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Toward the theory of semi-linear Beltrami equations

Year 2023, , 151 - 163, 15.09.2023
https://doi.org/10.33205/cma.1248692

Abstract

We study the semi-linear Beltrami equation $\omega_{\bar{z}}-\mu(z) \omega_z=\sigma(z)q(\omega(z))$ and show that it is closely related to the corresponding semi-linear equation of the form ${\rm div} A(z)\nabla\,U(z)=G(z) Q(U(z)).$ Applying the theory of completely continuous operators by Ahlfors-Bers and Leray-Schauder, we prove existence of regular solutions both to the semi-linear Beltrami equation and to the given above semi-linear equation in the divergent form, see Theorems 1.1 and 5.2. We also derive their representation through solutions of the semi-linear Vekua type equations and generalized analytic functions with sources. Finally, we apply Theorem 5.2 for several model equations describing physical phenomena in anisotropic and inhomogeneous media.

Supporting Institution

The European Federation of Academies of Sciences and Humanities (ALLEA)

Project Number

EFDS-FL2-08

Thanks

The first 3 authors are partially supported by the Grant EFDS-FL2-08 of the found of the European Federation of Academies of Sciences and Humanities (ALLEA)

References

  • L. Ahlfors: Lectures on Quasiconformal Mappings, Van Nostrand, New York (1966).
  • L. V. Ahlfors, L. Bers: Riemanns mapping theorem for variable metrics, Ann. Math., 72 (2) (1960), 385–404.
  • K. Astala, T. Iwaniec and G. J. Martin: Elliptic differential equations and quasiconformal mappings in the plane, Princeton Math. Ser., 48, Princeton Univ. Press, Princeton (2009).
  • R. Aris: The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, V. I–II, Clarendon Press, Oxford (1975).
  • G. I. Barenblatt, Ja. B. Zel’dovic, V. B. Librovich and G. M. Mahviladze: The mathematical theory of combustion and explosions, Consult. Bureau, New York (1985).
  • B. Bojarski, V. Gutlyanskii, O. Martio and V. Ryazanov: Infinitesimal geometry of quasiconformal and bi-lipschitz mappings in the plane, EMS Tracts in Mathematics, 19, European Mathematical Society, Zürich (2013).
  • L. E. J. Brouwer: Über Abbildungen von Mannigfaltigkeiten, Math. Ann., 71 (1912), 97–115.
  • J. I. Diaz: Nonlinear partial differential equations and free boundaries. V. I. Elliptic equations., Research Notes in Mathematics, 106, Pitman, Boston (1985).
  • G. Dinca, J. Mawhin: Brouwer degree - the core of nonlinear analysis, Progress in Nonlinear Differential Equations and their Applications, 95, Birkhäuser, Springer, Cham, (2021).
  • O. P. Dovhopiatyi, E. A. Sevost’yanov: On the existence of solutions of quasilinear Beltrami equations with two characteristics, Ukrainian Math. J., 74 (7) (2022), 1099–1112.
  • N. Dunford, J. T. Schwartz: Linear Operators. I. General Theory, Pure and Applied Mathematics, 7, Interscience Publishers, New York, London, (1958).
  • V. Gutlyanskii, O. Nesmelova and V. Ryazanov: On a model semilinear elliptic equation in the plane, J. Math. Sci. (USA), 220 (5) (2017), 603–614.
  • V. Gutlyanskii, O. Nesmelova and V. Ryazanov: On quasiconformal maps and semi-linear equations in the plane, J. Math. Sci. (USA), 229 (1) (2018), 7–29.
  • V. Gutlyanskii, O. Nesmelova and V. Ryazanov: To the theory of semi-linear equations in the plane, J. Math. Sci. (USA), 242 (6) (2019), 833–859.
  • V. Gutlyanskii, O. Nesmelova and V. Ryazanov: On a quasilinear Poisson equation in the plane, Anal. Math. Phys., 10 (1) (2020), Paper No. 6, 14 pp.
  • V. Gutlyanskii, O. Nesmelova and V. Ryazanov: Semi-linear equations and quasiconformal mappings, Complex Var. Elliptic Equ., 65 (5) (2020), 823–843.
  • [V. Gutlyanskii, O. Nesmelova, V. Ryazanov and A. Yefimushkin: On boundary-value problems for semi-linear equations in the plane, J. Math. Sci. (USA), 259 (1) (2021), 53–74.
  • V. Gutlyanskii, O. Nesmelova, V. Ryazanov and A. Yefimushkin: Logarithmic potential and generalized analytic functions, J. Math. Sci. (USA), 256 (6) (2021), 735–752.
  • J. Heinonen, T. Kilpelainen and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford-New York-Tokyo (1993).
  • E. Karapinar: A short survey on the recent fixed point results on b-metric spaces, Constr. Math. Anal. 1 (1)(2018), 15–44.
  • O. Lehto, K. J. Virtanen: Quasiconformal mappings in the plane, Springer-Verlag, Berlin, Heidelberg (1973).
  • J. Leray, Ju. Schauder: Topologie et equations fonctionnelles, Ann. Sci. Ecole Norm. Sup., 51 (3) (1934), 45–78 (in French).
  • M. Marcus, L. Veron: Nonlinear second order elliptic equations involving measures, De Gruyter Series in Nonlinear Analysis and Applications, 21, De Gruyter, Berlin (2014).
  • V. G. Maz’ja: Sobolev spaces, Springer-Verlag, Berlin (1985).
  • J. Mawhin: Leray-Schauder continuation theorems in the absence of a priori bounds, Topol. Methods Nonlinear Anal., 9 (1) (1997), 179–200.
  • H. K. Pathak: An introduction to nonlinear analysis and fixed point theory, Springer, Singapore (2018).
  • S. I. Pohozhaev: Concerning an equation in the theory of combustion, Math. Notes, 88 (1) (2010), 48–56.
  • V. Ryazanov: On Hilbert and Riemann problems for generalized analytic functions and applications, Anal. Math. Phys., 11 (1) (2021), Paper No. 5, 18 pp.
  • V. Ryazanov: Dirichlet problem with measurable data in rectifiable domains, Filomat, 36 (6) (2022), 2119–2127.
  • E. A. Sevost’yanov: On quasilinear Beltrami-type equations with degeneration, Math. Notes 90 (3) (2011), 431–438.
  • S. L. Sobolev: Applications of functional analysis in mathematical physics, Transl. of Math. Mon., 7, AMS, Providence, R.I. (1963).
  • W. Varnhorn: On the Poisson equation in exterior domains, Constr. Math. Anal. 5 (3) (2022), 134–140.
  • I. N. Vekua: Generalized analytic functions, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass (1962).
  • L. Veron: Local and global aspects of quasilinear degenerate elliptic equations. Quasilinear elliptic singular problems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2017).
Year 2023, , 151 - 163, 15.09.2023
https://doi.org/10.33205/cma.1248692

