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Applicable Analysis
An International Journal
Volume 101, 2022 - Issue 8
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Articles

Hölder estimates for the elliptic p(x)-Laplacian equation with the logarithmic function

Pages 3048-3064 | Received 30 Jan 2020, Accepted 01 Oct 2020, Published online: 21 Oct 2020

References

  • Coscia A, Mingione G. Hölder continuity of the gradient of p(x)-harmonic mappings. C R Acad Sci Paris Math. 1999;328(4):363–368.
  • Fan X. Global C1,α regularity for variable exponent elliptic equations in divergence form. J Differ Equ. 2007;235:397–417.
  • Lyaghfouri A. Hölder continuity of p(x)-superharmonic functions. Nonlinear Anal. 2010;73(8):2433–2444.
  • Zhang C, Zhou S. Hölder regularity for the gradients of solutions of the strong p(x)-Laplacian. J Math Anal Appl. 2012;389(2):1066–1077.
  • Acerbi E, Mingione G. Regularity results for a stationary electro-rheologicaluids. Arch Ration Mech Anal. 2002;164(3):213–259.
  • Byun S, Lee K, Oh J, et al. Regularity results of the thin obstacle problem for the p(x)-Laplacian. J Funct Anal. 2019;276:496–519.
  • Byun S, Ok J. On W1,q(x)-estimates for elliptic equations of p(x)-Laplacian type. J Math Pures Appl (9). 2016;106(3):512–545.
  • Byun S, Ok J, Youn Y. Global gradient estimates for spherical quasi-minimizers of integral functionals with p(x)-growth. Nonlinear Anal. 2018;177:186–208. part A.
  • Challal S, Lyaghfouri A. Gradient estimates for p(x)-harmonic functions. Manuscripta Math. 2010;131(3–4):403–414.
  • Zhang C, Zhou S. Global weighted estimates for quasilinear elliptic equations with non-standard growth. J Funct Anal. 2014;267:605–642.
  • Baroni P, Bögelein V. Calderón–Zygmund estimates for parabolic p(x,t)-Laplacian systems. Rev Mat Iberoam. 2014;30(4):1355–1386.
  • Bögelein V, Duzaar F. Hölder estimates for parabolic p(x,t)-Laplacian systems. Math Ann. 2012;354(3):907–938.
  • Bögelein V, Li Q. Very weak solutions of degenerate parabolic systems with non-standard p(x,t)-growth. Nonlinear Anal. 2014;98:190–225.
  • Byun S, Ok J. Nonlinear parabolic equations with variable exponent growth in nonsmooth domains. SIAM J Math Anal. 2016;48(5):3148–3190.
  • Ok J. Regularity for parabolic equations with time dependent growth. J Math Pures Appl (9). 2018;120:253–293.
  • Xu M, Chen Y. Hölder continuity of weak solutions for parabolic equations with nonstandard growth conditions. Acta Math Sin (Engl Ser). 2006;22(3):793–806.
  • Breit D, Cianchi A, Diening L, et al. Pointwise Calderón-Zygmund gradient estimates for the p-Laplace system. J Math Pures Appl (9). 2018;114:146–190.
  • DiBenedetto E, Manfredi J. On the higer integrability of the gradient of weak solutions of certain degenerate elliptic systems. Amer J Math. 1993;115:1107–1134.
  • Duzaar F, Mingione G. Gradient estimates via non-linear potentials. Amer J Math. 2011;133(4):1093–1149.
  • Kinnunen J, Zhou S. A local estimate for nonlinear equations with discontinuous coefficients. Comm Partial Differ Equ. 1999;24:2043–2068.
  • Palagachev D. Quasilinear elliptic equations with VMO coefficients. Trans Amer Math Soc. 1995;347:2481–2493.
  • Passarelli di Napoli A. A C1,α partial regularity result for non-autonomous convex integrals with discontinuous coefficients. NoDEA Nonlinear Differ Equ Appl. 2015;22(5):1319–1343.
  • Phuc NC. Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients. Ann Sc Norm Super Pisa Cl Sci (5). 2011;10(1):1–17.
