Riemann–Hilbert problem for the Cauchy–Riemann operator in lens and lune

References

• Dzhuraev A. Method of singular integral equations. Moscow: Nauka; 1987; Harlow: Longman; 1992. Russian.
   Google Scholar
• Dzhuraev A. Degenerate and other problems. Harlow: Longman; 1992.
   Google Scholar
• Muskhelishvili N. Singular integral equations. Groningen: Noordhoff; 1953.
   Google Scholar
• Wen GC, Begehr H. Boundary value problems for elliptic equations and systems. Harlow: Longman; 1990.
   Google Scholar
• Antipov Y. Diffraction of an obliquely incident electromagnetic wave by an impedance right-angled concave wedge. Quart J Mech Appl Math. 2014;67:1–42.
   Web of Science ®Google Scholar
• Aptekarev A, Van Assche W. Scalar and matrix Riemann-Hilbert approach to the strong asymptotics of Pad\’e approximants and complex orthogonal polynomials with varying weight. J Approx Theory. 2004;129:129–166.
   Web of Science ®Google Scholar
• Azhdari A, Obata M, Nemat-Nasser S. Alternative solution methods for crack problems in plane anisotropic elasticity, with examples. Int J Solids Struct. 2000;37:6433–6478.
   Web of Science ®Google Scholar
• Baldi P, Toland J. Steady periodic water waves under nonlinear elastic membranes. J Reine Angew Math. 2011;652:67–112.
   Web of Science ®Google Scholar
• Bertola M, Katsevich A, Tovbis A. On Sobolev instability of the interior problem of tomography. J Math Anal Appl. 2016;438:962–990.
   Web of Science ®Google Scholar
• Dal Corso F, Bigoni D, Gei M. The stress concentration near a rigid line inclusion in a prestressed, elastic material. Part I. Full-field solution and asymptotics. J Mech Phys Solids. 2008;56:815–838.
   Web of Science ®Google Scholar
• De Monvel B, Shepelsky D. Riemann--Hilbert problem in the inverse scattering for the Camassa--Holm equation on the line. In: Pinsky M, Birnir B, editors. Probability, geometry and integrable systems. Vol. 55. Mathematical Sciences Research Institute Publications. Cambridge: Cambridge University Press; 2008. p. 53–75.
   Google Scholar
• Hoell N, Bal G. Ray transforms on a conformal class of curves. Inverse Probl Imaging. 2014;8:103–125.
   Web of Science ®Google Scholar
• Pryce L, Morini L, Mishuris G. Weight function approach to study a crack propagating along a bimaterial interface under asymmetric loading in anisotropic solids. JoMMS. 2013;8:479–500.
   Web of Science ®Google Scholar
• Rogosin S. Hele-Shaw moving boundary value problem in a bounded domain local in time solvability. Complex Var Theory Appl. 2005;50:745–764.
   Google Scholar
• Abdymanapov S, Begehr H, Harutyunyan G, Tungatarov AB. Four boundary value problems for the Cauchy-Riemann equation in a quarter plane. In: Begehr H, Nicolosi F, editors. More progress in analysis. Proceedings of the 5th International ISAAC Congress, Catania, Italy, 2005. Singapore: World Scientific; 2009. p. 1137–1147.
   Google Scholar
• Akel M, Alabbad F. A Riemann--Hilbert boundary value problem in a triangle. J Math Anal Appl. 2015;424:523–536.
   Web of Science ®Google Scholar
• Akel M, Alabbad F. A Riemann--Hilbert boundary value problem in a bounded sector. Complex Var Elliptic Equ. 2014;60:493–509.
   Web of Science ®Google Scholar
• Akel M, Begehr H. Schwarz problem for first order elliptic systems in unbounded sectors. Eurasian Math J. 2014;5:6–24.
   Google Scholar
• Akel M, Hussein H. Two basic boundary value problems for inhomogeneous Cauchy--Riemann equation in an infinite sector. Adv Pure Appl Math. 2012;3:315–328.
   Google Scholar
• Aksoy U. Schwarz problem for complex differential equations [PhD thesis]. Ankara: METU.
   Google Scholar
• Alip M, Wong MW. Solutions of the Riemann--Hilbert--Poincaré problem and the Robin problem for the inhomogeneous Cauchy--Riemann equation. Proc Roy Soc Edinburgh Sect A. 2009;139:157–181.
   Google Scholar
• Begehr H. Complex analytic methods for partial differential equations: an introductory text. Singapore: World Scientific; 1994.
   Google Scholar
• Begehr H. Boundary value problems in complex analysis, I, II. Bol Assoc Mat Venez. 2005;XII:65–85; 217--250.
   Google Scholar
• Begehr H, Dai DQ. On the Riemann--Hilbert--Poincaré problem for analytic functions. Analysis. 2002;22:183–199.
   Google Scholar
• Begehr H, Gaertner E. Dirichlet problem for the inhomogeneous polyharmonic equation in the upper half plane. Georgian Math J. 2007;14:33–52.
   Web of Science ®Google Scholar
• Begehr H, Vaitekhovich T. Harmonic boundary value problems in half disc and half ring. Funct Approx Comment Math. 2009;40:251–282.
   Google Scholar
• Begehr H, Vaitekhovich T. Green functions, reflections and plane parqueting. Eurasian Math J. 2010;1:17–31; Eurasian Math J. 2011;2:139–142.
   Google Scholar
• Begehr H, Vaitekhovich T. How to find harmonic Green function in the plane. Complex Var Elliptic Equ. 2011;56:1169–1181.
   Web of Science ®Google Scholar
• Begehr H, Vaitekhovich T. Harmonic Dirichlet problem for some equilateral triangle. Complex Var Elliptic Equ. 2012;57:185–196.
   Web of Science ®Google Scholar
• Begehr H, Vaitekhovich T. Schwarz problem in lens and lune. Complex Var Elliptic Equ. 2014;59:76–84.
   Web of Science ®Google Scholar
• Begehr H, Wen GC. Nonlinear elliptic boundary value problems and their applications. Harlow: Addison Wesley Longman; 1996.
   Google Scholar
• Shupeyeva B. Harmonic boundary value problems in a quarter ring domain. Adv Pure Appl Math. 2012;3:393–419; Adv Pure Appl Math. 2013;4:103--105 (Erratum).
   Google Scholar
• Wang Y. Schwarz-type boundary value problem for the polyanalytic equation in the half unit disc. Complex Var Elliptic Equ. 2012;57:983–993.
   Web of Science ®Google Scholar
• Wang YF, Wang YJ. Schwarz-type problem of nonhomogeneous Cauchy--Riemann equation on a triangle. J Math Anal Appl. 2011;377:557–570.
   Web of Science ®Google Scholar
• Begehr H. Green function for a hyperbolic strip and a class of related plane domains. Appl Anal. 2014;93:2370–2385.
   Web of Science ®Google Scholar
• Vekua I. Generalized analytic functions. International series of monographs in pure and applied mathematics. Oxford: Pergamon Press; 1962.
   Google Scholar
• Gakhov F. Boundary value problems. Oxford: Pergamon Press; 1966.
   Google Scholar
• Courant R, Hilbert D. Methods of mathematical physics, I. New York (NY): Interscience Publishers; 1953.
   Google Scholar