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Mihai Mihăilescu; Vicenţiu Rădulescu
Neumann problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces
(Problèmes de Neumann associés aux opérateurs différentiels non homogènes dans les espaces d’Orlicz–Sobolev)
Annales de l'institut Fourier, 58 no. 6 (2008), p. 2087-2111, doi: 10.5802/aif.2407
Article: subscription required (your ip address: 54.242.188.217) | Reviews MR 2473630 | Zbl pre05367570
Class. Math.: 35D05, 35J60, 35J70, 58E05, 68T40, 76A02
Keywords: Nonhomogeneous differential operator, nonlinear partial differential equation, Neumann boundary value problem, Orlicz–Sobolev space
[1] Emilio Acerbi & Giuseppe Mingione, “Regularity results for a class of functionals with non-standard growth”, Arch. Ration. Mech. Anal. 156 (2001) no. 2, p. 121-140
Article | MR 1814973 | Zbl 0984.49020
[2] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975, Pure and Applied Mathematics, Vol. 65 MR 450957 | Zbl 0314.46030
[3] Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, 1983, Théorie et applications. [Theory and applications] Zbl 0511.46001
[4] Yunmei Chen, Stacey Levine & Murali Rao, “Variable exponent, linear growth functionals in image restoration”, SIAM J. Appl. Math. 66 (2006) no. 4, p. 1383-1406 (electronic)
Article | MR 2246061 | Zbl 1102.49010
[5] Ph. Clément, M. García-Huidobro, R. Manásevich & K. Schmitt, “Mountain pass type solutions for quasilinear elliptic equations”, Calc. Var. Partial Differential Equations 11 (2000) no. 1, p. 33-62
Article | MR 1777463 | Zbl 0959.35057
[6] Philippe Clément, Ben de Pagter, Guido Sweers & François de Thélin, “Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces”, Mediterr. J. Math. 1 (2004) no. 3, p. 241-267
Article | MR 2094464 | Zbl pre02216926
[7] Gabriele Dankert, Sobolev Embedding Theorems in Orlicz Spaces, thesis, University of Köln, 1966
[8] Lars Diening, Theorical and numerical results for electrorheological fluids, thesis, University of Freiburg, 2002 Zbl 1022.76001
[9] Lars Diening, “Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces”, Bull. Sci. Math. 129 (2005) no. 8, p. 657-700
Article | MR 2166733 | Zbl 1096.46013
[10] Thomas K. Donaldson & Neil S. Trudinger, “Orlicz-Sobolev spaces and imbedding theorems”, J. Functional Analysis 8 (1971), p. 52-75
Article | MR 301500 | Zbl 0216.15702
[11] D. E. Edmunds, J. Lang & A. Nekvinda, “On $L^{p(x)}$ norms”, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999) no. 1981, p. 219-225
Article | MR 1700499 | Zbl 0953.46018
[12] David E. Edmunds & Jiří Rákosník, “Density of smooth functions in $W^{k,p(x)}(\Omega )$”, Proc. Roy. Soc. London Ser. A 437 (1992) no. 1899, p. 229-236
Article | MR 1177754 | Zbl 0779.46027
[13] David E. Edmunds & Jiří Rákosník, “Sobolev embeddings with variable exponent”, Studia Math. 143 (2000) no. 3, p. 267-293
Article | MR 1815935 | Zbl 0974.46040
[14] I. Ekeland, “On the variational principle”, J. Math. Anal. Appl. 47 (1974), p. 324-353
Article | MR 346619 | Zbl 0286.49015
[15] Xianling Fan, Jishen Shen & Dun Zhao, “Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega )$”, J. Math. Anal. Appl. 262 (2001) no. 2, p. 749-760
Article | MR 1859337 | Zbl 0995.46023
[16] Xianling Fan & Dun Zhao, “On the spaces $L^{p(x)}(\Omega )$ and $W^{m,p(x)}(\Omega )$”, J. Math. Anal. Appl. 263 (2001) no. 2, p. 424-446
Article | MR 1866056 | Zbl 1028.46041
[17] Thomas C. Halsey, “Electrorheological Fluids”, Science 258 (1992) no. 5083, p. 761 -766
Article
[18] Ondrej Kováčik & Jiří Rákosník, “On spaces $L^{p(x)}$ and $W^{k,p(x)}$”, Czechoslovak Math. J. 41(116) (1991) no. 4, p. 592-618
Article | MR 1134951 | Zbl 0784.46029
[19] John Lamperti, “On the isometries of certain function-spaces”, Pacific J. Math. 8 (1958), p. 459-466
