On semilinear equations in the complex plane

Authors

  • V.Ya. Gutlyanskiĭ Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slov’yansk
  • O.V. Nesmelova Donbas State Pedagogical University, Slov’yansk
  • V.I. Ryazanov Bogdan Khmelnytsky National University of Cherkasy

DOI:

https://doi.org/10.15407/dopovidi2019.07.009

Keywords:

anisotropic and inhomogeneous media, conformal and quasiconformal mappings, Dirichlet problem, semilinear elliptic equations

Abstract

We study the Dirichlet problem for the semilinear partial differential equations div (A∇u) = f (u) in simply connected domains D of the complex plane C with continuous boundary data. We prove the existence of the weak solutions u in the class C ∩Wloc1,2 (D), if a Jordan domain D satisfies the quasihyperbolic boundary condition by Gehring—Martio. An example of such a domain that fails to satisfy the standard (A)-condition by Ladyzhenskaya—Ural'tseva and the known outer cone condition is given. Some applications of the results to various processes of diffusion and absorption in anisotropic and inhomogeneous media are presented.

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References

Gutlyanskii, V. Ya., Nesmelova, O. V. & Ryazanov, V. I. (2018). On the regularity of solutions of quasilinear Poisson equations. Dopov. Nac. Akad. Nauk. Ukr., No. 10, pp. 9-17. doi: https://doi.org/10.15407/dopovidi2018.10.009

Lehto, O. & Virtanen, K. I. (1973). Quasiconformal mappings in the plane, 2nd ed. Berlin, Heidelberg, New York: Springer. doi: https://doi.org/10.1007/978-3-642-65513-5

Gutlyanskii, V. Ya., Nesmelova, O. V. & Ryazanov, V. I. (2018). On quasiconformal maps and semilinear equations in the plane. J. Math. Sci., 229, No. 1, pp. 7-29. doi: https://doi.org/10.1007/s10958-018-3659-6

Gutlyanskii, V. Ya., Nesmelova, O. V. & Ryazanov, V. I. (2018). Semilinear equations in the plane with measurable data. Dopov. Nac. Akad. Nauk Ukr., No. 2, pp. 12-18. doi: https://doi.org/10.15407/dopovidi2018.02.012

Bojarski, B. V. (2009). Generalized solutions of a system of differential equations of the first order and elliptic type with discontinuous coefficients. Report Dept. Math. Stat. (Vol. 118). Jyväskylä: Univ. of Jyväskylä.

Astala, K. & Koskela, P. (1991). Quasiconformal mappings and global integrability of the derivative. J. Anal. Math, 57, pp. 203-220. doi: https://doi.org/10.1007/BF03041070

Becker, J. & Pommerenke, Ch. (1982). Hölder continuity of conformal mappings and nonquasiconformal Jordan curves. Comment. Math. Helv., 57, No. 2, pp. 221-225. doi: https://doi.org/10.1007/BF02565858

Gehring, F. W. & Martio, O. (1985). Lipschitz classes and quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A. I. Math., 10, pp. 203-219. doi: https://doi.org/10.5186/aasfm.1985.1022

Ladyzhenskaya, O.A. & Ural’tseva N.N. (1968). Linear and quasilinear elliptic equations. New York, London: Academic Press.

Gehring, F. W. & Martio, O. (1985). Quasiextremal distance domains and extension of quasiconformal mappings. J. Anal. Math., 45, pp. 181-206. doi: https://doi.org/10.1007/BF02792549

Diaz, J.I. (1985). Nonlinear partial differential equations and free boundaries. (Vol. 1). Elliptic equations. Research Notes in Mathematics, (Vol. 106). Boston: Pitman.

Marcus, M. & Veron, L. (2014). Nonlinear second order elliptic equations involving measures. De Gruyter Series in Nonlinear Analysis and Applications. (Vol. 21). Berlin: De Gruyter. doi: https://doi.org/10.1515/9783110305319

Aris, R. (1975).The mathematical theory of diffusion and reaction in permeable catalysts. (Vol. 1, 2). Oxford: Clarendon Press.

Bear, J. (1972). Dynamics of fluids in porous media. New York: Elsevier.

Pokhozhaev, S. I. (2010). On an equation of combustion theory. Math. Notes, 88, No. 1-2, pp. 48-56. doi: https://doi.org/10.1134/S0001434610070059

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Published

21.04.2024

How to Cite

Gutlyanskiĭ, V., Nesmelova, O., & Ryazanov, V. (2024). On semilinear equations in the complex plane . Reports of the National Academy of Sciences of Ukraine, (7), 9–16. https://doi.org/10.15407/dopovidi2019.07.009

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