Planar quasilinear  elliptic equations  with right-hand side in $L(\log L)^{\delta}$

Angela Alberico; Teresa Alberico; Carlo Sbordone

doi:10.3934/dcds.2011.31.1053

Article Contents

2011, Volume 31, Issue 4: 1053-1067. Doi: 10.3934/dcds.2011.31.1053

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Planar quasilinear elliptic equations with right-hand side in $L(\log L)^{\delta}$

1.
Consiglio Nazionale delle Ricerche, Istituto per le Applicazioni del Calcolo “Mauro Picone” - Sezione di Napoli, Via Pietro Castellino 111, 80131, Napoli
2.
Università degli Studi di Napoli “Federico II”, Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Via Cintia, 80126, Napoli

Received: November 2009

Revised: July 2010

Published: October 2011

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Abstract

For $\Omega\subset \mathbb{R}^2$ a bounded open set with $\mathcal{C}^1$ boundary, we study the regularity of the variational solution $v\in W_0^{1,2}(\Omega)$ to the quasilinear elliptic equation of Leray-Lions \begin{equation*} - \,\textrm{div}\, A(x, \nabla v) = f \end{equation*} when $f$ belongs to the Zygmund space $L(\log L)^{\delta}(\Omega)$, $\frac{1}{2} \leq \delta \leq 1$. We prove that $|\nabla v|$ belongs to the Lorentz space $L^{2, 1/\delta}(\Omega)$.

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Mathematics Subject Classification: Primary: 35B65; Secondary: 46E30.

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References

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