January  2017, 37(1): 15-31. doi: 10.3934/dcds.2017002

Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes

1. 

Departamento de Matemáticas, Universidad de Almería, Ctra. Sacramento s/n, La Cañada de San Urbano 04120, Almería, Spain

2. 

Departamento de Matemática Aplicada y Estadística, Campus Alfonso XIII, Universidad Politécnica de Cartagena, 30203, Murcia, Spain

* Corresponding author

Received  March 2016 Revised  August 2016 Published  November 2016

Fund Project: Research supported by MINECO-FEDER grant MTM2015-68210-P and Junta de Andalucía FQM-194 (first author) and FQM-116 (second author). Programa de Apoyo a la Investigación de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia, reference 19461/PI/14 (second author).

In this paper we consider the homogenization problem for quasilinear elliptic equations with singularities in the gradient, whose model is the following
$\begin{equation*}\begin{cases}\displaystyle -Δ u^\varepsilon + \frac{|\nabla u^\varepsilon|^2}{{(u^\varepsilon})^θ} = f (x)& \mbox{in} \; Ω^\varepsilon,\\u^\varepsilon = 0&\mbox{on} \; \partial Ω^\varepsilon,\\\end{cases}\end{equation*}$
where Ω is an open bounded set of
$\mathbb{R}^N$
,
$θ ∈ (0,1)$
and
$f$
is positive function that belongs to a certain Lebesgue's space. The homogenization of these equations is posed in a sequence of domains
$Ω^\varepsilon$
obtained by removing many small holes from a fixed domain Ω. We also give a corrector result.
Citation: José Carmona, Pedro J. Martínez-Aparicio. Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 15-31. doi: 10.3934/dcds.2017002
References:
[1]

D. ArcoyaJ. CarmonaT. LeonoriP. J. Martínez-AparicioL. Orsina and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations, J. Differential Equations, 246 (2009), 4006-4042.  doi: 10.1016/j.jde.2009.01.016.

[2]

D. ArcoyaJ. Carmona and P. J. Martínez-Aparicio, Bifurcation for Quasilinear Elliptic Singular BVP, Comm. Partial Differential Equations, 36 (2011), 670-692.  doi: 10.1080/03605302.2010.501835.

[3]

D. Arcoya, J. Carmona and P. J. Martínez-Aparicio, Comparison principle for elliptic equations in divergence with singular lower order terms having natural growth Commun. Contemp. Math. to appear. doi: 10.1142/S0219199716500139.

[4]

D. Arcoya and P. J. Martínez-Aparicio, Quasilinear equations with natural growth, Rev. Mat. Iberoam., 24 (2008), 597-616.  doi: 10.4171/RMI/548.

[5]

D. Arcoya and S. Segura de León, Uniqueness of solutions for some elliptic equations with a quadratic gradient term, ESAIM Control Optim. Calc. Var., 16 (2010), 327-336.  doi: 10.1051/cocv:2008072.

[6]

L. Boccardo, Dirichlet problems with singular and quadratic gradient lower order terms, ESAIM Control Optim. Calc. Var., 14 (2008), 411-426.  doi: 10.1051/cocv:2008031.

[7]

L. Boccardo and J. Casado-Díaz, Some properties of solutions of some semilinear elliptic singular problems and applications to the G-convergence, Asymptot. Anal., 86 (2014), 1-15. 

[8]

J. CarmonaP. J. Martínez-Aparicio and A. Suárez, Existence and non-existence of positive solutions for nonlinear elliptic singular equations with natural growth, Nonlinear Anal., 89 (2013), 157-169.  doi: 10.1016/j.na.2013.05.015.

[9]

J. Casado-Díaz, Homogenization of general quasi-linear Dirichlet problems with quadratic growth in perforated domains, J. Math. Pures Appl., 76 (1997), 431-476.  doi: 10.1016/S0021-7824(97)89958-8.

[10]

J. Casado-Díaz, Homogenization of a quasi-linear problem with quadratic growth in perforated domains: An example, Ann. Inst. H. Poincaré Anal. non linéaire, 14 (1997), 669-686.  doi: 10.1016/S0294-1449(97)80129-1.

[11]

D. Cioranescu and F. Murat, Un terme étrange venu d'ailleurs, Ⅰ et Ⅱ', In Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. Ⅱ and Vol. Ⅲ, ed. by H. Brezis and J. -L. Lions. Research Notes in Math. 60 and 70, Pitman, London, (1982), 98-138 and 154-178. English translation: D. Cioranescu and F. Murat, A strange term coming from nowhere. In Topics in mathematical modeling of composite materials, ed. by A. Cherkaev and R. V. Kohn. Progress in Nonlinear Differential Equations and their Applications 31, Birkhäuser, Boston, (1997), 44-93.

[12]

G. Dal Maso and A. Garroni, New results of the asymptotic behaviour of Dirichlet problems in perforated domains, Math. Models Methods Appl. Sci., 4 (1994), 373-407.  doi: 10.1142/S0218202594000224.

