Regularity estimates for quasilinear elliptic equations with variable growth involving measure data

Byun, Sun-Sig; Ok, Jihoon; Park, Jung-Tae

doi:10.1016/j.anihpc.2016.12.002

Byun, Sun-Sig ; Ok, Jihoon ; Park, Jung-Tae

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1639-1667.

Résumé

We investigate a quasilinear elliptic equation with variable growth in a bounded nonsmooth domain involving a signed Radon measure. We obtain an optimal global Calderón–Zygmund type estimate for such a measure data problem, by proving that the gradient of a very weak solution to the problem is as globally integrable as the first order maximal function of the associated measure, up to a correct power, under minimal regularity requirements on the nonlinearity, the variable exponent and the boundary of the domain.

MR Zbl

DOI : 10.1016/j.anihpc.2016.12.002

Classification : 35J92, 46F30, 42B37
Mots clés : Nonlinear elliptic equation, Measure data, Variable exponent, Calderón–Zygmund type estimate, Reifenberg flat domain

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     author = {Byun, Sun-Sig and Ok, Jihoon and Park, Jung-Tae},
     title = {Regularity estimates for quasilinear elliptic equations with variable growth involving measure data},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1639--1667},
     publisher = {Elsevier},
     volume = {34},
     number = {7},
     year = {2017},
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     zbl = {1374.35183},
     mrnumber = {3724751},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.12.002/}
}

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Byun, Sun-Sig; Ok, Jihoon; Park, Jung-Tae. Regularity estimates for quasilinear elliptic equations with variable growth involving measure data. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1639-1667. doi : 10.1016/j.anihpc.2016.12.002. http://www.numdam.org/articles/10.1016/j.anihpc.2016.12.002/

Bibliographie
Cité par

[1] Acerbi, E.; Mingione, G. Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., Volume 156 (2001) no. 2, pp. 121–140 | DOI | MR | Zbl

[2] Acerbi, E.; Mingione, G. Gradient estimates for the $p (x)$ -Laplacean system, J. Reine Angew. Math., Volume 584 (2005), pp. 117–148 | MR | Zbl

[3] Antontsev, S.; Rodrigues, J. On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara, Sez. 7: Sci. Mat., Volume 52 (2006) no. 1, pp. 19–36 | MR | Zbl

[4] Antontsev, S.; Shmarev, S. A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal., Volume 60 (2005) no. 3, pp. 515–545 | DOI | MR | Zbl

[5] Baroni, P.; Habermann, J. Elliptic interpolation estimates for non-standard growth operators, Ann. Acad. Sci. Fenn., Math., Volume 39 (2014) no. 1, pp. 119–162 | MR | Zbl

[6] Bendahmane, M.; Wittbold, P. Renormalized solutions for nonlinear elliptic equations with variable exponents and $L^{1}$ data, Nonlinear Anal., Volume 70 (2009) no. 2, pp. 567–583 | DOI | MR | Zbl

[7] Bénilan, P.; Boccardo, L.; Gallouët, T.; Gariepy, R.; Pierre, M.; Vázquez, J. An $L^{1}$ -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 22 (1995) no. 2, pp. 241–273 | Numdam | MR | Zbl

[8] Boccardo, L.; Gallouët, T. Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., Volume 87 (1989) no. 1, pp. 149–169 | DOI | MR | Zbl

[9] Boccardo, L.; Gallouët, T. Nonlinear elliptic equations with right-hand side measures, Commun. Partial Differ. Equ., Volume 17 (1992) no. 3–4, pp. 641–655 | MR | Zbl

[10] Boccardo, L.; Gallouët, T.; Orsina, L. Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 13 (1996) no. 5, pp. 539–551 | DOI | Numdam | MR | Zbl

[11] Bögelein, V.; Habermann, J. Gradient estimates via non standard potentials and continuity, Ann. Acad. Sci. Fenn., Math., Volume 35 (2010) no. 2, pp. 641–678 | MR | Zbl

