On the quasilinear Poisson equations in the complex plane

TitleOn the quasilinear Poisson equations in the complex plane
Publication TypeJournal Article
Year of Publication2020
AuthorsGutlyanskii, VYa., Nesmelova, OV, Ryazanov, VI
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
Date Published1/2020
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 \[L^{1}\left(D\right)\cap L_loc^p \left(D\right)\], p > 1. Then, applying the Leray— Schauder approach, we prove the existence of Höldercontinuous solutions U in the class \[W_loc^2\cdot^{p}\left(D\right)\] 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.
Keywordsanisotropic and inhomogeneous media, potential theory, quasiconformal mappings, quasilinear Poisson equations, semilinear equations

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