The strong minimum principle for quasisuperminimizers of non-standard growth
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 5, pp. 731-742.

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.

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.

DOI : https://doi.org/10.1016/j.anihpc.2011.06.001
Classification : 49N60,  35B50,  35J60
Mots clés : 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},
     pages = {731--742},
     publisher = {Elsevier},
     volume = {28},
     number = {5},
     year = {2011},
     doi = {10.1016/j.anihpc.2011.06.001},
     zbl = {1251.49028},
     mrnumber = {2838399},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.06.001/}
}
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, Tome 28 (2011) no. 5, pp. 731-742. doi : 10.1016/j.anihpc.2011.06.001. http://www.numdam.org/articles/10.1016/j.anihpc.2011.06.001/

[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 | EuDML 78076 | 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 | EuDML 78872 | 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) | EuDML 55377 | 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 | EuDML 13956 | 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 | EuDML 78948 | 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