[12] Byun, S.; Ok, J. On $W^{1, q (\cdot)}$ -estimates for elliptic equations of $p (x)$ -Laplacian type, J. Math. Pures Appl. (9), Volume 106 (2016) no. 3, pp. 512–545 | DOI | MR | Zbl

[13] Byun, S.; Ok, J.; Ryu, S. Global gradient estimates for elliptic equations of $p (x)$ -Laplacian type with BMO nonlinearity, J. Reine Angew. Math., Volume 715 (2016), pp. 1–38 | MR | Zbl

[14] Byun, S.; Wang, L. Elliptic equations with BMO coefficients in Reifenberg domains, Commun. Pure Appl. Math., Volume 57 (2004) no. 10, pp. 1283–1310 | DOI | MR | Zbl

[15] Caffarelli, L.; Cabré, X. Fully Nonlinear Elliptic Equations, Colloq. Publ. – Am. Math. Soc., vol. 43, American Mathematical Society, Providence, RI, 1995 | MR | Zbl

[16] Caffarelli, L.; Peral, I. On $W^{1, p}$ estimates for elliptic equations in divergence form, Commun. Pure Appl. Math., Volume 51 (1998) no. 1, pp. 1–21 | DOI | MR | Zbl

[17] Cekic, B.; Kalinin, A.; Mashiyev, R.; Avci, M. $L^{p (x)} (Ω)$ -estimates of vector fields and some applications to magnetostatics problems, J. Math. Anal. Appl., Volume 389 (2012) no. 2, pp. 838–851 | DOI | MR | Zbl

[18] Chen, Y.; Levine, S.; Rao, M. Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., Volume 66 (2006) no. 4, pp. 1383–1406 (electronic) | DOI | MR | Zbl

[19] Cruz-Uribe, D.; Fiorenza, A. Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Heidelberg, 2013 | MR | Zbl

[20] Dall'Aglio, A. Approximated solutions of equations with $L^{1}$ data. Application to the H-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl. (4), Volume 170 (1996), pp. 207–240 | DOI | MR | Zbl

[21] Dal Maso, G.; Murat, F.; Orsina, L.; Prignet, A. Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 28 (1999) no. 4, pp. 741–808 | Numdam | MR | Zbl

[22] DiBenedetto, E. $C^{1 + α}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., Volume 7 (1983) no. 8, pp. 827–850 | DOI | MR | Zbl

[23] Diening, L.; Harjulehto, P.; Hästö, P.; Růžička, M. Lebesgue and Sobolev Spaces with Variable Exponents, Lect. Notes Math., vol. 2017, Springer, Heidelberg, 2011 | MR | Zbl

[24] Duzaar, F.; Mingione, G. Gradient estimates via linear and nonlinear potentials, J. Funct. Anal., Volume 259 (2010) no. 11, pp. 2961–2998 | DOI | MR | Zbl

[25] Duzaar, F.; Mingione, G. Gradient estimates via non-linear potentials, Am. J. Math., Volume 133 (2011) no. 4, pp. 1093–1149 | DOI | MR | Zbl

[26] Fan, X. Global $C^{1, α}$ regularity for variable exponent elliptic equations in divergence form, J. Differ. Equ., Volume 235 (2007) no. 2, pp. 397–417 | MR | Zbl

[27] Giusti, E. Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003 | DOI | MR | Zbl

[28] Harjulehto, P.; Hästö, P.; Lê, Ú.; Nuortio, M. Overview of differential equations with non-standard growth, Nonlinear Anal., Volume 72 (2010) no. 12, pp. 4551–4574 | DOI | MR | Zbl

[29] Kuusi, T.; Mingione, G. Universal potential estimates, J. Funct. Anal., Volume 262 (2012) no. 10, pp. 4205–4269 | DOI | MR | Zbl

[30] Kuusi, T.; Mingione, G. Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal., Volume 207 (2013) no. 1, pp. 215–246 | DOI | MR | Zbl