Abstract

Project Number

EFDS-FL2-08

References

  • L. Ahlfors: Lectures on Quasiconformal Mappings, Van Nostrand, New York (1966).
  • L. V. Ahlfors, L. Bers: Riemanns mapping theorem for variable metrics, Ann. Math., 72 (2) (1960), 385–404.
  • K. Astala, T. Iwaniec and G. J. Martin: Elliptic differential equations and quasiconformal mappings in the plane, Princeton Math. Ser., 48, Princeton Univ. Press, Princeton (2009).
  • R. Aris: The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, V. I–II, Clarendon Press, Oxford (1975).
  • G. I. Barenblatt, Ja. B. Zel’dovic, V. B. Librovich and G. M. Mahviladze: The mathematical theory of combustion and explosions, Consult. Bureau, New York (1985).
  • B. Bojarski, V. Gutlyanskii, O. Martio and V. Ryazanov: Infinitesimal geometry of quasiconformal and bi-lipschitz mappings in the plane, EMS Tracts in Mathematics, 19, European Mathematical Society, Zürich (2013).
  • L. E. J. Brouwer: Über Abbildungen von Mannigfaltigkeiten, Math. Ann., 71 (1912), 97–115.
  • J. I. Diaz: Nonlinear partial differential equations and free boundaries. V. I. Elliptic equations., Research Notes in Mathematics, 106, Pitman, Boston (1985).
  • G. Dinca, J. Mawhin: Brouwer degree - the core of nonlinear analysis, Progress in Nonlinear Differential Equations and their Applications, 95, Birkhäuser, Springer, Cham, (2021).
  • O. P. Dovhopiatyi, E. A. Sevost’yanov: On the existence of solutions of quasilinear Beltrami equations with two characteristics, Ukrainian Math. J., 74 (7) (2022), 1099–1112.
  • N. Dunford, J. T. Schwartz: Linear Operators. I. General Theory, Pure and Applied Mathematics, 7, Interscience Publishers, New York, London, (1958).
  • V. Gutlyanskii, O. Nesmelova and V. Ryazanov: On a model semilinear elliptic equation in the plane, J. Math. Sci. (USA), 220 (5) (2017), 603–614.
  • V. Gutlyanskii, O. Nesmelova and V. Ryazanov: On quasiconformal maps and semi-linear equations in the plane, J. Math. Sci. (USA), 229 (1) (2018), 7–29.
  • V. Gutlyanskii, O. Nesmelova and V. Ryazanov: To the theory of semi-linear equations in the plane, J. Math. Sci. (USA), 242 (6) (2019), 833–859.
  • V. Gutlyanskii, O. Nesmelova and V. Ryazanov: On a quasilinear Poisson equation in the plane, Anal. Math. Phys., 10 (1) (2020), Paper No. 6, 14 pp.
  • V. Gutlyanskii, O. Nesmelova and V. Ryazanov: Semi-linear equations and quasiconformal mappings, Complex Var. Elliptic Equ., 65 (5) (2020), 823–843.
  • [V. Gutlyanskii, O. Nesmelova, V. Ryazanov and A. Yefimushkin: On boundary-value problems for semi-linear equations in the plane, J. Math. Sci. (USA), 259 (1) (2021), 53–74.
  • V. Gutlyanskii, O. Nesmelova, V. Ryazanov and A. Yefimushkin: Logarithmic potential and generalized analytic functions, J. Math. Sci. (USA), 256 (6) (2021), 735–752.
  • J. Heinonen, T. Kilpelainen and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford-New York-Tokyo (1993).
  • E. Karapinar: A short survey on the recent fixed point results on b-metric spaces, Constr. Math. Anal. 1 (1)(2018), 15–44.
  • O. Lehto, K. J. Virtanen: Quasiconformal mappings in the plane, Springer-Verlag, Berlin, Heidelberg (1973).
  • J. Leray, Ju. Schauder: Topologie et equations fonctionnelles, Ann. Sci. Ecole Norm. Sup., 51 (3) (1934), 45–78 (in French).
  • M. Marcus, L. Veron: Nonlinear second order elliptic equations involving measures, De Gruyter Series in Nonlinear Analysis and Applications, 21, De Gruyter, Berlin (2014).
  • V. G. Maz’ja: Sobolev spaces, Springer-Verlag, Berlin (1985).
  • J. Mawhin: Leray-Schauder continuation theorems in the absence of a priori bounds, Topol. Methods Nonlinear Anal., 9 (1) (1997), 179–200.
  • H. K. Pathak: An introduction to nonlinear analysis and fixed point theory, Springer, Singapore (2018).
  • S. I. Pohozhaev: Concerning an equation in the theory of combustion, Math. Notes, 88 (1) (2010), 48–56.
  • V. Ryazanov: On Hilbert and Riemann problems for generalized analytic functions and applications, Anal. Math. Phys., 11 (1) (2021), Paper No. 5, 18 pp.
  • V. Ryazanov: Dirichlet problem with measurable data in rectifiable domains, Filomat, 36 (6) (2022), 2119–2127.
  • E. A. Sevost’yanov: On quasilinear Beltrami-type equations with degeneration, Math. Notes 90 (3) (2011), 431–438.
  • S. L. Sobolev: Applications of functional analysis in mathematical physics, Transl. of Math. Mon., 7, AMS, Providence, R.I. (1963).
  • W. Varnhorn: On the Poisson equation in exterior domains, Constr. Math. Anal. 5 (3) (2022), 134–140.
  • I. N. Vekua: Generalized analytic functions, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass (1962).
  • L. Veron: Local and global aspects of quasilinear degenerate elliptic equations. Quasilinear elliptic singular problems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2017).
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences, Applied Mathematics (Other)
Journal Section Articles
Authors

