Gradient regularity for solutions to quasilinear elliptic equations in the plane

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• 作者：Linda Maria De Cave^a ; ^{linda.decave@sbai.uniroma1.it" class="auth_mail} ; Carlo Sbordone^b ; ^{sbordone@unina.it" class="auth_mail}
• 关键词：Elliptic equations ; Zygmund spaces ; Gradient regularity
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：15 September 2014
• 年：2014
• 卷：417
• 期：2
• 页码：537-551
• 全文大小：342 K

文摘

We investigate the Dirichlet problem for a quasilinear elliptic equation in a planar domain 惟  , when f   belongs to the Zygmund space $L{(\log L)}^{\frac{1}{2}}{(\log \log L)}^{系}(惟)$, 0<系<1$0<系<1$. We prove that the gradient of the variational solution $v\in W_0^{1,2}(惟)$ belongs to the space $L^2{(\log \log L)}^{2系}(惟;{\mathbb{R}}^2)$. A main tool is a result on the regularity of the gradient of the solution 蠁   to the Dirichlet problem where 22d3d18dc13c045f723526fdb0ec56">$\underset{滩}{蠂}\in L^2{(\log \log L)}^{-尾}(惟;{\mathbb{R}}^2)$, d35bbabc81f9154" title="Click to view the MathML source">尾>0$尾>0$. Namely, if the mapping a:惟×R²→R²$a:惟\times {\mathbb{R}}^2\to {\mathbb{R}}^2$ satisfies the Leray–Lions type conditions, then we prove the estimates