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Local strong solutions to the compressible viscous magnetohydrodynamic equations
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 |
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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