American Institute of Mathematical Sciences

Advanced
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

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 & Continuous Dynamical Systems - A, 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 (1988).   Google Scholar

[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.  doi: 10.4171/RMI/281.  Google Scholar

[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, (1983).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jfa.2009.03.002.  Google Scholar

[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.  doi: 10.3934/dcds.2010.27.1037.  Google Scholar

[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, (2007).   Google Scholar

[8]

David Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, (1998).   Google Scholar

[9]

J. M. Rakotoson, A few natural extension of the regularity of a very weak solution,, Differential and Integral Equations, 24 (2011), 1125.   Google Scholar

[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], (2008).   Google Scholar

[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, (1999).   Google Scholar

show all references

References:
[1]

Colin Bennett and Robert Sharpley, "Interpolation of Operators,", Pure and Applied Mathematics, 129 (1988).   Google Scholar

[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.  doi: 10.4171/RMI/281.  Google Scholar

[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, (1983).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jfa.2009.03.002.  Google Scholar

[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.  doi: 10.3934/dcds.2010.27.1037.  Google Scholar

[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, (2007).   Google Scholar

[8]

David Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, (1998).   Google Scholar

[9]

J. M. Rakotoson, A few natural extension of the regularity of a very weak solution,, Differential and Integral Equations, 24 (2011), 1125.   Google Scholar

[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], (2008).   Google Scholar

[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, (1999).   Google Scholar

[1]

Jesus Idelfonso Díaz, 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 & Continuous Dynamical Systems - A, 2010, 27 (3) : 1037-1058. doi: 10.3934/dcds.2010.27.1037
[2]

Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
[3]

Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure & Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465
[4]

Elder Jesús Villamizar-Roa, Henry Lamos-Díaz, Gilberto Arenas-Díaz. Very weak solutions for the magnetohydrodynamic type equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 957-972. doi: 10.3934/dcdsb.2008.10.957
[5]

Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521
[6]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
[7]

Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271
[8]

Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[9]

Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621
[10]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[11]

Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631
[12]

Verena Bögelein, Frank Duzaar, Ugo Gianazza. Very weak solutions of singular porous medium equations with measure data. Communications on Pure & Applied Analysis, 2015, 14 (1) : 23-49. doi: 10.3934/cpaa.2015.14.23
[13]

Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661
[14]

Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115
[15]

Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027
[16]

Evgeny Galakhov, Olga Salieva. Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets. Conference Publications, 2015, 2015 (special) : 489-494. doi: 10.3934/proc.2015.0489
[17]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[18]

Nicola Abatangelo. Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5555-5607. doi: 10.3934/dcds.2015.35.5555
[19]

Mingzhu Wu, Zuodong Yang. Existence of boundary blow-up solutions for a class of quasiliner elliptic systems for the subcritical case. Communications on Pure & Applied Analysis, 2007, 6 (2) : 531-540. doi: 10.3934/cpaa.2007.6.531
[20]

Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences