February 2019, 39(2): 1135-1155. doi: 10.3934/dcds.2019048

Extremal functions for an embedding from some anisotropic space, involving the "one Laplacian"

UMR 8088, University of Cergy Pontoise, 2 avenue Adolphe Chauvain, Cergy, France

* Corresponding author: Françoise Demengel

Received  April 2018 Revised  July 2018 Published  November 2018

In this paper, we prove the existence of extremal functions for the best constant of embedding from anisotropic space, allowing some of the Sobolev exponents to be equal to $1$. We prove also that the extremal functions satisfy a partial differential equation involving the $1$ Laplacian.

Citation: Françoise Demengel, Thomas Dumas. Extremal functions for an embedding from some anisotropic space, involving the "one Laplacian". Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1135-1155. doi: 10.3934/dcds.2019048
References:
[1]

A. AlvinoV. FeroneG. Trombetti and P.-L. Lions, Convex symmetrization and applications, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 14 (1997), 275-293. doi: 10.1016/S0294-1449(97)80147-3.

[2]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318. doi: 10.1007/BF01781073.

[3]

T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry, 11 (1976), 573-598.

[4]

L. BoccardoP. Marcellini and C. Sbordone, $L^∞$ -regularity for variational problems with sharp nonstandard growth conditions, Boll. Un. Mat. Ital. A, 4 (1990), 219-225.

[5]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.

[6]

D. Cordero-ErausquinB. Nazaret and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Advances in Mathematics, 182 (2004), 307-332. doi: 10.1016/S0001-8708(03)00080-X.

[7]

G. CupiniP. Marcellini and E. Mascolo, Regularity under sharp anisotropic general growth conditions, Discrete and Continuous Dynamical Systems, 11 (2009), 67-86. doi: 10.3934/dcdsb.2009.11.67.

[8]

F. Demengel, On some nonlinear partial differential equations involving the "1"-Laplacian and critical Sobolev exponent, ESAIM: Control, Optimisation and Calculus of Variations, 4 (1999), 667-686. doi: 10.1051/cocv:1999126.

[9]

F. Demengel, Some Existence's results for non coercive "$1$-Laplacian" operator, Asymptotic Analysis, 43 (2005), 287-322.

[10]

F. Demengel, Functions locally $1$-harmonic, Applicable Analysis, 83 (2004), 865-896. doi: 10.1080/00036810310001621369.

[11]

F. Demengel and R. Temam, Functions of a measure and its applications, Indiana Math. Journal, 33 (1984), 673-709. doi: 10.1512/iumj.1984.33.33036.

[12]

J. Dieudonné, Eléments D'analyse, 2, Gauthiers Villars 1968.

[13]

T. Dumas, Existence de Solutions Pour des Équations Apparentées au 1 Laplacien Anisotrope, Thèse d'Université, Université de Cergy, 2018.

[14]

L. EspositoF. Leonetti and G. Mingione, Higher integrability for minimizers of integral functionals with $(p, q)$ growth, J. Differential Equations, 157 (1999), 414-438. doi: 10.1006/jdeq.1998.3614.

[15]

L. EspositoF. Leonetti and G. Mingione, Sharp regularity for functionals with $(p, q)$ growth, J. Differential Equations, 204 (2004), 5-55. doi: 10.1016/j.jde.2003.11.007.

[16]

I. FragalaF. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equation, Ann. I. H. Poincaré Anal., 21 (2004), 715-734. doi: 10.1016/j.anihpc.2003.12.001.

[17]

N. Fusco and C. Sbordone, Local boundedness of minimizers in a limit case, Manuscripta Math., 69 (1990), 19-25. doi: 10.1007/BF02567909.

[18]

M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math., 59 (1987), 245-248. doi: 10.1007/BF01158049.

[19]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhauser, 1984. doi: 10.1007/978-1-4684-9486-0.

[20]

A. El Hamidi and J. M. Rakotoson, Compactness and quasilinear problems with critical exponents, Differential Integral Equations, 18 (2005), 1201-1220.

