference:
|
[32] J. Král: The Fredholm method in potential theory.Trans. Amer. Math. Soc. 125 (1966), 511–547. MR 0209503, 10.2307/1994580 |
| Reference:
|
[33] J. Král, W.L. Wendland: Some example concerning applicability of the Fredholm-Radon method in potential theory.Aplikace matematiky 31 (1986), 293–308. MR 0854323 |
| Reference:
|
[34] N.L. Landkof: Fundamentals of Modern Potential Theory.Izdat. Nauka, Moscow, 1966. (Russian) MR 0214795 |
| Reference:
|
[35] V.G. Maz’ya: Boundary integral equations. Sovremennyje problemy matematiki, fundamental’nyje napravlenija, t. 27.Viniti, Moskva, 1988. (Russian) |
| Reference:
|
[36] V.G. Maz’ya: Boundary integral equations, Encyclopedia of Mathematical Sciences.vol. 27, Springer-Verlag, 1991. |
| Reference:
|
[37] V.G. Maz’ya, B.A. Plamenevsky: The first boundary-value problem for the classical equations of the mathematical physics in domain with piece-wise smooth boundary. I, II.Z. Anal. Anwend. 2 (1983), no. 4, 335–359, No. 6, 523–551. (Russian) MR 0719176, 10.4171/ZAA/83 |
| Reference:
|
[38] W. McLean: Boundary integral methods for the Laplace equation. Thesis.Australian National University, 1985. MR 0825529 |
| Reference:
|
[39] D. Medková: On the convergence of Neumann series for noncompact operators.Czechoslovak Math. J. 41(116) (1991), 312–316. MR 1105448 |
| Reference:
|
[40] D. Medková: Invariance of the Fredholm radius of the Neumann operator.Čas. pěst. mat. 115 (1990), no. 2, 147–164. MR 1054002 |
| Reference:
|
[41] D. Medková: On essential norm of Neumann operator.Mathematica Bohemica 117 (1992), no. 4, 393–408. MR 1197288 |
| Reference:
|
[42] S.G. Michlin: Integralnyje uravnenija i ich prilozhenija k nekotorym problemam mekhaniki, matematicheskoj fiziki i tekhniki.Moskva, 1949. |
| Reference:
|
[43] I. Netuka: The Robin problem in potential theory.Comment. Math. Univ. Carolinae 12 (1971), 205–211. Zbl 0215.42602, MR 0287021 |
| Reference:
|
[44] I. Netuka: Generalized Robin problem in potential theory.Czechoslovak Math. J. 22(97) (1972), 312–324. Zbl 0241.31008, MR 0294673 |
| Reference:
|
[45] 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:
|
[46] I. Netuka: The third boundary value problem in potential theory.Czechoslovak Math. J. 22(97) (1972), 554–580. Zbl 0242.31007, MR 0313528 |
| Reference:
|
[47] J. Plemelj: Potentialtheoretische Untersuchungen.Leipzig, 1911. |
| Reference:
|
[48] J. Radon: Über lineare Funktionaltransformationen und Funktionalgleichungen. Collected Works.vol. 1, 1987. |
| Reference:
|
[49] J. Radon: Über Randwertaufgaben beim logarithmischen Potential. Collected Works.vol. 1, 1987. |
| Reference:
|
[50] 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), no. 1–4, 135–177. MR 1293594, 10.1080/00036819208840093 |
| Reference:
|
[51] S. Rempel: Corner singularity for transmission problems in three dimensions.Integral Equations and Operator Theory 12 (1989), 835–854. Zbl 0695.47046, MR 1018215, 10.1007/BF01196880 |
| Reference:
|
[52] S. Rempel and G. Schmidt: Eigenvalues for spherical domains with corners via boundary integral equations.Integral Equations Oper. Theory 14 (1991), 229–250. MR 1090703, 10.1007/BF01199907 |
| Reference:
|
[53] F. Riesz, B. Sz. Nagy: Leçons d’analyse fonctionelle.Budapest, 1952. |
| Reference:
|
[54] S. Saks: Theory of the Integral.Hafner Publishing Comp., New York, 1937. Zbl 0017.30004 |
| Reference:
|
[55] V.D. Sapožnikova: Solution of the third boundary value problem by the method of potential theory for regions with irregular boundaries (Russian), Problems Math. Anal. Boundary Value Problems Integr. Equations (Russian), 35–44.Izdat. Leningrad. Univ., Leningrad, 1966. MR 0213597 |
| Reference:
|
[56] M. Schechter: Principles of Functional Analysis.Academic Press, 1973. MR 0467221 |
| Reference:
|
[57] L. Schwartz: Théorie des distributions.Hermann, Paris, 1950. Zbl 0037.07301, MR 0209834 |
| Reference:
|
[58] N. Suzuki: On the convergence of Neumann series in Banach space.Math. Ann. 220 (1976), 143–146. Zbl 0304.47016, MR 0412855, 10.1007/BF01351698 |
| Reference:
|
[59] A.E. Taylor: Introduction to Functional Analysis.New York, 1967. |
| Reference:
|
[60] G.C. Verchota: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains.Journal of Functional Analysis 59 (1984), 572–611. Zbl 0589.31005, MR 0769382, 10.1016/0022-1236(84)90066-1 |
| Reference:
|
[61] K. Yosida: Functional Analysis.Springer-Verlag, Berlin, 1965. Zbl 0126.11504 |
| . |