[13]

G. Dal Maso and F. Murat, Asymptotic behavior and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 239-290. 

[14]

G. Dal Maso and F. Murat, Asymptotic behavior and correctors for linear Dirichlet problems with simultaneously varying operators and domains, Ann. Inst. H. Poincaré Anal. non linéaire, 21 (2004), 445-486.  doi: 10.1016/j.anihpc.2003.05.001.

[15]

P. Donato and D. Giachetti, Homogenization of some nonlinear elliptic problems, Int. J. Pure Appl. Math., 73 (2011), 349-378. 

[16]

D. Giachetti, P. J. Martínez-Aparicio and F. Murat, A semilinear elliptic equation with a mild singularity at u=0: Existence and homogenization J. Math. Pures Appl. to appear. doi: 10.1016/j.matpur.2016.04.007.

[17]

D. Giachetti, P. J. Martínez-Aparicio and F. Murat, Definition, existence, stability and uniqueness of the solution to a semi-linear elliptic problem with a strong singularity at u = 0, preprint.

[18]

D. Giachetti, P. J. Martínez-Aparicio and F. Murat, Homogenization of a Dirichlet semi-linear elliptic problem with a strong singularity at u = 0 in a domain with many small holes, preprint.

[19]

D. Giachetti and F. Murat, Elliptic problems with lower order terms having singular behaviour, Boll. Unione Mat. Ital., 2 (2009), 349-370. 

[20]

J. Leray and J. L. Lions, Quelques résultats de Višik sur les problémes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. 

[21]

V. A. Marčenko and E. Ya. Khruslov, Kraevye Zadachi v Oblastyakh s Melkozernistoi Granitsei, (Russian) Boundary value problems in domains with a fine-grained boundary, Izdat. "Naukova Dumka", Kiev, 1974.

[22]

G. Stampacchia, Èquations elliptiques du second ordre à coefficients discontinus, in Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965) Les Presses de l'Université de Montréal, Montreal, Que. , 1966.

show all references

References:
[1]

D. ArcoyaJ. CarmonaT. LeonoriP. J. Martínez-AparicioL. Orsina and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations, J. Differential Equations, 246 (2009), 4006-4042.  doi: 10.1016/j.jde.2009.01.016.

[2]

D. ArcoyaJ. Carmona and P. J. Martínez-Aparicio, Bifurcation for Quasilinear Elliptic Singular BVP, Comm. Partial Differential Equations, 36 (2011), 670-692.  doi: 10.1080/03605302.2010.501835.

[3]

D. Arcoya, J. Carmona and P. J. Martínez-Aparicio, Comparison principle for elliptic equations in divergence with singular lower order terms having natural growth Commun. Contemp. Math. to appear. doi: 10.1142/S0219199716500139.

[4]

D. Arcoya and P. J. Martínez-Aparicio, Quasilinear equations with natural growth, Rev. Mat. Iberoam., 24 (2008), 597-616.  doi: 10.4171/RMI/548.

[5]

D. Arcoya and S. Segura de León, Uniqueness of solutions for some elliptic equations with a quadratic gradient term, ESAIM Control Optim. Calc. Var., 16 (2010), 327-336.  doi: 10.1051/cocv:2008072.

[6]

L. Boccardo, Dirichlet problems with singular and quadratic gradient lower order terms, ESAIM Control Optim. Calc. Var., 14 (2008), 411-426.  doi: 10.1051/cocv:2008031.

[7]

L. Boccardo and J. Casado-Díaz, Some properties of solutions of some semilinear elliptic singular problems and applications to the G-convergence, Asymptot. Anal., 86 (2014), 1-15. 

[8]

J. CarmonaP. J. Martínez-Aparicio and A. Suárez, Existence and non-existence of positive solutions for nonlinear elliptic singular equations with natural growth, Nonlinear Anal., 89 (2013), 157-169.  doi: 10.1016/j.na.2013.05.015.

[9]

J. Casado-Díaz, Homogenization of general quasi-linear Dirichlet problems with quadratic growth in perforated domains, J. Math. Pures Appl., 76 (1997), 431-476.  doi: 10.1016/S0021-7824(97)89958-8.

[10]

J. Casado-Díaz, Homogenization of a quasi-linear problem with quadratic growth in perforated domains: An example, Ann. Inst. H. Poincaré Anal. non linéaire, 14 (1997), 669-686.  doi: 10.1016/S0294-1449(97)80129-1.

[11]

D. Cioranescu and F. Murat, Un terme étrange venu d'ailleurs, Ⅰ et Ⅱ', In Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. Ⅱ and Vol. Ⅲ, ed. by H. Brezis and J. -L. Lions. Research Notes in Math. 60 and 70, Pitman, London, (1982), 98-138 and 154-178. English translation: D. Cioranescu and F. Murat, A strange term coming from nowhere. In Topics in mathematical modeling of composite materials, ed. by A. Cherkaev and R. V. Kohn. Progress in Nonlinear Differential Equations and their Applications 31, Birkhäuser, Boston, (1997), 44-93.

