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August  2018, 38(8): 4231-4242. doi: 10.3934/dcds.2018184

Second order regularity for degenerate nonlinear elliptic equations

Annamaria Canino 1, , Elisa De Giorgio 2, and  Berardino Sciunzi 1,

1. 

Dipartimento di Matematica e Informatica, UNICAL, Ponte Pietro Bucci 31B, 87036 Arcavacata di Rende, Cosenza, Italy
2. 

Dipartimento di Fisica, UNICAL, Ponte Pietro Bucci 33B, 87036 Arcavacata di Rende, Cosenza, Italy

Received  March 2018 Revised  April 2018 Published  May 2018

We investigate the second order regularity of solutions to degenerate nonlinear elliptic equations.

Keywords: Nonlinear elliptic equations, regularity theory.
Mathematics Subject Classification: 35J70, 35J62, 35B06.
Citation: Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184
References:
[1]

A. Canino, P. Le and B. Sciunzi, Local $W_{loc}^{2, m\left(\cdot \right)}$ regularity for p(·)-Laplace equations, Manuscripta Mathematica, 140 (2013), 481-496.  doi: 10.1007/s00229-012-0549-y.  Google Scholar

[2]

L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, J. Differential Equations, 206 (2004), 483-515.  doi: 10.1016/j.jde.2004.05.012.  Google Scholar

[3]

E. Di Benedetto, $C^{1+α}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[4]

T. Kuusi and G. Mingione, Universal potential estimates, J. Funct. Anal., 262 (2012), 4205-4269.  doi: 10.1016/j.jfa.2012.02.018.  Google Scholar

[5]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar

[6]

C. Mercuri, G. Riey and B. Sciunzi, A regularity result for the p-Laplacian near uniform ellipticity, Siam J. Math. Anal., 48 (2016), 2059-2075.  doi: 10.1137/16M1058546.  Google Scholar

[7]

G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations, Applications of Mathematics, 51 (2006), 355-426.  doi: 10.1007/s10778-006-0110-3.  Google Scholar

[8]

G. Mingione, The Calderon-Zygmund theory for elliptic problems with measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(5), 6 (2007), 195-261.   Google Scholar

[9] P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Boston, 2007.   Google Scholar
[10]

B. Sciunzi, Some results on the qualitative properties of positive solutions of quasilinear elliptic equations, NoDEA. Nonlinear Differential Equations and Applications, 14 (2007), 315-334.  doi: 10.1007/s00030-007-5047-7.  Google Scholar

[11]

B. Sciunzi, Regularity and comparison principles for p-Laplace equations with vanishing source term, Comm. Cont. Math., 16 (2014), 450013, 20pp.  doi: 10.1142/S0219199714500138.  Google Scholar

[12]

E. Teixeira, Regularity for quasilinear equations on degenerate singular sets, Math. Ann., 358 (2014), 241-256.  doi: 10.1007/s00208-013-0959-5.  Google Scholar

[13]

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.  Google Scholar

show all references

References:
[1]

A. Canino, P. Le and B. Sciunzi, Local $W_{loc}^{2, m\left(\cdot \right)}$ regularity for p(·)-Laplace equations, Manuscripta Mathematica, 140 (2013), 481-496.  doi: 10.1007/s00229-012-0549-y.  Google Scholar

[2]

L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, J. Differential Equations, 206 (2004), 483-515.  doi: 10.1016/j.jde.2004.05.012.  Google Scholar

[3]

E. Di Benedetto, $C^{1+α}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[4]

T. Kuusi and G. Mingione, Universal potential estimates, J. Funct. Anal., 262 (2012), 4205-4269.  doi: 10.1016/j.jfa.2012.02.018.  Google Scholar

[5]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar

[6]

C. Mercuri, G. Riey and B. Sciunzi, A regularity result for the p-Laplacian near uniform ellipticity, Siam J. Math. Anal., 48 (2016), 2059-2075.  doi: 10.1137/16M1058546.  Google Scholar

[7]

G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations, Applications of Mathematics, 51 (2006), 355-426.  doi: 10.1007/s10778-006-0110-3.  Google Scholar

[8]

G. Mingione, The Calderon-Zygmund theory for elliptic problems with measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(5), 6 (2007), 195-261.   Google Scholar

[9] P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Boston, 2007.   Google Scholar
[10]

B. Sciunzi, Some results on the qualitative properties of positive solutions of quasilinear elliptic equations, NoDEA. Nonlinear Differential Equations and Applications, 14 (2007), 315-334.  doi: 10.1007/s00030-007-5047-7.  Google Scholar

[11]

B. Sciunzi, Regularity and comparison principles for p-Laplace equations with vanishing source term, Comm. Cont. Math., 16 (2014), 450013, 20pp.  doi: 10.1142/S0219199714500138.  Google Scholar

[12]

E. Teixeira, Regularity for quasilinear equations on degenerate singular sets, Math. Ann., 358 (2014), 241-256.  doi: 10.1007/s00208-013-0959-5.  Google Scholar

[13]

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.  Google Scholar

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