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To the theory of semilinear equations in the plane

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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.

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slavyansk, Ukraine

    Vladimir Gutlyanskiĭ, Olga Nesmelova & Vladimir Ryazanov

  2. Bogdan Khmelnytsky National University of Cherkasy, Cherkasy, Ukraine

    Vladimir Ryazanov

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  1. Vladimir Gutlyanskiĭ
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  2. Olga Nesmelova
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  3. Vladimir Ryazanov
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Correspondence to Vladimir Gutlyanskiĭ.

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Dedicated to the memory of Professor Bogdan Bojarski

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 16, No. 1, pp. 105–140 January–March, 2019.

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Gutlyanskiĭ, V., Nesmelova, O. & Ryazanov, V. To the theory of semilinear equations in the plane. J Math Sci 242, 833–859 (2019). https://doi.org/10.1007/s10958-019-04519-z

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