On the regularity of solutions of quasilinear Poisson equations

1Gutlyanskii, VYa., 1Nesmelova, OV, 1Ryazanov, VI
1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk
Dopov. Nac. akad. nauk Ukr. 2018, 10:9-17
https://doi.org/10.15407/dopovidi2018.10.009
Section: Mathematics
Language: English
Abstract: 
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 $\alpha = (p − 2) / p$ .
Keywords: Dirichlet problem, logarithmic and Newtonian potentials, potential theory, quasiconformal mappings, quasilinear Poisson equation, Sobolev classes
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