Article | MR 105017 | Zbl 0085.09702
[20] Paolo Marcellini, “Regularity and existence of solutions of elliptic equations with $p,q$-growth conditions”, J. Differential Equations 90 (1991) no. 1, p. 1-30
Article | MR 1094446 | Zbl 0724.35043
[21] Mihai Mihăilescu, Patrizia Pucci & Vicenţiu Rădulescu, “Nonhomogeneous boundary value problems in anisotropic Sobolev spaces”, C. R. Math. Acad. Sci. Paris 345 (2007) no. 10, p. 561-566 MR 2374465 | Zbl 1127.35020
[22] Mihai Mihăilescu & Vicenţiu Rădulescu, “A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006) no. 2073, p. 2625-2641
Article | MR 2253555 | Zbl pre05278124
[23] Mihai Mihăilescu & Vicenţiu Rădulescu, “Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting”, J. Math. Anal. Appl. 330 (2007) no. 1, p. 416-432
Article | MR 2302933 | Zbl pre05141592
[24] Mihai Mihăilescu & Vicenţiu Rădulescu, “On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent”, Proc. Amer. Math. Soc. 135 (2007) no. 9, p. 2929-2937 (electronic)
Article | MR 2317971 | Zbl pre05165473
[25] Mihai Mihăilescu & Vicenţiu Rădulescu, “Continuous spectrum for a class of nonhomogeneous differential operators”, Manuscripta Math. 125 (2008) no. 2, p. 157-167
Article | MR 2373080 | Zbl 1138.35070
[26] J. Musielak & W. Orlicz, “On modular spaces”, Studia Math. 18 (1959), p. 49-65
Article | MR 101487 | Zbl 0086.08901
[27] Julian Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics 1034, Springer-Verlag, 1983 MR 724434 | Zbl 0557.46020
[28] Hidegorô Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., 1950 MR 38565 | Zbl 0041.23401
[29] R. O’Neill, “Fractional integration in Orlicz spaces”, Trans. Amer. Math. Soc. 115 (1965), p. 300-328
Article | Zbl 0132.09201
[30] Władysław Orlicz, “Über konjugierte Exponentenfolgen”, Studia Math. 3 (1931), p. 200-211
Article | Zbl 0003.25203
[31] K. R. Rajagopal & M. Růžička, “Mathematical modelling of electrorheological fluids”, Cont. Mech. Term. 13 (2001), p. 59-78
Article | Zbl 0971.76100
[32] M. Růžička, “Electrorheological fluids: modeling and mathematical theory”, Sūrikaisekikenkyūsho Kōkyūroku (2000) no. 1146, p. 16-38, Mathematical analysis of liquids and gases (Japanese) (Kyoto, 1999) MR 1788852 | Zbl 0968.76531
[33] Michael Struwe, Variational methods, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 34, Springer-Verlag, 1996, Applications to nonlinear partial differential equations and Hamiltonian systems MR 1411681 | Zbl 0864.49001
[34] V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory”, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986) no. 4, p. 675-710, 877 MR 864171 | Zbl 0599.49031
[35] V. V. Zhikov, “Meyer-type estimates for solving the nonlinear Stokes system”, Differ. Uravn. 33 (1997) no. 1, p. 107-114, 143 MR 1607245 | Zbl 0911.35089
Mihai Mihăilescu; Vicenţiu Rădulescu
Neumann problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces
(Problèmes de Neumann associés aux opérateurs différentiels non homogènes dans les espaces d’Orlicz–Sobolev)
Annales de l'institut Fourier, 58 no. 6 (2008), p. 2087-2111, doi: 10.5802/aif.2407
Article: subscription required (your ip address: 54.242.188.217) | Reviews MR 2473630 | Zbl pre05367570
Class. Math.: 35D05, 35J60, 35J70, 58E05, 68T40, 76A02
Keywords: Nonhomogeneous differential operator, nonlinear partial differential equation, Neumann boundary value problem, Orlicz–Sobolev space
Résumé - Abstract
We study a nonlinear Neumann boundary value problem associated to a nonhomogeneous differential operator. Taking into account the competition between the nonlinearity and the bifurcation parameter, we establish sufficient conditions for the existence of nontrivial solutions in a related Orlicz–Sobolev space.
