Global weighted estimates for the gradient of solutions to nonlinear elliptic equations
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 2, pp. 291-313.

We consider nonlinear elliptic equations of p-Laplacian type that are not necessarily of variation form when the nonlinearity is allowed to be discontinuous and the boundary of the domain can go beyond the Lipschitz category. Under smallness in the BMO nonlinearity and sufficient flatness of the Reifenberg domain, we obtain the global weighted L q estimates with q(p,) for the gradient of weak solutions.

DOI: 10.1016/j.anihpc.2012.08.001
Classification: 35J60, 35R05, 46E30, 46E35
Mots-clés : Gradient estimate, Weighted $ {L}^{p}$ space, Nonlinear elliptic equation, BMO space, Reifenberg domain
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     title = {Global weighted estimates for the gradient of solutions to nonlinear elliptic equations},
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Byun, Sun-Sig; Ryu, Seungjin. Global weighted estimates for the gradient of solutions to nonlinear elliptic equations. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 2, pp. 291-313. doi : 10.1016/j.anihpc.2012.08.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2012.08.001/

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

[2] E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 no. 2 (2007), 285-320 | MR | Zbl

[3] V. Bögelein, F. Duzaar, G. Mingione, The boundary regularity of non-linear parabolic systems. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 no. 1 (2010), 145-200 | Numdam | MR | Zbl

[4] V. Bögelein, F. Duzaar, G. Mingione, The boundary regularity of non-linear parabolic systems. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 no. 1 (2010), 201-255 | Numdam | MR | Zbl

[5] S. Byun, Gradient estimates in Orlicz spaces for nonlinear elliptic equations with BMO nonlinearity in nonsmooth domains, Forum Math. 23 (2011), 693-711 | MR | Zbl

[6] S. Byun, L. Wang, Parabolic equations in time dependent Reifenberg domains, Adv. Math. 212 no. 2 (2007), 797-818 | MR | Zbl

[7] S. Byun, L. Wang, Gradient estimates for elliptic systems in non-smooth domains, Math. Ann. 341 no. 3 (2008), 629-650 | MR | Zbl

[8] S. Byun, L. Wang, Elliptic equations with BMO nonlinearity in Reifenberg domains, Adv. Math. 219 no. 6 (2008), 1937-1971 | MR | Zbl

[9] S. Byun, L. Wang, Nonlinear gradient estimates for elliptic equations of general type, Calc. Var. Partial Differential Equations (2011), 1-17, http://dx.doi.org/10.1007/s00526-011-0463-2

[10] S. Byun, F. Yao, S. Zhou, Gradient estimates in Orlicz space for nonlinear elliptic equations, J. Funct. Anal. 255 no. 8 (2008), 1851-1873 | MR | Zbl

[11] V. Bögelein, M. Parviainen, Self-improving property of nonlinear higher order parabolic systems near the boundary, NoDEA Nonlinear Differential Equations Appl. 17 no. 1 (2010), 21-54 | MR | Zbl

[12] L.A. Caffarelli, I. Peral, On W 1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), 1-21 | MR | Zbl

[13] L. Esposito, G. Mingione, C. Trombetti, On the Lipschitz regularity for certain elliptic problems, Forum Math. 18 no. 2 (2006), 263-292 | MR | Zbl

[14] G. Hong, L. Wang, A geometric approach to the topological disk theorem for Reifenberg, Pacific J. Math. 233 no. 2 (2007), 321-339 | MR | Zbl

[15] F. John, L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426 | MR | Zbl

[16] V. Kokilashvili, M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific Publishing Co., Inc., River Edge, NJ (1991) | MR | Zbl

[17] T. Kilpeläinen, P. Koskela, Global integrability of the gradients of solutions to partial differential equations, Nonlinear Anal. 23 no. 7 (1994), 899-909 | MR | Zbl

[18] O.A. Ladyzhenskaya, N.N. UralʼTseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, London (1968) | MR | Zbl

[19] G. Lieberman, Boundary regularity for linear and quasilinear variational inequalities, Proc. Roy. Soc. Edinburgh Sect. A 112 no. 3–4 (1989), 319-326 | MR

[20] T. Mengesha, N.C. Phuc, Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations 250 no. 5 (2011), 2485-2507 | MR | Zbl

[21] T. Mengesha, N.C. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Ration. Mech. Anal. 203 no. 1 (2012), 189-216, http://dx.doi.org/10.1007/s00205-011-0446-7 | MR | Zbl

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

[23] D. Palagachev, Quasilinear elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 347 (1995), 2481-2493 | MR | Zbl

[24] D. Palagachev, L. Softova, A priori estimates and precise regularity for parabolic systems with discontinuous data, Discrete Contin. Dyn. Syst. 13 (2005), 721-742 | MR | Zbl

[25] M. Parviainen, Global higher integrability for parabolic quasiminimizers in nonsmooth domains, Calc. Var. Partial Differential Equations 31 no. 1 (2008), 75-98 | MR | Zbl

[26] M. Parviainen, Reverse Hölder inequalities for singular parabolic equations near the boundary, J. Differential Equations 246 no. 2 (2009), 512-540 | MR | Zbl

[27] N.C. Phuc, Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 no. 1 (2011), 1-17 | Numdam | MR | Zbl

[28] E. Reinfenberg, Solutions of the plateau problem for m-dimensional surfaces of varying topological type, Acta Math. (1960), 1-92 | MR

[29] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Pure Appl. Math. vol. 123, Academic Press, Inc., Orlando, FL (1986) | MR | Zbl

[30] T. Toro, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc. (1997), 1087-1094 | MR | Zbl

[31] F. Yao, Y. Sun, S. Zhou, Gradient estimates in Orlicz spaces for quasilinear elliptic equation, Nonlinear Anal. 69 no. 8 (2008), 2553-2565 | MR | Zbl

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