The strong minimum principle for quasisuperminimizers of non-standard growth
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 5, p. 731-742
We prove the strong minimum principle for non-negative quasisuperminimizers of the variable exponent Dirichlet energy integral under the assumption that the exponent has modulus of continuity slightly more general than Lipschitz. The proof is based on a new version of the weak Harnack estimate.
Nous prouvons le fort principe du minimum pour des quasisuperminimizeurs non-négatifs de problème de Dirichlet de lʼexposant variable en supposant que lʼexposant a le module de continuité un peu plus général que Lipschitz. La démonstration est fondée sur une nouvelle version de la faible inégalité de Harnack.
DOI : https://doi.org/10.1016/j.anihpc.2011.06.001
Classification:  49N60,  35B50,  35J60
Keywords: Non-standard growth, Variable exponent, Dirichlet energy, Maximum principle, Minimum principle, Weak Harnack inequality, De Giorgi method
@article{AIHPC_2011__28_5_731_0,
     author = {Harjulehto, P. and H\"ast\"o, P. and Latvala, V. and Toivanen, O.},
     title = {The strong minimum principle for quasisuperminimizers of non-standard growth},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {28},
     number = {5},
     year = {2011},
     pages = {731-742},
     doi = {10.1016/j.anihpc.2011.06.001},
     zbl = {1251.49028},
     mrnumber = {2838399},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2011__28_5_731_0}
}
Harjulehto, P.; Hästö, P.; Latvala, V.; Toivanen, O. The strong minimum principle for quasisuperminimizers of non-standard growth. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 5, pp. 731-742. doi : 10.1016/j.anihpc.2011.06.001. http://www.numdam.org/item/AIHPC_2011__28_5_731_0/

[1] E. Acerbi, G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal. 156 no. 2 (2001), 121-140 | MR 1814973 | Zbl 0984.49020

[2] T. Adamowicz, P. Hästö, Mappings of finite distortion and p(·)-harmonic functions, Int. Math. Res. Not. IMRN (2010), 1940-1965 | MR 2646346 | Zbl 1206.35134

[3] Yu. Alkhutov, The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition, Differ. Equ. 33 no. 12 (1997), 1653-1663 | MR 1669915 | Zbl 0949.35048

[4] E. Dibenedetto, N.S. Trudinger, Harnack inequalities for quasiminima of variational integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 295-308 | Numdam | MR 778976 | Zbl 0565.35012

[5] L. Diening, Maximal function on generalized Lebesgue spaces L p(·) , Math. Inequal. Appl. 7 no. 2 (2004), 245-253 | MR 2057643 | Zbl 1071.42014

[6] L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. vol. 2017, Springer-Verlag, Berlin (2011) | MR 2790542 | Zbl 1222.46002

[7] X.-L. Fan, Global C 1,α regularity for variable exponent elliptic equations in divergence form, J. Differential Equations 235 no. 2 (2007), 397-417 | Zbl 1143.35040

[8] X.-L. Fan, D. Zhao, The quasi-minimizer of integral functionals with m(x) growth conditions, Nonlinear Anal. 39 (2001), 807-816 | MR 1736389 | Zbl 0943.49029

[9] X.-L. Fan, Y.Z. Zhao, Q.-H. Zhang, A strong maximum principle for p(x)-Laplace equations, Chinese J. Contemp. Math. 24 (2003), 277-282 | MR 2016638

[10] R. Fortini, D. Mugnai, P. Pucci, Maximum principles for anisotropic elliptic inequalities, Nonlinear Anal. 70 no. 8 (2009), 2917-2929 | MR 2509379 | Zbl 1169.35314

[11] J. García-Melián, J.D. Rossi, J.C. Sabina De Lis, Large solutions for the Laplacian with a power nonlinearity given by a variable exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 3 (2009), 889-902 | Numdam | MR 2526407 | Zbl 1177.35072

[12] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore (2003) | MR 1962933 | Zbl 1028.49001

[13] P. Harjulehto, P. Hästö, U. Lê, M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 no. 12 (2010), 4551-4574 | MR 2639204 | Zbl 1188.35072

[14] P. Harjulehto, J. Kinnunen, T. Lukkari, Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. Value Probl. (2007) | MR 2291928 | Zbl 1161.35020

[15] P. Harjulehto, T. Kuusi, T. Lukkari, N. Marola, M. Parviainen, Harnackʼs inequality for quasiminimizers with non-standard growth conditions, J. Math. Anal. Appl. 344 no. 1 (2008), 504-520 | MR 2416324 | Zbl 1145.49023

[16] N. Kôno, On generalized Takagi functions, Acta Math. Hungar. 49 no. 3–4 (1987), 315-324 | MR 891041 | Zbl 0627.26004

[17] O. Kováčik, J. Rákosník, On spaces L p(x) and W 1,p(x) , Czechoslovak Math. J. 41 no. 116 (1991), 592-618 | MR 1134951 | Zbl 0784.46029

[18] N.V. Krylov, M.V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 161-175 | MR 563790 | Zbl 0464.35035

[19] V. Latvala, A theorem on fine connectedness, Potential Anal. 12 no. 1 (2000), 221-232 | MR 1752852 | Zbl 0952.31007

[20] F. Li, Z. Li, L. Pi, Variable exponent functionals in image restoration, Appl. Math. Comput. 216 (2010), 870-882 | MR 2606995 | Zbl 1186.94010

[21] T. Lukkari, F.-Y. Maeda, N. Marola, Wolff potential estimates for elliptic equations with nonstandard growth and applications, Forum Math. 22 no. 6 (2010), 1061-1087 | MR 2735887 | Zbl 1203.35099

[22] J.J. Manfredi, J.D. Rossi, J.M. Urbano, p(x)-harmonic function with unbounded exponent in a subdomain, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 6 (2009), 2581-2595 | Numdam | MR 2569909 | Zbl 1180.35242

[23] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. vol. 1748, Springer-Verlag, Berlin (2000) | MR 1810360 | Zbl 0968.76531

[24] M. Sanchón, J. Urbano, Entropy solutions for the p(x)-Laplace equation, Trans. Amer. Math. Soc. 361 no. 12 (2009), 6387-6405 | MR 2538597 | Zbl 1181.35121

[25] P. Wittbold, A. Zimmermann, Existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and L 1 -data, Nonlinear Anal. 72 no. 6 (2010), 2990-3008 | MR 2580154 | Zbl 1185.35088

[26] C. Zhang, S. Zhou, Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L 1 data, J. Differential Equations 248 no. 6 (2010), 1376-1400 | MR 2593046 | Zbl 1195.35097

[27] Q. Zhang, Y. Wang, Z. Qiu, Existence of solutions and boundary asymptotic behavior of p(r)-Laplacian equation multi-point boundary value problems, Nonlinear Anal. 72 no. 6 (2010), 2950-2973 | MR 2580151 | Zbl 1188.34021