Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Eigenvalues of holomorphic functions for the third boundary condition


Authors: Alip Mohammed, Dennis A. Siginer and Fahir Talay Akyildiz
Journal: Quart. Appl. Math. 73 (2015), 553-574
MSC (2010): Primary 30E25, 35J35, 35J45
DOI: https://doi.org/10.1090/qam/1419
Published electronically: June 16, 2015
MathSciNet review: 3400759
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Abstract: The eigenvalue problem of holomorphic functions on the unit disc for the third boundary condition with general coefficient is studied using Fourier analysis. With a general anti-polynomial coefficient a variable number of additional boundary conditions need to be imposed to determine the eigenvalue uniquely. An additional boundary condition is required to obtain a unique eigenvalue when the coefficient includes an essential singularity rather than a pole. In either case explicit solutions are derived.


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  • [1] Giles Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer. Funct. Anal. Optim. 25 (2004), no. 3-4, 321-348. MR 2072072 (2005d:35187), https://doi.org/10.1081/NFA-120039655
  • [2] Catherine Bandle, Isoperimetric inequalities and applications, Monographs and Studies in Mathematics, vol. 7, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980. MR 572958 (81e:35095)
  • [3] Heinrich G. W. Begehr, Complex analytic methods for partial differential equations, An introductory text. World Scientific Publishing Co., Inc., River Edge, NJ, 1994. MR 1314196 (95m:35031)
  • [4] Heinrich Begehr and Dao-Qing Dai, On the Riemann-Hilbert-Poincaré problem for analytic functions, Analysis (Munich) 22 (2002), no. 2, 183-199. MR 1916424 (2003e:30055), https://doi.org/10.1524/anly.2002.22.2.183
  • [5] Heinrich G. W. Begehr and Abduhamid Dzhuraev, An introduction to several complex variables and partial differential equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 88, Longman, Harlow, 1997. MR 1605577 (99g:32001)
  • [6] H. Begehr and G. Harutjunjan, Robin boundary value problem for the Poisson equation, J. Anal. Appl. 4 (2006), no. 3, 201-213. MR 2237442 (2007d:30033)
  • [7] H. Begehr and G. C. Wen, Oblique derivative problems for elliptic systems of second order equations in infinite domains, Z. Anal. Anwendungen 18 (1999), no. 2, 193-204. MR 1701349 (2000f:35049), https://doi.org/10.4171/ZAA/877
  • [8] H. Begehr and T. Vaitekhovich, Modified harmonic Robin function, Complex Var. Elliptic Equ. 58 (2013), no. 4, 483-496. MR 3038742, https://doi.org/10.1080/17476933.2011.625092
  • [9] Yankis R. Linares and Carmen J. Vanegas, A Robin boundary value problem in the upper half plane for the Bitsadze equation, J. Math. Anal. Appl. 419 (2014), no. 1, 200-217. MR 3217144, https://doi.org/10.1016/j.jmaa.2014.04.056
  • [10] J. M. Cushing, Nonlinear Steklov problems on the unit circle, J. Math. Anal. Appl. 38 (1972), 766-783. MR 0304885 (46 #4017)
  • [11] J. M. Cushing, Nonlinear Steklov problems on the unit circle. II. And a hydrodynamical application, J. Math. Anal. Appl. 39 (1972), 267-278; errata, ibid. 41 (1973), 536-537. MR 0318668 (47 #7215)
  • [12] Dao-Qing Dai and Ming-Sheng Liu, Fourier method for Riemann-Hilbert-Poincaré problem of analytic function, In honor of Professor Erwin Kreyszig on the occasion of his 80th birthday. Complex Var. Theory Appl. 47 (2002), no. 8, 645-652. MR 1916534 (2003e:30056), https://doi.org/10.1080/02781070290010210
  • [13] Dao-Qing Dai and Ming-Sheng Liu, Spectrum of the Riemann-Hilbert-Poincaré problem for analytic functions, Complex Var. Theory Appl. 50 (2005), no. 7-11, 497-505. MR 2155422 (2006j:30071), https://doi.org/10.1080/02781070500086552
