To the theory of semilinear equations in the plane
- Authors: Gutlyanskiĭ V.1, Nesmelova O.1, Ryazanov V.1,2
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Affiliations:
- Institute of Applied Mathematics and Mechanics of the NAS of Ukraine
- Bogdan Khmelnytsky National University of Cherkasy
- Issue: Vol 242, No 6 (2019)
- Pages: 833-859
- Section: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/243046
- DOI: https://doi.org/10.1007/s10958-019-04519-z
- ID: 243046
Cite item
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.
About the authors
Vladimir Gutlyanskiĭ
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine
Author for correspondence.
Email: vgutlyanskii@gmail.com
Ukraine, Slavyansk
Olga Nesmelova
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine
Email: vgutlyanskii@gmail.com
Ukraine, Slavyansk
Vladimir Ryazanov
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine; Bogdan Khmelnytsky National University of Cherkasy
Email: vgutlyanskii@gmail.com
Ukraine, Slavyansk; Cherkasy