[21]

A. El Hamidi and J. M. Rakotoson, Extremal functions for the anisotropic Sobolev inequalities, Ann. I.H. Poincaré, Analyse non Linéaire, 24 (2007), 741-756. doi: 10.1016/j.anihpc.2006.06.003.

[22]

S. N. Kruzhkov and I. M. Kolodii, On the theory of embedding of anisotropic Sobolev spaces, Uspekhi Mat. Nauk, 38 (1983), 207-208.

[23]

S. N. Kruzhkov and A. G. Korolev, On the imbedding theory of anisotropic function spaces, (Russian), Dokl. Akad. Nauk SSSR, 285 (1985), 1054-1057.

[24]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ., and Ⅱ, Ann. Inst. H. Poincar Anal. Non Lin ire 1, 1 (1985), 223-283.

[25]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoamericana 1, 1 (1985), 145-201. doi: 10.4171/RMI/6.

[26]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 2, Rev. Mat. Iberoamericana 1, 1 (1985), 45-121. doi: 10.4171/RMI/12.

[27]

P. Marcellini, Regularity of minimizers of integrals in the calculus of variations with non standard growth conditions, Arch. Rational Mech. Anal., 105 (1989), 267-284. doi: 10.1007/BF00251503.

[28]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p- q$ growth conditions, J. Differential Equations, 90 (1991), 1-30. doi: 10.1016/0022-0396(91)90158-6.

[29]

A. MercaldoJ. D. RossiS. Segura de León and C. Trombetti, Anisotropic p, q-Laplacian equations when p goes to 1, Nonlinear Analysis, 73 (2010), 3546-3560. doi: 10.1016/j.na.2010.07.030.

[30]

S. M. Nikolskii, On imbedding, continuation and approximation theorems for differentiable functions of several variables, Uspehi Mat. Nauk., 6 (1961), 63-114.

[31]

G. Strang and R. Temam, Duality and relaxation in the variational problems of plasticity, J. Mécanique, 19 (1980), 493-527.

[32]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.

[33]

R. Temam, Mathematical Problems in Plasticity, Gauthiers Villars, 1983.

[34]

M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3-24.

[35]

J. Vetois, Decay estimates and a vanishing phenomenon for the solutions of critical anisotropic equations, Adv. Math., 284 (2015), 122-158. doi: 10.1016/j.aim.2015.04.029.

show all references

References:
[1]

A. AlvinoV. FeroneG. Trombetti and P.-L. Lions, Convex symmetrization and applications, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 14 (1997), 275-293. doi: 10.1016/S0294-1449(97)80147-3.

[2]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318. doi: 10.1007/BF01781073.

[3]

T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry, 11 (1976), 573-598.

[4]

L. BoccardoP. Marcellini and C. Sbordone, $L^∞$ -regularity for variational problems with sharp nonstandard growth conditions, Boll. Un. Mat. Ital. A, 4 (1990), 219-225.

[5]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.

[6]

D. Cordero-ErausquinB. Nazaret and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Advances in Mathematics, 182 (2004), 307-332. doi: 10.1016/S0001-8708(03)00080-X.

[7]

G. CupiniP. Marcellini and E. Mascolo, Regularity under sharp anisotropic general growth conditions, Discrete and Continuous Dynamical Systems, 11 (2009), 67-86. doi: 10.3934/dcdsb.2009.11.67.

[8]

F. Demengel, On some nonlinear partial differential equations involving the "1"-Laplacian and critical Sobolev exponent, ESAIM: Control, Optimisation and Calculus of Variations, 4 (1999), 667-686. doi: 10.1051/cocv:1999126.

[9]

F. Demengel, Some Existence's results for non coercive "$1$-Laplacian" operator, Asymptotic Analysis, 43 (2005), 287-322.

[10]

F. Demengel, Functions locally $1$-harmonic, Applicable Analysis, 83 (2004), 865-896. doi: 10.1080/00036810310001621369.

[11]

F. Demengel and R. Temam, Functions of a measure and its applications, Indiana Math. Journal, 33 (1984), 673-709. doi: 10.1512/iumj.1984.33.33036.

