The strong minimum principle for quasisuperminimizers of non-standard growth

Harjulehto, P.; Hästö, P.; Latvala, V.; Toivanen, O.

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.

[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 | Zbl

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

[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 | Zbl

[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 | Numdam | MR | Zbl

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

[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 | Zbl

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

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

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

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

[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 | Numdam | MR | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

[17] O. Kováčik, J. Rákosník, On spaces $L^{p\left(x\right)}$ and $W^{1,p\left(x\right)}$, Czechoslovak Math. J. 41 no. 116 (1991), 592-618 | EuDML | MR | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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