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

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May 2013, 33(5): 1809-1818. doi: 10.3934/dcds.2013.33.1809

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, France
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 December 2012

Abstract
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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 $Ω$.

Keywords: monotone rearrangement, distance to the boundary, gradient blow-up., Very weak solutions, linear PDE, regularity.

Mathematics Subject Classification: 35B44, 35A01, 35J67, 35J2.

Citation: Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809

References:

[1]	Colin Bennett and Robert Sharpley, "Interpolation of Operators," Pure and Applied Mathematics, 129. Academic Press, Inc., Boston, MA, 1988. xiv+469 pp.
[2]	Marie-Francoise Bidaut-Véron and Laurent Vivier, An elliptic semilinear equation with source term involving boundary measures: the subcritical case, Rev. Mat. Iberoamericana, 16 (2000), 477-513. doi: 10.4171/RMI/281.
[3]	Haïm Brezis, "Analyse Fonctionnelle," [Functional analysis] Théorie et applications. [Theory and applications] Collection Mathématiques Appliquées. pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree] Masson, Paris, 1983. xiv+234 pp.
[4]	Haïm Brezis, Thierry Cazenave, Yvan Martel and Arthur Ramiandrisoa, Blow up for $u_t-\Delta u=g(u)$ revisited, Adv. Differential Equations, 1 (1996), 73-90.
[5]	J. I. Díaz and J. M. Rakotoson, On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary, J. Funct. Anal., 257 (2009), 807-831. doi: 10.1016/j.jfa.2009.03.002.
[6]	Jesus Idelfonso Díaz and Jean Michel Rakotoson, On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary, Discrete Contin. Dyn. Syst., 27 (2010), 1037-1058. doi: 10.3934/dcds.2010.27.1037.
[7]	Françoise Demengel and Gilbert Demengel, "Espaces Fonctionnels," (French) [Functional spaces] Utilisation dans la résolution des équations aux dérivées partielles. [Application to the solution of partial differential equations] Savoirs Actuels (Les Ulis). [Current Scholarship (Les Ulis)] EDP Sciences, Les Ulis; CNRS Éditions, Paris, 2007. xii+467 pp.
[8]	David Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp.
[9]	J. M. Rakotoson, A few natural extension of the regularity of a very weak solution, Differential and Integral Equations, 24 (2011), 1125-1140.
[10]	Jean-Michel Rakotoson, "Réarrangement Relatif," (French. French summary) [Relative rearrangement] Un instrument d'estimations dans les problèmes aux limites. [An estimation tool for limit problems] Mathématiques & Applications (Berlin) [Mathematics & Applications], 64. Springer, Berlin, 2008. xvi+293 pp.
[11]	Jean-Émile Rakotoson and Jean-Michel Rakotoson, "Analyse Fonctionnelle Appliquée aux Équations aux Dérivées Partielles," [Functional analysis applied to partial differential equations] Mathématiques. [Mathematics] Presses Universitaires de France, Paris, 1999. 232 pp.

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References:

[1]	Colin Bennett and Robert Sharpley, "Interpolation of Operators," Pure and Applied Mathematics, 129. Academic Press, Inc., Boston, MA, 1988. xiv+469 pp.
[2]	Marie-Francoise Bidaut-Véron and Laurent Vivier, An elliptic semilinear equation with source term involving boundary measures: the subcritical case, Rev. Mat. Iberoamericana, 16 (2000), 477-513. doi: 10.4171/RMI/281.
[3]	Haïm Brezis, "Analyse Fonctionnelle," [Functional analysis] Théorie et applications. [Theory and applications] Collection Mathématiques Appliquées. pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree] Masson, Paris, 1983. xiv+234 pp.
[4]	Haïm Brezis, Thierry Cazenave, Yvan Martel and Arthur Ramiandrisoa, Blow up for $u_t-\Delta u=g(u)$ revisited, Adv. Differential Equations, 1 (1996), 73-90.
[5]	J. I. Díaz and J. M. Rakotoson, On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary, J. Funct. Anal., 257 (2009), 807-831. doi: 10.1016/j.jfa.2009.03.002.
[6]	Jesus Idelfonso Díaz and Jean Michel Rakotoson, On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary, Discrete Contin. Dyn. Syst., 27 (2010), 1037-1058. doi: 10.3934/dcds.2010.27.1037.
[7]	Françoise Demengel and Gilbert Demengel, "Espaces Fonctionnels," (French) [Functional spaces] Utilisation dans la résolution des équations aux dérivées partielles. [Application to the solution of partial differential equations] Savoirs Actuels (Les Ulis). [Current Scholarship (Les Ulis)] EDP Sciences, Les Ulis; CNRS Éditions, Paris, 2007. xii+467 pp.
[8]	David Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp.
[9]	J. M. Rakotoson, A few natural extension of the regularity of a very weak solution, Differential and Integral Equations, 24 (2011), 1125-1140.
[10]	Jean-Michel Rakotoson, "Réarrangement Relatif," (French. French summary) [Relative rearrangement] Un instrument d'estimations dans les problèmes aux limites. [An estimation tool for limit problems] Mathématiques & Applications (Berlin) [Mathematics & Applications], 64. Springer, Berlin, 2008. xvi+293 pp.
[11]	Jean-Émile Rakotoson and Jean-Michel Rakotoson, "Analyse Fonctionnelle Appliquée aux Équations aux Dérivées Partielles," [Functional analysis applied to partial differential equations] Mathématiques. [Mathematics] Presses Universitaires de France, Paris, 1999. 232 pp.

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