Continuity of solutions to the $G$-Laplace equation involving measures
We establish local continuity of solutions to the $G$-Laplace equation involving measures, i.e., \begin{align*} -\text{div}\ \bigg(\frac{g(|\nabla u|)}{|\nabla u|}\nabla u\bigg)=\mu,\end{align*} where $\mu$ is a nonnegative Radon measure satisfying $\mu (B_{r}(x_{0}))\leq Cr^{m}$ for any ball $B_{r}...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2019-06-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=7373 |
| Summary: | We establish local continuity of solutions to the $G$-Laplace equation involving measures, i.e.,
\begin{align*}
-\text{div}\ \bigg(\frac{g(|\nabla u|)}{|\nabla u|}\nabla
u\bigg)=\mu,\end{align*}
where $\mu$ is a nonnegative Radon measure satisfying $\mu (B_{r}(x_{0}))\leq
Cr^{m}$ for any ball $B_{r}(x_{0})\subset\subset \Omega$
with $r\leq 1$ and $m>n-1-\delta\geq 0$. The function $g$ is supposed to be nonnegative and $C^{1}$-continuous
on $[0,+\infty)$, satisfying $g(0)=0$ and \begin{align*}\delta\leq \frac{tg'(t)}{g(t)}\leq g_{0}, \forall t>0\end{align*} with positive constants $\delta$ and $g_{0}$, which generalizes the structural conditions of Ladyzhenskaya–Ural'tseva for an elliptic operator. |
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| ISSN: | 1417-3875 |