[31] Kuusi, T.; Mingione, G. Guide to nonlinear potential estimates, Bull. Math. Sci., Volume 4 (2014) no. 1, pp. 1–82 | DOI | MR | Zbl

[32] Kuusi, T.; Mingione, G. Nonlinear potential theory of elliptic systems, Nonlinear Anal., Volume 138 (2016), pp. 277–299 | DOI | MR | Zbl

[33] Lemenant, A.; Milakis, E.; Spinolo, L. On the extension property of Reifenberg-flat domains, Ann. Acad. Sci. Fenn., Math., Volume 39 (2014) no. 1, pp. 51–71 | MR | Zbl

[34] Lewis, J.; Nyström, K. Regularity and free boundary regularity for the p-Laplace operator in Reifenberg flat and Ahlfors regular domains, J. Am. Math. Soc., Volume 25 (2012) no. 3, pp. 827–862 | DOI | MR | Zbl

[35] Lieberman, G. Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., Volume 12 (1988) no. 11, pp. 1203–1219 | DOI | MR | Zbl

[36] Lukkari, T. Elliptic equations with nonstandard growth involving measures, Hiroshima Math. J., Volume 38 (2008) no. 1, pp. 155–176 | DOI | MR | Zbl

[37] Mingione, G. Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math., Volume 51 (2006) no. 4, pp. 355–426 | DOI | MR | Zbl

[38] Mingione, G. The Calderón–Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 6 (2007) no. 2, pp. 195–261 | Numdam | MR | Zbl

[39] Mingione, G. Gradient estimates below the duality exponent, Math. Ann., Volume 346 (2010) no. 3, pp. 571–627 | DOI | MR | Zbl

[40] Mingione, G. Gradient potential estimates, J. Eur. Math. Soc., Volume 13 (2011) no. 2, pp. 459–486 | MR | Zbl

[41] Mingione, G. Nonlinear measure data problems, Milan J. Math., Volume 79 (2011) no. 2, pp. 429–496 | DOI | MR | Zbl

[42] Nguyen, Q. Global estimates for quasilinear parabolic equations on Reifenberg flat domains and its applications to Riccati type parabolic equations with distributional data, Calc. Var. Partial Differ. Equ., Volume 54 (2015) no. 4, pp. 3927–3948 | DOI | MR | Zbl

[43] Phuc, N. Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations, Adv. Math., Volume 250 (2014), pp. 387–419 | DOI | MR | Zbl

[44] Prignet, A. Remarks on existence and uniqueness of solutions of elliptic problems with right-hand side measures, Rend. Mat. Appl. (7), Volume 15 (1995) no. 3, pp. 321–337 | MR | Zbl

[45] Reifenberg, E. Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta Math., Volume 104 (1960), pp. 1–92 | DOI | MR | Zbl

[46] Růžička, M. Electrorheological Fluids: Modeling and Mathematical Theory, Lect. Notes Math., vol. 1748, Springer-Verlag, Berlin, 2000 | DOI | MR | Zbl

[47] Sanchón, M.; Urbano, J. Entropy solutions for the $p (x)$ -Laplace equation, Trans. Am. Math. Soc., Volume 361 (2009) no. 12, pp. 6387–6405 | DOI | MR | Zbl

[48] Serrin, J. Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Super. Pisa (3), Volume 18 (1964), pp. 385–387 | Numdam | MR | Zbl

[49] Showalter, R. Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surv. Monogr., vol. 49, American Mathematical Society, Providence, RI, 1997 | MR | Zbl

[50] Toro, T. Doubling and flatness: geometry of measures, Not. Am. Math. Soc., Volume 44 (1997) no. 9, pp. 1087–1094 | MR | Zbl

[51] Wang, L. A geometric approach to the Calderón–Zygmund estimates, Acta Math. Sin. Engl. Ser., Volume 19 (2003) no. 2, pp. 381–396 | DOI | MR | Zbl

[52] Zhikov, V. Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 50 (1986) no. 4, pp. 675–710 (in Russian) | MR | Zbl

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