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Journal of Mathematical Sciences

, Volume 242, Issue 6, pp 833–859 | Cite as

To the theory of semilinear equations in the plane

  • Vladimir GutlyanskiĭEmail author
  • Olga Nesmelova
  • Vladimir Ryazanov
Article
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Abstract

In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z), whereas its reaction term f(u) is a continuous non-linear function. Assuming that f(t)/t → 0 as t → ∞, we establish a theorem on existence of weak \( C\left(\overline{D}\right)\cap {W}_{\mathrm{loc}}^{1,2}(D) \) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components. As consequences, we give applications to some concrete model semilinear equations of mathematical physics, arising from modeling processes in anisotropic and inhomogeneous media. With a view to the further development of the theory of boundary-value problems for the semilinear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity.

Keywords

Semilinear elliptic equations quasilinear Poisson equations anisotropic and inhomogeneous media conformal and quasiconformal mappings 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Vladimir Gutlyanskiĭ
    • 1
    Email author
  • Olga Nesmelova
    • 1
  • Vladimir Ryazanov
    • 1
    • 2
  1. 1.Institute of Applied Mathematics and Mechanics of the NAS of UkraineSlavyanskUkraine
  2. 2.Bogdan Khmelnytsky National University of CherkasyCherkasyUkraine

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