Bibliography
Article | MR 1814973 | Zbl 0984.49020
[2] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975, Pure and Applied Mathematics, Vol. 65 MR 450957 | Zbl 0314.46030
[3] Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, 1983, Théorie et applications. [Theory and applications] Zbl 0511.46001
[4] Yunmei Chen, Stacey Levine & Murali Rao, “Variable exponent, linear growth functionals in image restoration”, SIAM J. Appl. Math. 66 (2006) no. 4, p. 1383-1406 (electronic)
Article | MR 2246061 | Zbl 1102.49010
[5] Ph. Clément, M. García-Huidobro, R. Manásevich & K. Schmitt, “Mountain pass type solutions for quasilinear elliptic equations”, Calc. Var. Partial Differential Equations 11 (2000) no. 1, p. 33-62
Article | MR 1777463 | Zbl 0959.35057
[6] Philippe Clément, Ben de Pagter, Guido Sweers & François de Thélin, “Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces”, Mediterr. J. Math. 1 (2004) no. 3, p. 241-267
Article | MR 2094464 | Zbl pre02216926
[7] Gabriele Dankert, Sobolev Embedding Theorems in Orlicz Spaces, thesis, University of Köln, 1966
[8] Lars Diening, Theorical and numerical results for electrorheological fluids, thesis, University of Freiburg, 2002 Zbl 1022.76001
[9] Lars Diening, “Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces”, Bull. Sci. Math. 129 (2005) no. 8, p. 657-700
Article | MR 2166733 | Zbl 1096.46013
[10] Thomas K. Donaldson & Neil S. Trudinger, “Orlicz-Sobolev spaces and imbedding theorems”, J. Functional Analysis 8 (1971), p. 52-75
Article | MR 301500 | Zbl 0216.15702
[11] D. E. Edmunds, J. Lang & A. Nekvinda, “On $L^{p(x)}$ norms”, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999) no. 1981, p. 219-225
Article | MR 1700499 | Zbl 0953.46018
[12] David E. Edmunds & Jiří Rákosník, “Density of smooth functions in $W^{k,p(x)}(\Omega )$”, Proc. Roy. Soc. London Ser. A 437 (1992) no. 1899, p. 229-236
Article | MR 1177754 | Zbl 0779.46027
[13] David E. Edmunds & Jiří Rákosník, “Sobolev embeddings with variable exponent”, Studia Math. 143 (2000) no. 3, p. 267-293
Article | MR 1815935 | Zbl 0974.46040
[14] I. Ekeland, “On the variational principle”, J. Math. Anal. Appl. 47 (1974), p. 324-353
Article | MR 346619 | Zbl 0286.49015
[15] Xianling Fan, Jishen Shen & Dun Zhao, “Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega )$”, J. Math. Anal. Appl. 262 (2001) no. 2, p. 749-760
Article | MR 1859337 | Zbl 0995.46023
[16] Xianling Fan & Dun Zhao, “On the spaces $L^{p(x)}(\Omega )$ and $W^{m,p(x)}(\Omega )$”, J. Math. Anal. Appl. 263 (2001) no. 2, p. 424-446
Article | MR 1866056 | Zbl 1028.46041
[17] Thomas C. Halsey, “Electrorheological Fluids”, Science 258 (1992) no. 5083, p. 761 -766
Article
[18] Ondrej Kováčik & Jiří Rákosník, “On spaces $L^{p(x)}$ and $W^{k,p(x)}$”, Czechoslovak Math. J. 41(116) (1991) no. 4, p. 592-618
Article | MR 1134951 | Zbl 0784.46029
