On the quasilinear Poisson equations in the complex plane
DOI:
https://doi.org/10.15407/dopovidi2020.01.003Keywords:
anisotropic and inhomogeneous media, potential theory, quasiconformal mappings, quasilinear Poisson equations, semilinear equationsAbstract
First, we study the existence and regularity of solutions for the linear Poisson equations ∆U(z) = g(z) in bounded domains D of the complex plane £ with charges g in the classes L1(D)∩Llocp(D) , p > 1. Then, applying the Leray— Schauder approach, we prove the existence of Höldercontinuous solutions U in the class Wloc2,p(D) for the quasilinear Poisson equations of the form ∆U(z) = h(z)⋅ f (U(z)) with h in the same classes as g and continuous functions f : R → R such that f (t) / t → 0 as t → ∞. These results can be applied to various problems of mathematical physics.
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