Global weighted estimates for the gradient of solutions to nonlinear elliptic equations
Byun, Sun-Sig ; Ryu, Seungjin
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 291-313 / Harvested from Numdam
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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.

Publié le : 2013-01-01
DOI : https://doi.org/10.1016/j.anihpc.2012.08.001
Classification:  35J60,  35R05,  46E30,  46E35 
@article{AIHPC_2013__30_2_291_0,
     author = {Byun, Sun-Sig and Ryu, Seungjin},
     title = {Global weighted estimates for the gradient of solutions to nonlinear elliptic equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {30},
     year = {2013},
     pages = {291-313},
     doi = {10.1016/j.anihpc.2012.08.001},
     mrnumber = {3035978},
     zbl = {1292.35127},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2013__30_2_291_0}
}

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, Tome 30 (2013) pp. 291-313. doi : 10.1016/j.anihpc.2012.08.001. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_2_291_0/

[1] E. Acerbi, G. Mingione, Gradient estimates for the p(x)-Laplacean system, J. Reine Angew. Math. 584 (2005), 117-148 | MR 2155087 | Zbl 1093.76003

[2] E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 no. 2 (2007), 285-320 | MR 2286632 | Zbl 1113.35105

[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 2580508 | Zbl 1194.35085

[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 2580509 | Zbl 1194.35086

[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 2820386 | Zbl 1241.35064

[6] S. Byun, L. Wang, Parabolic equations in time dependent Reifenberg domains, Adv. Math. 212 no. 2 (2007), 797-818 | MR 2329320 | Zbl 1117.35080

[7] S. Byun, L. Wang, Gradient estimates for elliptic systems in non-smooth domains, Math. Ann. 341 no. 3 (2008), 629-650 | MR 2399163 | Zbl 1143.35095

[8] S. Byun, L. Wang, Elliptic equations with BMO nonlinearity in Reifenberg domains, Adv. Math. 219 no. 6 (2008), 1937-1971 | MR 2456271 | Zbl 1221.35427

[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 2462578 | Zbl 1156.35038

[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 2596493 | Zbl 1194.35087

[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 1486629 | Zbl 0906.35030

[13] L. Esposito, G. Mingione, C. Trombetti, On the Lipschitz regularity for certain elliptic problems, Forum Math. 18 no. 2 (2006), 263-292 | MR 2218421 | Zbl 1261.35058

[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 2366379 | Zbl 1155.49037

[15] F. John, L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426 | MR 131498 | Zbl 0102.04302

[16] V. Kokilashvili, M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific Publishing Co., Inc., River Edge, NJ (1991) | MR 1156767 | Zbl 0751.46021

[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 1302151 | Zbl 0820.35064

[18] O.A. Ladyzhenskaya, N.N. UralʼTseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, London (1968) | MR 244627 | Zbl 0164.13002

[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 1014660

[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 2756073 | Zbl 1210.35094

[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 2864410 | Zbl 1255.35113

[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 | Numdam | MR 2352517 | Zbl 1178.35168

[23] D. Palagachev, Quasilinear elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 347 (1995), 2481-2493 | MR 1308019 | Zbl 0833.35048

[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 2153140 | Zbl 1091.35035

[25] M. Parviainen, Global higher integrability for parabolic quasiminimizers in nonsmooth domains, Calc. Var. Partial Differential Equations 31 no. 1 (2008), 75-98 | MR 2342615 | Zbl 1173.35036

[26] M. Parviainen, Reverse Hölder inequalities for singular parabolic equations near the boundary, J. Differential Equations 246 no. 2 (2009), 512-540 | MR 2468726 | Zbl 1173.35037

[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 | MR 2829320 | Zbl 1228.35260

[28] E. Reinfenberg, Solutions of the plateau problem for m-dimensional surfaces of varying topological type, Acta Math. (1960), 1-92 | MR 114145

[29] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Pure Appl. Math. vol. 123, Academic Press, Inc., Orlando, FL (1986) | MR 869816 | Zbl 0621.42001

[30] T. Toro, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc. (1997), 1087-1094 | MR 1470167 | Zbl 0909.31006

[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 2446351 | Zbl 1158.35039