[12]

G. Dal Maso and A. Garroni, New results of the asymptotic behaviour of Dirichlet problems in perforated domains, Math. Models Methods Appl. Sci., 4 (1994), 373-407.  doi: 10.1142/S0218202594000224.

[13]

G. Dal Maso and F. Murat, Asymptotic behavior and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 239-290. 

[14]

G. Dal Maso and F. Murat, Asymptotic behavior and correctors for linear Dirichlet problems with simultaneously varying operators and domains, Ann. Inst. H. Poincaré Anal. non linéaire, 21 (2004), 445-486.  doi: 10.1016/j.anihpc.2003.05.001.

[15]

P. Donato and D. Giachetti, Homogenization of some nonlinear elliptic problems, Int. J. Pure Appl. Math., 73 (2011), 349-378. 

[16]

D. Giachetti, P. J. Martínez-Aparicio and F. Murat, A semilinear elliptic equation with a mild singularity at u=0: Existence and homogenization J. Math. Pures Appl. to appear. doi: 10.1016/j.matpur.2016.04.007.

[17]

D. Giachetti, P. J. Martínez-Aparicio and F. Murat, Definition, existence, stability and uniqueness of the solution to a semi-linear elliptic problem with a strong singularity at u = 0, preprint.

[18]

D. Giachetti, P. J. Martínez-Aparicio and F. Murat, Homogenization of a Dirichlet semi-linear elliptic problem with a strong singularity at u = 0 in a domain with many small holes, preprint.

[19]

D. Giachetti and F. Murat, Elliptic problems with lower order terms having singular behaviour, Boll. Unione Mat. Ital., 2 (2009), 349-370. 

[20]

J. Leray and J. L. Lions, Quelques résultats de Višik sur les problémes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. 

[21]

V. A. Marčenko and E. Ya. Khruslov, Kraevye Zadachi v Oblastyakh s Melkozernistoi Granitsei, (Russian) Boundary value problems in domains with a fine-grained boundary, Izdat. "Naukova Dumka", Kiev, 1974.

[22]

G. Stampacchia, Èquations elliptiques du second ordre à coefficients discontinus, in Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965) Les Presses de l'Université de Montréal, Montreal, Que. , 1966.

[1]

Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128

[2]

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

[3]

Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747

[4]

Zhaoli Liu, Jiabao Su. Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 617-634. doi: 10.3934/dcds.2004.10.617

[5]

Shuangjie Peng. Remarks on singular critical growth elliptic equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 707-719. doi: 10.3934/dcds.2006.14.707

[6]

Daniela Giachetti, Francesco Petitta, Sergio Segura de León. Elliptic equations having a singular quadratic gradient term and a changing sign datum. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1875-1895. doi: 10.3934/cpaa.2012.11.1875

[7]

Minh-Phuong Tran, Thanh-Nhan Nguyen. Pointwise gradient bounds for a class of very singular quasilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4461-4476. doi: 10.3934/dcds.2021043

[8]

José M. Arrieta, Ariadne Nogueira, Marcone C. Pereira. Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4217-4246. doi: 10.3934/dcdsb.2019079

[9]

Olivier Guibé, Anna Mercaldo. Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms. Communications on Pure and Applied Analysis, 2008, 7 (1) : 163-192. doi: 10.3934/cpaa.2008.7.163

[10]

Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 933-982. doi: 10.3934/dcds.2020067

[11]

Sami Aouaoui. On some local-nonlocal elliptic equation involving nonlinear terms with exponential growth. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1767-1784. doi: 10.3934/cpaa.2017086

[12]

Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897

[13]

Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227

[14]

Shenzhou Zheng, Xueliang Zheng, Zhaosheng Feng. Optimal regularity for $A$-harmonic type equations under the natural growth. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 669-685. doi: 10.3934/dcdsb.2011.16.669

[15]

Y. Efendiev, B. Popov. On homogenization of nonlinear hyperbolic equations. Communications on Pure and Applied Analysis, 2005, 4 (2) : 295-309. doi: 10.3934/cpaa.2005.4.295

[16]

Andrzej Świȩch. Pointwise properties of $ L^p $-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2945-2962. doi: 10.3934/dcds.2020156

[17]

Massimiliano Berti, M. Matzeu, Enrico Valdinoci. On periodic elliptic equations with gradient dependence. Communications on Pure and Applied Analysis, 2008, 7 (3) : 601-615. doi: 10.3934/cpaa.2008.7.601

[18]

Yanjun Liu, Chungen Liu. Ground state solution and multiple solutions to elliptic equations with exponential growth and singular term. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2819-2838. doi: 10.3934/cpaa.2020123

[19]

Gregory A. Chechkin, Vladimir V. Chepyzhov, Leonid S. Pankratov. Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1133-1154. doi: 10.3934/dcdsb.2018145

[20]

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

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (189)
  • HTML views (70)
  • Cited by (1)

Other articles
by authors

[Back to Top]