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Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation
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 |
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. |
show all references
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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