The Dirichlet problem for the Poisson type equations in the plane

1Gutlyanskii, VYa.
1Nesmelova, OV
1Ryazanov, VI
1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk
Dopov. Nac. akad. nauk Ukr. 2020, 5:10-16
Section: Mathematics
Language: English

We present a new approach to the study of 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. We establish a theorem on the existence of weak C(D)∩W1.2loc (D) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components and give applications to equations of mathematical physics in anisotropic media.

Keywords: anisotropic and inho mogeneous media, Dirichlet problem, quasiconformal maps, quasilinear Poisson equations, semilinear elliptic equations

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