On the regularity of solutions of quasilinear Poisson equations

TitleOn the regularity of solutions of quasilinear Poisson equations
Publication TypeJournal Article
Year of Publication2018
AuthorsGutlyanskii, VYa., Nesmelova, OV, Ryazanov, VI
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
Date Published10/2018
We study the Dirichlet problem for quasilinear partial differential equations of the form $\Delta u(z) = h(z)f (u(z))$ in the unit disk $\mathbb{D} \subset  \mathbb{C}$ with continuous boundary data. Here, the function $h : \mathbb{D} \rightarrow  \mathbb{R}$ belongs to the class $L^{p}(\mathbb{D})$, $p > 1$, and the continuous function $f : \mathbb{R} \rightarrow  \mathbb{R}$ is assumed to have the nondecreasing $| f |$ of $| t |$ and such that $f (t) / t\rightarrow 0$ as $t \rightarrow \infty$. We prove the existence of a continuous solution $u$ of the problem in the Sobolev class ${W_{loc}}^{2, p}(\mathbb{D})$. Moreover, we show that if $p > 2$, then $u \in {C_{loc}}^{1, \alpha }(\mathbb{D})$ with $\alpha = (p − 2) / p$ .
KeywordsDirichlet problem, logarithmic and Newtonian potentials, potential theory, quasiconformal mappings, quasilinear Poisson equation, Sobolev classes
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