  • Wang L. Compactness methods for certain degenerate elliptic equations. J Differ Equ. 1994;107(2):341–350.
  • Baroni P. Riesz potential estimates for a general class of quasilinear equations. Calc Var Partial Differ Equ. 2015;53(3–4):803–846.
  • Breit D, De Maria B, Passarelli di Napoli A. Regularity for non-autonomous functionals with almost linear growth. Manuscripta Math. 2011;136(1–2):83–114.
  • Byun S, Cho Y. Nonlinear gradient estimates for generalized elliptic equations with nonstandard growth in nons-mooth domains. Nonlinear Anal. 2016;140:145–165.
  • Cianchi A, Maz'ya V. Global Lipschitz regularity for a class of quasilinear elliptic equations. Comm Partial Differ Equ. 2011;36(1):100–133.
  • Cianchi A, Maz'ya V. Global boundedness of the gradient for a class of nonlinear elliptic systems. Arch Ration Mech Anal. 2014;212(1):129–177.
  • Cianchi A, Maz'ya V. Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems. J Eur Math Soc. 2014;16(3):571–595.
  • Diening L, Stroffolini B, Verde A. The ϕ-harmonic approximation and the regularity of ϕ-harmonic maps. J Differ Equ. 2012;253(7):1943–1958.
  • Diening L, Stroffolini B, Verde A. Everywhere regularity of functionals with ϕ-growth. Manuscripta Math. 2009;129(4):449–481.
  • Giannetti F, Passarelli di Napoli A, Tachikawa A. Partial and full boundary regularity for non-autonomous functionals with ϕ-growth conditions. Forum Math. 2019;31(4):1027–1050.
  • Ok J. Gradient estimates for elliptic equations with Lp(⋅)log⁡L growth. Calc Var Partial Differ Equ. 2016;55(2):121.Art. 26, 30 pp.
  • Giannetti F, Passarelli di Napoli A. Regularity results for a new class of functionals with non-standard growth conditions. J Differ Equ. 2013;254(3):1280–1305.
  • Diening L, Harjulehto P, Hästö P, et al. Lebesgue and Sobolev spaces with variable exponents. Heidelberg: Springer; 2011. (Lecture Notes in Mathematics; 2017).
  • Diening L, R˙užička M. Calderón-Zygmund operators on generalized Lebesgue spaces Lp(⋅) and problems related to fluid dynamics. J Reine Angew Math. 2003;563:197–220.
  • Fan X, Shen J, Zhao D. Sobolev embedding theorems for spaces Wk,p(x)(Ω). J Math Anal Appl. 2001;262:749–760.
  • Fan X, Zhao D. On the spaces Lp(x)(Ω) and Wm,p(x)(Ω). J Math Anal Appl. 2001;263:424–446.
  • Harjulehto P. Variable exponent Sobolev spaces with zero boundary values. Math Bohem. 2007;132:125–136.
  • Acerbi E, Mingione G. Gradient estimates for the p(x)-Laplacean system. J Reine Angew Math. 2005;584:117–148.
  • Diening L, Schwarzacher S. Global gradient estimates for the p(x)-Laplacian. Nonlinear Anal. 2014;106:70–85.
  • Yao F, Zhou S. Calderón-Zygmund estimates for a class of quasilinear elliptic equations. J Funct Anal. 2017;272:1524–1552.
  • Diening L, Ettwein F. Fractional estimates for non-differentiable elliptic systems with general growth. Forum Math. 2008;20(3):523–556.
  • Giaquinta M. Multiple integrals in the calculus of variations and nonlinear elliptic systems. New Jersey, Princeton: Princeton University Press; 1983.
  • Lieberman GM. The natural generalization of the natural conditions of Ladyzenskaja and Ural'tzeva for elliptic equations. Comm Partial Differ Equ. 1991;16:311–361.
  • Acerbi E, Mingione G. Regularity results for a class of functionals with nonstandard growth. Arch Ration Mech Anal. 2001;156:121–140.

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