On semilinear equations in the complex plane
DOI:
https://doi.org/10.15407/dopovidi2019.07.009Keywords:
anisotropic and inhomogeneous media, conformal and quasiconformal mappings, Dirichlet problem, semilinear elliptic equationsAbstract
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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