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

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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

Fengping Yao ^1, and Shulin Zhou ^2,

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

Abstract
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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.

Keywords: quasilinear, $C^{1, elliptic., divergence, \alpha}$ regularity.

Mathematics Subject Classification: 35J20, 35J60, 35J9.

Citation: Fengping Yao, Shulin Zhou. Interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations. Discrete and 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, New York, 2003.
[2]	S. Byun, F. Yao and S. Zhou, Gradient Estimates in Orlicz space for nonlinear elliptic Equations, J. Funct. Anal., 255 (2008), 1851-1873. doi: 10.1016/j.jfa.2008.09.007.
[3]	L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213. doi: 10.2307/1971480.
[4]	Y. Chen and L. Wu, Second Order Elliptic Partial Differential Equations and Elliptic Systems, American Mathematical Society, Providence, RI, 1998.
[5]	A. Cianchi and V. Maz'ya, Global Lipschitz regularity for a class of quasilinear elliptic equations, Comm. Partial Differential Equations, 36 (2011), 100-133. doi: 10.1080/03605301003657843.
[6]	E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5.
[7]	F. Duzaar and G. Mingione, Local Lipschitz regularity for degenerate elliptic systems, Ann. Inst. H. Poincaré, 27 (2010), 1361-1396. doi: 10.1016/j.anihpc.2010.07.002.
[8]	F. Duzaar and G. Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal., 259 (2010), 2961-2998. doi: 10.1016/j.jfa.2010.08.006.
[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), 356-373. doi: 10.1016/0022-0396(82)90033-X.
[10]	M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, 1983.
[11]	D. Gilbarg and N. Trudinger, Elliptic Partial Diferential Equations of Second Order (3rd edition), Springer-Verlag, Berlin, 1998.
[12]	J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J., 32 (1983), 849-858. doi: 10.1512/iumj.1983.32.32058.
[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), 311-361. doi: 10.1080/03605309108820761.
[14]	J. Malý, D. Swanson and W. Ziemer, Fine behavior of functions whose gradients are in an Orlicz space, Studia Math., 190 (2009), 33-71. doi: 10.4064/sm190-1-2.
[15]	M. Shaw and L. Wang, Hölder and Lp estimates for Db on CR manifolds of arbitrary codimension, Math. Ann., 331 (2005), 297-343. doi: 10.1007/s00208-004-0583-5.
[16]	P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.
[17]	K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.
[18]	L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations, 107 (1994), 341-350. doi: 10.1006/jdeq.1994.1016.
[19]	L. Wang, Hölder estimates for subelliptic operators, J. Funct. Anal., 199 (2003), 228-242. doi: 10.1016/S0022-1236(03)00093-4.
[20]	L. Wang, F. Yao, S. Zhou and H. Jia, Optimal regularity for the poisson equation, Proc. Amer. Math. Soc., 137 (2009), 2037-2047. doi: 10.1090/S0002-9939-09-09805-0.

show all references

References:

[1]	R. A. Adams and J. J. F. Fournier, Sobolev Spaces (2nd edition), Academic Press, New York, 2003.
[2]	S. Byun, F. Yao and S. Zhou, Gradient Estimates in Orlicz space for nonlinear elliptic Equations, J. Funct. Anal., 255 (2008), 1851-1873. doi: 10.1016/j.jfa.2008.09.007.
[3]	L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213. doi: 10.2307/1971480.
[4]	Y. Chen and L. Wu, Second Order Elliptic Partial Differential Equations and Elliptic Systems, American Mathematical Society, Providence, RI, 1998.
[5]	A. Cianchi and V. Maz'ya, Global Lipschitz regularity for a class of quasilinear elliptic equations, Comm. Partial Differential Equations, 36 (2011), 100-133. doi: 10.1080/03605301003657843.
[6]	E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5.
[7]	F. Duzaar and G. Mingione, Local Lipschitz regularity for degenerate elliptic systems, Ann. Inst. H. Poincaré, 27 (2010), 1361-1396. doi: 10.1016/j.anihpc.2010.07.002.
[8]	F. Duzaar and G. Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal., 259 (2010), 2961-2998. doi: 10.1016/j.jfa.2010.08.006.
[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), 356-373. doi: 10.1016/0022-0396(82)90033-X.
[10]	M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, 1983.
[11]	D. Gilbarg and N. Trudinger, Elliptic Partial Diferential Equations of Second Order (3rd edition), Springer-Verlag, Berlin, 1998.
[12]	J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J., 32 (1983), 849-858. doi: 10.1512/iumj.1983.32.32058.
[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), 311-361. doi: 10.1080/03605309108820761.
[14]	J. Malý, D. Swanson and W. Ziemer, Fine behavior of functions whose gradients are in an Orlicz space, Studia Math., 190 (2009), 33-71. doi: 10.4064/sm190-1-2.
[15]	M. Shaw and L. Wang, Hölder and Lp estimates for Db on CR manifolds of arbitrary codimension, Math. Ann., 331 (2005), 297-343. doi: 10.1007/s00208-004-0583-5.
[16]	P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.
[17]	K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.
[18]	L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations, 107 (1994), 341-350. doi: 10.1006/jdeq.1994.1016.
[19]	L. Wang, Hölder estimates for subelliptic operators, J. Funct. Anal., 199 (2003), 228-242. doi: 10.1016/S0022-1236(03)00093-4.
[20]	L. Wang, F. Yao, S. Zhou and H. Jia, Optimal regularity for the poisson equation, Proc. Amer. Math. Soc., 137 (2009), 2037-2047. doi: 10.1090/S0002-9939-09-09805-0.

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