American Institute of Mathematical Sciences

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

 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 $Ω$.
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] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [2] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [3] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267 [4] Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107 [5] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [6] Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 [7] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [8] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [9] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [10] Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331 [11] Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 [12] Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266 [13] Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052 [14] Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268 [15] Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115 [16] Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117 [17] Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101 [18] Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $\Lambda$-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328 [19] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [20] Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108

2019 Impact Factor: 1.338