×

zbMATH — the first resource for mathematics

\(\mathbb R\)-linear problem for multiply connected domains and alternating method of Schwarz. (English. Russian original) Zbl 1276.30052
J. Math. Sci., New York 189, No. 1, 68-77 (2013); translation from Sovrem. Mat. Prilozh. 77 (2012).
The authors apply a version of the alternating method of Schwarz to a system of boundary value problems involving several disjoint domains \(D_k\) in the complex plane \(\mathbb C\). Suppose that each \(D_k\) is bounded by a smooth simple curve \(L_k\), oriented counterclockwise. Let \(a,b,c\) be Hölder continuous functions on \(L=\bigcup L_k\) such that \(a\neq 0\). Also denote \(D=\overline{\mathbb C}\setminus \bigcup \overline{D_k}\). The \(\mathbb R\)-linear conjugation problem is to find a function \(\varphi\) that is holomorphic in the complement of \(L\), is continuous up to the boundary of each domain \(D, D_k\), and satisfies the equation \(\varphi^+=a\varphi^{-} + b \overline{\varphi^{-}} +c\) on \(L\). Here \(\varphi^{+}\) and \(\varphi^{-}\) stand for the boundary values of \(\varphi\) on \(L\) as approached from \(D\) and from \(D_k\), respectively. In addition, \(\varphi\) must vanish at infinity.
The following result (Theorem 1 in the paper) generalizes a 1960 theorem by the first-named author. Suppose in addition that \(|b|<a\) on \(L\), and let \(\kappa\) be the winding number of \(a\) on \(\partial D\). If \(\kappa\geq 0\), then the problem stated above has a solution. Furthermore, the homogeneous problem (\(c=0\)) has \(2\kappa\) \(\mathbb R\)-linearly independent solutions vanishing at infinity. If \(\kappa<0\), then the solution does not necessarily exist.
The article includes a historical overview of this and related boundary value problems, supported by a long list of references.

MSC:
30E25 Boundary value problems in the complex plane
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] I. A. Aleksandrov and A. S. Sorokin, ”The problem of Schwarz for multiply connected domains,” Sib. Mat. Zh., 13, No. 5, 971–1001 (1972). · Zbl 0249.30035
[2] F. A. Apel’tsin, ”A generalization of D. A. Grave’s method for plane boundary-value problems in harmonic potentials theory,” Comput. Math. Model., 11, No. 1, 1–14 (2000). · Zbl 1062.30010
[3] B. Bojarski, ”On a boundary value problem of the theory of analytic functions,” Dokl. Akad. Nauk SSSR, 119, 199–202 (1958).
[4] B. Bojarski, ”On generalized Hilbert boundary-value problem,” Soobsch. Akad. Nauk Gruz. SSR, 25, No. 4, 385–390 (1960).
[5] T. K. DeLillo and E. H. Kropf, ”Slit maps and Schwarz–Christoffel maps for multiply connected domain,” Electr. Trans. Numer. Anal., 36, 195–223 (2010). · Zbl 1207.30007
[6] L. E. Dunduchenko, ”On the Schwarz formula for an n-connected domain,” Dopov. Akad. Nauk URSR, 5, 1386–1389 (1966) · Zbl 0149.03503
[7] A. Dzhuraev, Method of Singular Integral Equations, Pitman Monogr., Wiley, New York (1992). · Zbl 0817.35002
[8] F. D. Gakhov, Boundary-Value Problems [in Russian], Moscow, Nauka, (1977). · Zbl 0449.30030
[9] G. M. Golusin, ”Solution of basic plane problems of mathetical physics for the case of the Laplace equation and multiply connected domains bounded by circles (method of functional equations),” Mat. Sb., 41, No. 2, 246–276 (1934).
[10] G. M. Golusin, ”Solution of the spatial Dirichlet problem for the Laplace equation and for domains enbounded by finite number of spheres,” Mat. Sb., 41, No. 2, 277–283 (1934).
[11] G. M. Golusin, ”Solution of the plane heat-conduction problem for multiply connected domains enclosed by circles in the case of isolated layer,” Mat. Sb., 42, No. 2, 191–198 (1935). · Zbl 0012.35601
[12] L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis, Groningen, Noordhoff (1958). · Zbl 0083.35301
[13] I. I. Komyak, ”Closed form solution to a two-dimensional singular integral equation with analytic coefficients,” Dokl. Akad. Nauk BSSR, 21, No. 7, 583–586 (1977). · Zbl 0354.45009
[14] I. I. Komyak, ”On two-dimensional singular integral equations with analytic coefficients,” Differ. Uravn., 16, No. 5, 908–916 (1980). · Zbl 0445.45004
[15] M. A. Krasnosel’skii et al., Approximate Methods for Solution of Operator Equations, Wolters-Noordhoff Publ., Groningen (1972).
[16] G. S. Litvinchuk and I. M. Spitkovsky, Factorization of Measurable Matrix Functions, Operator Theory: Advances and Applications, 25, Birkhäuser, Basel (1987).
[17] A. I. Markushevich, ”On a boundary-value problem of the theory of analytic function,” Uch. Zap. MGU, 1, No. 100, 20–30 (1946).
[18] L. G. Mikhailov, ”On a boundary-value problem,” Dokl. Akad. Nauk SSSR, 139, No. 2, 294–297 (1961).
[19] L. G. Mikhailov, ”A New Class of Singular Integral Equations, Wolters-Noordhoff Publ., Groningen (1970). · Zbl 0198.15301
[20] S. G. Mikhlin, Integral Equations, Pergamon Press, New York (1964). · Zbl 0117.31902
[21] V. V. Mityushev, ”Solution of the Hilbert boundary-value problem for a multiply connected domain,” Slupskie Prace Mat.-Przyr., 9a, 37–69 (1994). · Zbl 0818.30026
[22] V. V. Mityushev, ”Convergence of the Poincaré series for classical Schottky groups,” Proc. Amer. Math. Soc., 126, No. 8, 2399–2406 (1998). · Zbl 0899.30029
[23] V. V. Mityushev, ”Hilbert boundary-value problem for multiply connected domains,” Complex Variables, 35, 283–295 (1998). · Zbl 0906.30033
[24] V. Mityushev, ”Transport properties of doubly periodic array of circular cylinder and optimal design problem,” Appl. Math Optim., 44, 17–31 (2001). · Zbl 0982.30019
[25] V. Mityushev, ”Conductivity of a two-dimensional composite containing elliptical inclusions,” Proc. Roy. Soc. London, A465, 2991–3010 (2009). · Zbl 1180.82185
[26] V. Mityushev, E. Pesetskaya, and S. Rogosin, ”Analytical methods for heat conduction in composites and porous media,” in: Cellular and Porous Materials: Thermal Properties Simulation and Prediction (A. ”Ochsner, G. E. Murch, and M. J. S. de Lemos, eds.), Wiley (2008). · Zbl 1150.74038
[27] V. V. Mityushev and S. V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary-Value Problems for Analytic Functions Theory, Chapman &amp; Hall/CRC, Boca Raton (2000). · Zbl 0957.30002
[28] N. I. Muskhelishvili, ”On the problem of torsion and bending of beams constituted from different materials,” Izv. Akad. Nauk SSSR, 7, 907–945 (1932). · JFM 58.1279.03
[29] N. I. Muskhelishvili, Singular Integral Equations [in Russian], Nauka, Moscow (1968). · Zbl 0174.16202
[30] N. I. Muskhelishvili, Some Basic Problems of Mathematical Elasticity Theory [in Russian], Nauka, Moscow (1966). · Zbl 0151.36201
[31] H. Poincaré, Oeuvres, Vol. 2, Gauthier-Villar, Paris (1916); Vol. 4, Gauthier-Villars, Paris (1950); Vol. 9, Gauthier-Villar, Paris (1954).
[32] B. Smith, P. Björstad, and W. Gropp, Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge Univ. Press (1996).
[33] T. Vaitekhovich, Boundary-value problems for complex partial differential equations in a ring domain, Ph.D. thesis, FU Berlin (2006). · Zbl 1174.30047
[34] I. N. Vekua, Generalized Analytic Functions [in Russian], Nauka, Moscow (1988). · Zbl 0698.47036
[35] I. N. Vekua and A. K. Rukhadze, ”The problem of the torsion of circular cylinder reinforced by transversal circular beam,” Izv. Akad. Nauk SSSR, 3, 373–386 (1933). · JFM 59.1426.05
[36] I. N. Vekua and A. K. Rukhadze, ”Torsion and transversal bending of the beam compounded by two materials restricted by confocal ellipses,” Prikl. Mat. Mekh., 1, No. 2, 167–178 (1933).
[37] V. A. Zmorovich, ”On a generalization of the Schwarz integral formula on n-connected domains,” Dop. Akad. Nauk URSR, 5, 489–492 (1958). · Zbl 0079.29402
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.