[12]

J. Dieudonné, Eléments D'analyse, 2, Gauthiers Villars 1968.

[13]

T. Dumas, Existence de Solutions Pour des Équations Apparentées au 1 Laplacien Anisotrope, Thèse d'Université, Université de Cergy, 2018.

[14]

L. EspositoF. Leonetti and G. Mingione, Higher integrability for minimizers of integral functionals with $(p, q)$ growth, J. Differential Equations, 157 (1999), 414-438. doi: 10.1006/jdeq.1998.3614.

[15]

L. EspositoF. Leonetti and G. Mingione, Sharp regularity for functionals with $(p, q)$ growth, J. Differential Equations, 204 (2004), 5-55. doi: 10.1016/j.jde.2003.11.007.

[16]

I. FragalaF. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equation, Ann. I. H. Poincaré Anal., 21 (2004), 715-734. doi: 10.1016/j.anihpc.2003.12.001.

[17]

N. Fusco and C. Sbordone, Local boundedness of minimizers in a limit case, Manuscripta Math., 69 (1990), 19-25. doi: 10.1007/BF02567909.

[18]

M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math., 59 (1987), 245-248. doi: 10.1007/BF01158049.

[19]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhauser, 1984. doi: 10.1007/978-1-4684-9486-0.

[20]

A. El Hamidi and J. M. Rakotoson, Compactness and quasilinear problems with critical exponents, Differential Integral Equations, 18 (2005), 1201-1220.

[21]

A. El Hamidi and J. M. Rakotoson, Extremal functions for the anisotropic Sobolev inequalities, Ann. I.H. Poincaré, Analyse non Linéaire, 24 (2007), 741-756. doi: 10.1016/j.anihpc.2006.06.003.

[22]

S. N. Kruzhkov and I. M. Kolodii, On the theory of embedding of anisotropic Sobolev spaces, Uspekhi Mat. Nauk, 38 (1983), 207-208.

[23]

S. N. Kruzhkov and A. G. Korolev, On the imbedding theory of anisotropic function spaces, (Russian), Dokl. Akad. Nauk SSSR, 285 (1985), 1054-1057.

[24]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ., and Ⅱ, Ann. Inst. H. Poincar Anal. Non Lin ire 1, 1 (1985), 223-283.

[25]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoamericana 1, 1 (1985), 145-201. doi: 10.4171/RMI/6.

[26]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 2, Rev. Mat. Iberoamericana 1, 1 (1985), 45-121. doi: 10.4171/RMI/12.

[27]

P. Marcellini, Regularity of minimizers of integrals in the calculus of variations with non standard growth conditions, Arch. Rational Mech. Anal., 105 (1989), 267-284. doi: 10.1007/BF00251503.

[28]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p- q$ growth conditions, J. Differential Equations, 90 (1991), 1-30. doi: 10.1016/0022-0396(91)90158-6.

[29]

A. MercaldoJ. D. RossiS. Segura de León and C. Trombetti, Anisotropic p, q-Laplacian equations when p goes to 1, Nonlinear Analysis, 73 (2010), 3546-3560. doi: 10.1016/j.na.2010.07.030.

[30]

S. M. Nikolskii, On imbedding, continuation and approximation theorems for differentiable functions of several variables, Uspehi Mat. Nauk., 6 (1961), 63-114.

[31]

G. Strang and R. Temam, Duality and relaxation in the variational problems of plasticity, J. Mécanique, 19 (1980), 493-527.

[32]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.

[33]

R. Temam, Mathematical Problems in Plasticity, Gauthiers Villars, 1983.

[34]

M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3-24.

[35]

J. Vetois, Decay estimates and a vanishing phenomenon for the solutions of critical anisotropic equations, Adv. Math., 284 (2015), 122-158. doi: 10.1016/j.aim.2015.04.029.

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