The variable exponent periodic boundary value problems on the Lyapunov open curve

References

[1] H. Begehr and X. Li, Approximate solution of periodic Riemann boundary value problem for analytic functions, J. Comput. Appl. Math. 134 (2001), no. 1–2, 85–93. 10.1016/S0377-0427(00)00530-6Search in Google Scholar

[2] L. Bers, Theory of Pseudoanalytic Functions, New York University, New York, 1953. Search in Google Scholar

[3] L. Chien-ke, On fundamental problems of plane elasticity with periodic stresses, Acta Mech. Sinica 7 (1964), 316–327. Search in Google Scholar

[4] D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Appl. Numer. Harmon. Analy, Birkhäuser/Springer, Heidelberg, 2013. 10.1007/978-3-0348-0548-3Search in Google Scholar

[5] P. Dang, J. Du and T. Qian, Boundary value problems for periodic analytic functions, Bound. Value Probl. 2025 (2015), Paper No. 143. 10.1186/s13661-015-0407-4Search in Google Scholar

[6] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011. 10.1007/978-3-642-18363-8Search in Google Scholar

[7] P. Duan, M. Ku and Y. Wang, Riemann boundary value problems for multiplicative doubly quasi-periodic meta-analytic functions, Complex Var. Elliptic Equ. 68 (2023), no. 6, 892–917. 10.1080/17476933.2021.2024813Search in Google Scholar

[8] X. Fan and D. Zhao, On the spaces L p ⁢ ( x ) ⁢ ( Ω ) and W m , p ⁢ ( x ) ⁢ ( Ω ) , J. Math. Anal. Appl. 263 (2001), no. 2, 424–446. 10.1006/jmaa.2000.7617Search in Google Scholar

[9] F. D. Gakhov, Boundary Value Problems, Pergamon Press, Oxford, 1966. 10.1016/B978-0-08-010067-8.50007-4Search in Google Scholar

[10] H. Han, H. Liu and Y. Wang, Riemann boundary-value problem for doubly-periodic bianalytic functions, Bound. Value Probl. 2018 (2018), Paper No. 88. 10.1186/s13661-018-1005-zSearch in Google Scholar

[11] P. Harjulehto, P. Hästö, U. V. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), no. 12, 4551–4574. 10.1016/j.na.2010.02.033Search in Google Scholar

[12] F. He and Y. Hu, The nonlinear Riemann boundary value problem with square roots for Lyapunov open curve in variable exponent spaces, Complex Var. Elliptic Equ. 70 (2025), no. 9, 1485–1501. 10.1080/17476933.2024.2410944Search in Google Scholar

[13] F. He and X. Zhou, Riemann–Hilbert problem for variable exponent poly-Smirnov space on a piecewise Lyapunov boundary, Complex Var. Elliptic Equ. 69 (2024), no. 9, 1437–1461. 10.1080/17476933.2023.2226063Search in Google Scholar

[14] D. Hilbert, Über eine Anwendung der Integralgleichungen auf ein Problem Funktionentheorie, Verhandlungen des 3. Internationalen Mathematiker-Kongresses, Teubner, Leipzig (1905), 233–240. Search in Google Scholar

[15] B. A. Kats, The Riemann problem on a closed Jordan curve, Izv. Vyssh. Uchebn. Zaved. Mat. (1983), no. 4, 68–80. Search in Google Scholar

[16] G. Khuskivadze, V. Kokilashvili and V. Paatashvili, Boundary value problems for analytic and harmonic functions in domains with nonsmooth boundaries. Applications to conformal mappings, Mem. Differ. Equ. Math. Phys. 14 (1998), 1–195. Search in Google Scholar

[17] V. Kokilashvili and V. Paatashvili, The Dirichlet problem for harmonic functions in the Smirnov class with variable exponent, Georgian Math. J. 14 (2007), no. 2, 289–299. 10.1515/GMJ.2007.289Search in Google Scholar

[18] V. Kokilashvili and V. Paatashvili, The Riemann–Hilbert problem in weighted classes of Cauchy type integrals with density from L p ⁢ ( ⋅ ) ⁢ ( Γ ) , Complex Anal. Oper. Theory 2 (2008), no. 4, 569–591. 10.1007/s11785-008-0067-9Search in Google Scholar

[19] V. Kokilashvili and V. Paatashvili, The Dirichlet problem for harmonic functions from variable exponent Smirnov classes in domains with piecewise smooth boundary, J. Math. Sci. (N. Y.) 172 (2011), no. 3, 401–421. 10.1007/s10958-010-0201-xSearch in Google Scholar

