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Article

Keywords:
boundary value problem; integral equations; Fredholm type equation
Summary:
A mixed boundary value problem on a doubly connected domain in the complex plane is investigated. The solution is given in an integral form using reflection mapping. The reflection mapping makes it possible to reduce the problem to an integral equation considered only on a part of the boundary of the domain.
References:
[1] K. Astala: Calderón's problem for Lipschitz classes and the dimension of quasicircles. Rev. Mat. Iberoamericana 4 (1988), 469-486. DOI 10.4171/RMI/81 | MR 1048585 | Zbl 0703.30036
[2] Ju. D. Burago V. G. Maz'ja: Some problems of potential theory and theory of functions for domains with nonregular boundaries. Zapiski Naučnych Seminarov LOMI 3 (1967). (In Russian.)
[3] E. Dontová: Reflection and the Dirichlet and Neumann problems. Thesis, Prague, 1990. (In Czech.)
[4] E. Dontová: Reflection and the Dirichlet problem on doubly connected regions. Časopis Pěst. Mat. 113 (1988), 122-147. MR 0949040
[5] E. Dontová: Reflection and the Neumann problem in doubly connected regions. Časopis Pěst. Mat. 113 (1988), 148-168. MR 0949041
[6] H. Federer: Geometric Measure Theory. Springer-Verlag, 1969. MR 0257325 | Zbl 0176.00801
[7] H. Federer: The Gauss-Green theorem. Trans. Amer. Math. Soc. 58 (1945), 44-76. DOI 10.1090/S0002-9947-1945-0013786-6 | MR 0013786 | Zbl 0060.14102
[8] C. Jacob: Sur le problème de Dirichlet dans un domaine plan multiplement connexe et ses applications a l'Hydrodynamique. J. Math. Pures Appl. (9) 18 (1939), 363-383. MR 0001692
[9] J. Král: The Fredholm radius of an operator in potential theory. Czechoslovak Math. J. 15 (1965), 454-473, 565-588. MR 0190363
[10] J. Král: Integral Operators in Potential Theory. Lecture Notes in Math. Vol. 823, Springer-Verlag, 1980. DOI 10.1007/BFb0091035 | MR 0590244
[11] J. Král: Boundary regularity and normal derivatives of logarithmic potentials. Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), 241-258. MR 0906210
[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 D. Medková: Angular limits of the integrals of the Cauchy type. Preprint 47/1994, MU AV ČR. MR 1479307
[14] J. Král D. Medková: Angular limits of double layer potentials. Czechoslovak Math. J. 45 (1995), 267-292. MR 1331464
[15] V. G. Maz'ja: Boundary Integral Equations. Analysis IV, Encyclopaedia of Mathematical Sciences Vol. 27, Springer-Verlag, 1991.
[16] N. I. Muschelišvili: On the fundamental mixed boundary value problem of logarithmic potential for multiply connected domains. Soobščenija Akad. Nauk Gruzinskoj SSR 2 (1941), no. 4, 309-313. (In Russian.) MR 0010234
[17] S. Saks: Theory of the Integral. Dover Publications, New York. 1964. MR 0167578
[18] J. M. Sloss: Global reflection for a class of simple closed curves. Pacific J. Math. 52 (1974), 247-260. DOI 10.2140/pjm.1974.52.247 | MR 0379807 | Zbl 0243.30004
[19] J. M. Sloss: The plane Dirichlet problem for certain multiply connected regions. J. Anal. Math. 28 (1975), 86-100. DOI 10.1007/BF02786808 | Zbl 0325.31004
[20] J. M. Sloss: A new integral equation for certain plane Dirichlet problems. SIAM J. Math. 6 (1975), 998-1006. DOI 10.1137/0506088 | MR 0437784 | Zbl 0323.35025
[21] J. M. Sloss J. C. Bruch: Harmonic approximation with Dirichlet data on doubly connected regions. SIAM J. Numer. Anal. 14, (1974), 994-1005. DOI 10.1137/0714067 | MR 0478687
[22] J. Veselý: On the mixed boundary problem of the theory of analytic functions. Časopis Pěst. Mat. 91 (1966), 320-336. (In Czech.) MR 0206294
[23] W. L. Wendland: Boundary element methods and their asymptotic convergence. Lecture Notes of the CISM, Summer-School on Theoretical acoustic and numerical techniques, Int. Centre Mech. Sci., Udine (P. Filippi, ed.). Springer-Verlag, Wien, 1983, pp. 137-216. MR 0762829 | Zbl 0618.65109
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