Title | On the regularity of solutions of quasilinear Poisson equations |
Publication Type | Journal Article |
Year of Publication | 2018 |
Authors | Gutlyanskii, VYa., Nesmelova, OV, Ryazanov, VI |
Abbreviated Key Title | Dopov. Nac. akad. nauk Ukr. |
DOI | 10.15407/dopovidi2018.10.009 |
Issue | 10 |
Section | Mathematics |
Pagination | 9-17 |
Date Published | 10/2018 |
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 $\alpha = (p − 2) / p$ .
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Keywords | Dirichlet problem, logarithmic and Newtonian potentials, potential theory, quasiconformal mappings, quasilinear Poisson equation, Sobolev classes |
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