[19] John Lamperti, “On the isometries of certain function-spaces”, Pacific J. Math. 8 (1958), p. 459-466
Article | MR 105017 | Zbl 0085.09702
[20] Paolo Marcellini, “Regularity and existence of solutions of elliptic equations with $p,q$-growth conditions”, J. Differential Equations 90 (1991) no. 1, p. 1-30
Article | MR 1094446 | Zbl 0724.35043
[21] Mihai Mihăilescu, Patrizia Pucci & Vicenţiu Rădulescu, “Nonhomogeneous boundary value problems in anisotropic Sobolev spaces”, C. R. Math. Acad. Sci. Paris 345 (2007) no. 10, p. 561-566 MR 2374465 | Zbl 1127.35020
[22] Mihai Mihăilescu & Vicenţiu Rădulescu, “A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006) no. 2073, p. 2625-2641
Article | MR 2253555 | Zbl pre05278124
[23] Mihai Mihăilescu & Vicenţiu Rădulescu, “Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting”, J. Math. Anal. Appl. 330 (2007) no. 1, p. 416-432
Article | MR 2302933 | Zbl pre05141592
[24] Mihai Mihăilescu & Vicenţiu Rădulescu, “On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent”, Proc. Amer. Math. Soc. 135 (2007) no. 9, p. 2929-2937 (electronic)
Article | MR 2317971 | Zbl pre05165473
[25] Mihai Mihăilescu & Vicenţiu Rădulescu, “Continuous spectrum for a class of nonhomogeneous differential operators”, Manuscripta Math. 125 (2008) no. 2, p. 157-167
Article | MR 2373080 | Zbl 1138.35070
[26] J. Musielak & W. Orlicz, “On modular spaces”, Studia Math. 18 (1959), p. 49-65
Article | MR 101487 | Zbl 0086.08901
[27] Julian Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics 1034, Springer-Verlag, 1983 MR 724434 | Zbl 0557.46020
[28] Hidegorô Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., 1950 MR 38565 | Zbl 0041.23401
[29] R. O’Neill, “Fractional integration in Orlicz spaces”, Trans. Amer. Math. Soc. 115 (1965), p. 300-328
Article | Zbl 0132.09201
[30] Władysław Orlicz, “Über konjugierte Exponentenfolgen”, Studia Math. 3 (1931), p. 200-211
Article | Zbl 0003.25203
[31] K. R. Rajagopal & M. Růžička, “Mathematical modelling of electrorheological fluids”, Cont. Mech. Term. 13 (2001), p. 59-78
Article | Zbl 0971.76100
[32] M. Růžička, “Electrorheological fluids: modeling and mathematical theory”, Sūrikaisekikenkyūsho Kōkyūroku (2000) no. 1146, p. 16-38, Mathematical analysis of liquids and gases (Japanese) (Kyoto, 1999) MR 1788852 | Zbl 0968.76531
[33] Michael Struwe, Variational methods, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 34, Springer-Verlag, 1996, Applications to nonlinear partial differential equations and Hamiltonian systems MR 1411681 | Zbl 0864.49001
[34] V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory”, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986) no. 4, p. 675-710, 877 MR 864171 | Zbl 0599.49031
[35] V. V. Zhikov, “Meyer-type estimates for solving the nonlinear Stokes system”, Differ. Uravn. 33 (1997) no. 1, p. 107-114, 143 MR 1607245 | Zbl 0911.35089
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310