Vladimir Gutlyanskii 0000-0002-8691-4617

Olga Nesmelova 0000-0003-2542-5980

Vladimir Ryazanov 0000-0002-4503-4939

Eduard Yakubov 0000-0001-8366-6323

Project Number EFDS-FL2-08
Early Pub Date August 18, 2023
Publication Date September 15, 2023
Published in Issue Year 2023

Cite

APA Gutlyanskii, V., Nesmelova, O., Ryazanov, V., Yakubov, E. (2023). Toward the theory of semi-linear Beltrami equations. Constructive Mathematical Analysis, 6(3), 151-163. https://doi.org/10.33205/cma.1248692
AMA Gutlyanskii V, Nesmelova O, Ryazanov V, Yakubov E. Toward the theory of semi-linear Beltrami equations. CMA. September 2023;6(3):151-163. doi:10.33205/cma.1248692
Chicago Gutlyanskii, Vladimir, Olga Nesmelova, Vladimir Ryazanov, and Eduard Yakubov. “Toward the Theory of Semi-Linear Beltrami Equations”. Constructive Mathematical Analysis 6, no. 3 (September 2023): 151-63. https://doi.org/10.33205/cma.1248692.
EndNote Gutlyanskii V, Nesmelova O, Ryazanov V, Yakubov E (September 1, 2023) Toward the theory of semi-linear Beltrami equations. Constructive Mathematical Analysis 6 3 151–163.
IEEE V. Gutlyanskii, O. Nesmelova, V. Ryazanov, and E. Yakubov, “Toward the theory of semi-linear Beltrami equations”, CMA, vol. 6, no. 3, pp. 151–163, 2023, doi: 10.33205/cma.1248692.
ISNAD Gutlyanskii, Vladimir et al. “Toward the Theory of Semi-Linear Beltrami Equations”. Constructive Mathematical Analysis 6/3 (September 2023), 151-163. https://doi.org/10.33205/cma.1248692.
JAMA Gutlyanskii V, Nesmelova O, Ryazanov V, Yakubov E. Toward the theory of semi-linear Beltrami equations. CMA. 2023;6:151–163.
MLA Gutlyanskii, Vladimir et al. “Toward the Theory of Semi-Linear Beltrami Equations”. Constructive Mathematical Analysis, vol. 6, no. 3, 2023, pp. 151-63, doi:10.33205/cma.1248692.
Vancouver Gutlyanskii V, Nesmelova O, Ryazanov V, Yakubov E. Toward the theory of semi-linear Beltrami equations. CMA. 2023;6(3):151-63.