  • [14] M. Felici, B. Sapoval and M. Filoche, Renormalized random walk study of oxygen absorption in the human lung, Phys. Rev. Lett. 92 (2003), 068101-1-068101-4.
  • [15] F. D. Gakhov, Boundary value problems, translation edited by I. N. Sneddon, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. MR 0198152 (33 #6311)
  • [16] Denis S. Grebenkov, Partially reflected Brownian motion: a stochastic approach to transport phenomena, Focus on probability theory, Nova Sci. Publ., New York, 2006, pp. 135-169. MR 2553673 (2010m:60275)
  • [17] Karl Gustafson and Takehisa Abe, The third boundary condition--was it Robin's?, Math. Intelligencer 20 (1998), no. 1, 63-71. MR 1601764 (99e:01016), https://doi.org/10.1007/BF03024402
  • [18] D. Kochan, D. Krejčiřík, R. Novák, and P. Siegl, The Pauli equation with complex boundary conditions, J. Phys. A 45 (2012), no. 44, 444019, 14. MR 2991886, https://doi.org/10.1088/1751-8113/45/44/444019
  • [19] Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625 (2002e:32001)
  • [20] Jian Ke Lu, Boundary value problems for analytic functions, Series in Pure Mathematics, vol. 16, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1279172 (95j:30037)
  • [21] D. Medková and P. Krutitskii, Neumann and Robin problems in a cracked domain with jump conditions on cracks, J. Math. Anal. Appl. 301 (2005), no. 1, 99-114. MR 2105923 (2006i:35117), https://doi.org/10.1016/j.jmaa.2004.06.062
  • [22] Alip Mohammed and M. W. Wong, Solutions of the Riemann-Hilbert-Poincaré problem and the Robin problem for the inhomogeneous Cauchy-Riemann equation, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), no. 1, 157-181. MR 2487037 (2009k:30047), https://doi.org/10.1017/S0308210507000108
  • [23] Alip Mohammed, The Riemann-Hilbert-Poincaré problem for holomorphic functions in polydiscs, Progress in analysis, Vol. I, II (Berlin, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 721-727. MR 2032745
  • [24] W. Stekloff, Sur les problèmes fondamentaux de la physique mathématique (suite et fin), Ann. Sci. École Norm. Sup. (3) 19 (1902), 455-490 (French). MR 1509018
  • [25] N. E. Tovmasyan, Non-regular differential equations and calculations of electromagnetic fields, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. MR 1747796 (2001d:35186)
  • [26] L. v. Wolfersdorf, On the theory of nonlinear generalized Poincaré problems for harmonic functions, Continuum mechanics and related problems of analysis (Tbilisi, 1991), ``Metsniereba'', Tbilisi, 1993, pp. 330-337. MR 1379836

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Additional Information

Alip Mohammed
Affiliation: Department of Mathematics, Arts and Sciences, The Petroleum Institute, Abu Dhabi, United Arab Emirates
Email: aalifu@pi.ac.ae

Dennis A. Siginer
Affiliation: Department of Mechanical Engineering and Department of Applied Mathematics, Botswana International University of Science and Technology Palapye, Botswana; Centro de Investigación en Creatividad y Educación Superior, Universidad de Santiago de Chile, Santiago, Chile
Email: siginerd@biust.ac.bw, dennis.siginer@usach.cl

Fahir Talay Akyildiz
Affiliation: Department of Mathematics, College of Arts and Sciences, Gaziantep University, Gaziantep, Turkey
Email: fakyildiz@gantep.edu.tr

DOI: https://doi.org/10.1090/qam/1419
Keywords: Eigenvalue, boundary value problems, Riemann-Hilbert-Poincar\'e problem, the third boundary condition, holomorphic functions, Fourier series, Fuchsian differential equations
Received by editor(s): December 4, 2013
Received by editor(s) in revised form: September 29, 2014
Published electronically: June 16, 2015
Article copyright: © Copyright 2015 Brown University

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