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
Section: Mathematics
Language: English

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

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