The Dirichlet problem for the Poisson type equations in the plane
DOI:
https://doi.org/10.15407/dopovidi2020.05.010Keywords:
anisotropic and inho mogeneous media, Dirichlet problem, quasiconformal maps, quasilinear Poisson equations, semilinear elliptic equationsAbstract
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.
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