[20] V. Kokilashvili and V. Paatashvili, Boundary Value Problems for Analytic and Harmonic Functions in Nonstandard Banach Function Spaces, Nova Science, New York, 2012. Search in Google Scholar

[21] V. Kokilashvili and V. Paatashvili, The Dirichlet and Riemann–Hilbert problems in Smirnov classes with variable exponent in doubly connected domains, J. Math. Sci. (N.Y.) 198 (2014), no. 6, 735–746. 10.1007/s10958-014-1822-2Search in Google Scholar

[22] V. Kokilashvili and V. Paatashvili, On the Riemann–Hilbert boundary value problem for generalized analytic functions in the framework of variable exponent spaces, Math. Methods Appl. Sci. 40 (2017), no. 18, 7267–7286. 10.1002/mma.4528Search in Google Scholar

[23] V. Kokilashvili and V. Paatashvili, Riemann–Hilbert problem in the class of Cauchy type integrals with densities of grand Lebesgue spaces, Complex Var. Elliptic Equ. 63 (2018), no. 9, 1233–1257. 10.1080/17476933.2017.1357705Search in Google Scholar

[24] V. Kokilashvili, V. Paatashvili and S. Samko, Boundary value problems for analytic functions in the class of Cauchy-type integrals with density in L p ⁢ ( ⋅ ) ⁢ ( Γ ) , Bound. Value Probl. 2005 (2005), no. 1, 43–71. 10.1155/BVP.2005.43Search in Google Scholar

[25] O. Kováčik and J. Rákosník, On spaces L p ⁢ ( x ) and W k , p ⁢ ( x ) , Czechoslovak Math. J. 41(116) (1991), no. 4, 592–618. 10.21136/CMJ.1991.102493Search in Google Scholar

[26] C.-K. Lu, The periodic Riemann boundary value problem and its applications to the theory of elasticity, Acta Math. Sinica 13, 343–388. Search in Google Scholar

[27] J. K. Lu, Boundary Value Problems for Analytic Functions, Ser. Pure Math. 16, World Scientific, River Edge, 1993. Search in Google Scholar

[28] T. R. Muradov, On Riemann boundary value problem in Hardy classes with variable summability exponent, Int. J. Math. Anal. (Ruse) 6 (2012), no. 13–16, 743–751. Search in Google Scholar

[29] N. I. Muskhelishvili, Singular Integral Equations. Boundary Problems of Function Theory and Their Application to Mathematical Physics, P. Noordhoff, Groningen, 1953. Search in Google Scholar

[30] E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748. 10.1016/j.jfa.2012.01.004Search in Google Scholar

[31] W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), no. 1, 200–211. 10.4064/sm-3-1-200-211Search in Google Scholar

[32] V. Paatashvili, On some properties of analytic functions from Smirnov class with a variable exponent, Mem. Differ. Equ. Math. Phys. 56 (2012), 73–88. Search in Google Scholar

[33] B. Riemann, Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse, Dissertation, Göttingen, 1851. Search in Google Scholar

[34] R. T. Seeley, Singular integrals and boundary value problems, Amer. J. Math. 88 (1966), 781–809. 10.2307/2373078Search in Google Scholar

[35] I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London, 1962. Search in Google Scholar

[36] S. Wang and F. He, On the variable exponent Riemann boundary value problem for Liapunov open curve, J. Geom. Anal. 33 (2023), no. 2, Paper No. 62. 10.1007/s12220-022-01113-9Search in Google Scholar

[37] X. Wang and J. Du, Periodic Riemann boundary value problem, Complex Var. Elliptic Equ. 68 (2023), no. 7, 1077–1092. 10.1080/17476933.2022.2035929Search in Google Scholar

[38] Y. Wang and Y. Wang, On Riemann problems for single-periodic polyanalytic functions, Math. Nachr. 287 (2014), no. 16, 1886–1915. 10.1002/mana.201300124Search in Google Scholar

[39] X. Zhou and F. He, On compound Riemann Hilbert boundary value problem in the framework of variable exponent spaces, J. Math. Anal. Appl. 496 (2021), no. 1, Article ID 124795. 10.1016/j.jmaa.2020.124795Search in Google Scholar

[40] E. I. Zverovič, Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces (in Russian), Uspehi Mat. Nauk 26 (1971), no. 1(157), 113–179; translation in Russian Math. Surveys 26 (1971), no. 1, 117–192. Search in Google Scholar