Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation

Frédéric Abergel; Jean-Michel Rakotoson

doi:10.3934/dcds.2013.33.1809

Article Contents

2013, Volume 33, Issue 5: 1809-1818. Doi: 10.3934/dcds.2013.33.1809

This issue Previous Article Formal Poincaré-Dulac renormalization for holomorphic germs Next Article Characterizations of $\omega$-limit sets in topologically hyperbolic systems

Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation

Frédéric Abergel^1, and
Jean-Michel Rakotoson^2,

1.
Laboratoire de Mathématiques Appliquées aux Systémes, École Centrale Paris Grande voie des Vignes, 92295 Châtenay-Malabry Cedex
2.
UMR 6086 CNRS. Laboratoire de Mathématiques - Université de Poitiers - SP2MI, Boulevard Marie et Pierre Curie, Téléport 2, BP30179 - 86962 Futuroscope Chasseneuil Cedex

Received: December 2011

Revised: February 2012

Published: May 2013

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Abstract

It is known that the very weak solution of $-∫_Ω u\Deltaφ dx=∫_Ω fφ dx$, $∀φ∈ C^2(\overline{Ω}),$ $φ=0$ on $∂Ω$, $u\in L^1(Ω)$ has its gradient in $Ł^1(Ω)$ whenever $f∈ L^1(Ω;δ(1+|Lnδ|))$, $δ(x)$ being the distance of $x∈Ω$ to the boundary. In this paper, we show that if $f≥0$ is not in this weighted space $L^1(Ω;δ(1+|Lnδ|))$, then its gradient blows up in $L(\log L)$ at least. Moreover, we show that there exist a domain $Ω$ of class $C^\infty$ and a function $f∈ L^1_+(Ω,δ)$ such that the associated very weak solution has its gradient being non integrable on $Ω$.

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Mathematics Subject Classification: 35B44, 35A01, 35J67, 35J25.

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