On semilinear equations in the complex plane

1Gutlyanskii, VYa.
1Nesmelova, OV
1Ryazanov, VI
1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk
Dopov. Nac. akad. nauk Ukr. 2019, 7:9-16
https://doi.org/10.15407/dopovidi2019.07.009
Section: Mathematics
Language: English
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.

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

1. 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
2. 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
3. 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
4. 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
5. 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.
6. 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
7. 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
8. 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
9. Ladyzhenskaya, O.A. & Ural’tseva N.N. (1968). Linear and quasilinear elliptic equations. New York, London: Academic Press.
10. 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
11. Diaz, J.I. (1985). Nonlinear partial differential equations and free boundaries. (Vol. 1). Elliptic equations. Research Notes in Mathematics, (Vol. 106). Boston: Pitman.
12. 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
13. Aris, R. (1975).The mathematical theory of diffusion and reaction in permeable catalysts. (Vol. 1, 2). Oxford: Clarendon Press.
14. Bear, J. (1972). Dynamics of fluids in porous media. New York: Elsevier.
15. 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