July  2016, 21(5): 1635-1649. doi: 10.3934/dcdsb.2016015

Interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations

1. 

Department of Mathematics,Shanghai University, Shanghai 200444, China

2. 

LMAM, School of Mathematical Sciences, Peking University, Bejing 100871

Received  September 2013 Revised  March 2014 Published  April 2016

In this paper we present a new proof for the interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations, whose prototype is the $p$-Laplace equation.
Citation: Fengping Yao, Shulin Zhou. Interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1635-1649. doi: 10.3934/dcdsb.2016015
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces (2nd edition),, Academic Press, (2003). Google Scholar

[2]

S. Byun, F. Yao and S. Zhou, Gradient Estimates in Orlicz space for nonlinear elliptic Equations,, J. Funct. Anal., 255 (2008), 1851. doi: 10.1016/j.jfa.2008.09.007. Google Scholar

[3]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations,, Ann. of Math., 130 (1989), 189. doi: 10.2307/1971480. Google Scholar

[4]

Y. Chen and L. Wu, Second Order Elliptic Partial Differential Equations and Elliptic Systems,, American Mathematical Society, (1998). Google Scholar

[5]

A. Cianchi and V. Maz'ya, Global Lipschitz regularity for a class of quasilinear elliptic equations,, Comm. Partial Differential Equations, 36 (2011), 100. doi: 10.1080/03605301003657843. Google Scholar

[6]

E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827. doi: 10.1016/0362-546X(83)90061-5. Google Scholar

[7]

F. Duzaar and G. Mingione, Local Lipschitz regularity for degenerate elliptic systems,, Ann. Inst. H. Poincaré, 27 (2010), 1361. doi: 10.1016/j.anihpc.2010.07.002. Google Scholar

[8]

F. Duzaar and G. Mingione, Gradient estimates via linear and nonlinear potentials,, J. Funct. Anal., 259 (2010), 2961. doi: 10.1016/j.jfa.2010.08.006. Google Scholar

[9]

L. C. Evans, A new proof of local $C^{1,\alpha}$ regularity for solutions of certain degenerate elliptic p.d.e.,, J. Differential Equations, 45 (1982), 356. doi: 10.1016/0022-0396(82)90033-X. Google Scholar

[10]

M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,, Princeton University Press, (1983). Google Scholar

[11]

D. Gilbarg and N. Trudinger, Elliptic Partial Diferential Equations of Second Order (3rd edition),, Springer-Verlag, (1998). Google Scholar

[12]

J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations,, Indiana Univ. Math. J., 32 (1983), 849. doi: 10.1512/iumj.1983.32.32058. Google Scholar

[13]

G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations,, Comm. Partial Differential Equations, 16 (1991), 311. doi: 10.1080/03605309108820761. Google Scholar

[14]

J. Malý, D. Swanson and W. Ziemer, Fine behavior of functions whose gradients are in an Orlicz space,, Studia Math., 190 (2009), 33. doi: 10.4064/sm190-1-2. Google Scholar

[15]

M. Shaw and L. Wang, Hölder and Lp estimates for Db on CR manifolds of arbitrary codimension,, Math. Ann., 331 (2005), 297. doi: 10.1007/s00208-004-0583-5. Google Scholar

[16]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, J. Differential Equations, 51 (1984), 126. doi: 10.1016/0022-0396(84)90105-0. Google Scholar

[17]

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems,, Acta Math., 138 (1977), 219. doi: 10.1007/BF02392316. Google Scholar

[18]

L. Wang, Compactness methods for certain degenerate elliptic equations,, J. Differential Equations, 107 (1994), 341. doi: 10.1006/jdeq.1994.1016. Google Scholar

[19]

L. Wang, Hölder estimates for subelliptic operators,, J. Funct. Anal., 199 (2003), 228. doi: 10.1016/S0022-1236(03)00093-4. Google Scholar

[20]

L. Wang, F. Yao, S. Zhou and H. Jia, Optimal regularity for the poisson equation,, Proc. Amer. Math. Soc., 137 (2009), 2037. doi: 10.1090/S0002-9939-09-09805-0. Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces (2nd edition),, Academic Press, (2003). Google Scholar

[2]

S. Byun, F. Yao and S. Zhou, Gradient Estimates in Orlicz space for nonlinear elliptic Equations,, J. Funct. Anal., 255 (2008), 1851. doi: 10.1016/j.jfa.2008.09.007. Google Scholar

[3]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations,, Ann. of Math., 130 (1989), 189. doi: 10.2307/1971480. Google Scholar

[4]

Y. Chen and L. Wu, Second Order Elliptic Partial Differential Equations and Elliptic Systems,, American Mathematical Society, (1998). Google Scholar

[5]

A. Cianchi and V. Maz'ya, Global Lipschitz regularity for a class of quasilinear elliptic equations,, Comm. Partial Differential Equations, 36 (2011), 100. doi: 10.1080/03605301003657843. Google Scholar

[6]

E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827. doi: 10.1016/0362-546X(83)90061-5. Google Scholar

[7]

F. Duzaar and G. Mingione, Local Lipschitz regularity for degenerate elliptic systems,, Ann. Inst. H. Poincaré, 27 (2010), 1361. doi: 10.1016/j.anihpc.2010.07.002. Google Scholar

[8]

F. Duzaar and G. Mingione, Gradient estimates via linear and nonlinear potentials,, J. Funct. Anal., 259 (2010), 2961. doi: 10.1016/j.jfa.2010.08.006. Google Scholar

[9]

L. C. Evans, A new proof of local $C^{1,\alpha}$ regularity for solutions of certain degenerate elliptic p.d.e.,, J. Differential Equations, 45 (1982), 356. doi: 10.1016/0022-0396(82)90033-X. Google Scholar

[10]

M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,, Princeton University Press, (1983). Google Scholar

[11]

D. Gilbarg and N. Trudinger, Elliptic Partial Diferential Equations of Second Order (3rd edition),, Springer-Verlag, (1998). Google Scholar

[12]

J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations,, Indiana Univ. Math. J., 32 (1983), 849. doi: 10.1512/iumj.1983.32.32058. Google Scholar

[13]

G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations,, Comm. Partial Differential Equations, 16 (1991), 311. doi: 10.1080/03605309108820761. Google Scholar

[14]

J. Malý, D. Swanson and W. Ziemer, Fine behavior of functions whose gradients are in an Orlicz space,, Studia Math., 190 (2009), 33. doi: 10.4064/sm190-1-2. Google Scholar

[15]

M. Shaw and L. Wang, Hölder and Lp estimates for Db on CR manifolds of arbitrary codimension,, Math. Ann., 331 (2005), 297. doi: 10.1007/s00208-004-0583-5. Google Scholar

[16]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, J. Differential Equations, 51 (1984), 126. doi: 10.1016/0022-0396(84)90105-0. Google Scholar

[17]

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems,, Acta Math., 138 (1977), 219. doi: 10.1007/BF02392316. Google Scholar

[18]

L. Wang, Compactness methods for certain degenerate elliptic equations,, J. Differential Equations, 107 (1994), 341. doi: 10.1006/jdeq.1994.1016. Google Scholar

[19]

L. Wang, Hölder estimates for subelliptic operators,, J. Funct. Anal., 199 (2003), 228. doi: 10.1016/S0022-1236(03)00093-4. Google Scholar

[20]

L. Wang, F. Yao, S. Zhou and H. Jia, Optimal regularity for the poisson equation,, Proc. Amer. Math. Soc., 137 (2009), 2037. doi: 10.1090/S0002-9939-09-09805-0. Google Scholar

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