- Previous Article
- DCDS Home
- This Issue
-
Next Article
Convergence rates in homogenization of higher-order parabolic systems
Second order regularity for degenerate nonlinear elliptic equations
| 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 |
We investigate the second order regularity of solutions to degenerate nonlinear elliptic equations.
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. |
| [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. |
| [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. |
| [4] |
T. Kuusi and G. Mingione,
Universal potential estimates, J. Funct. Anal., 262 (2012), 4205-4269.
doi: 10.1016/j.jfa.2012.02.018. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [9] |
P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Boston, 2007.
|
| [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. |
| [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. |
| [12] |
E. Teixeira,
Regularity for quasilinear equations on degenerate singular sets, Math. Ann., 358 (2014), 241-256.
doi: 10.1007/s00208-013-0959-5. |
| [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. |
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. |
| [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. |
| [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. |
| [4] |
T. Kuusi and G. Mingione,
Universal potential estimates, J. Funct. Anal., 262 (2012), 4205-4269.
doi: 10.1016/j.jfa.2012.02.018. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [9] |
P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Boston, 2007.
|
| [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. |
| [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. |
| [12] |
E. Teixeira,
Regularity for quasilinear equations on degenerate singular sets, Math. Ann., 358 (2014), 241-256.
doi: 10.1007/s00208-013-0959-5. |
| [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. |
| [1] |
Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 |
| [2] |
Xavier Cabré. Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 331-359. doi: 10.3934/dcds.2002.8.331 |
| [3] |
Guji Tian, Xu-Jia Wang. Partial regularity for elliptic equations. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 899-913. doi: 10.3934/dcds.2010.28.899 |
| [4] |
Sun-Sig Byun, Hongbin Chen, Mijoung Kim, Lihe Wang. Lp regularity theory for linear elliptic systems. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 121-134. doi: 10.3934/dcds.2007.18.121 |
| [5] |
Huilian Jia, Lihe Wang, Fengping Yao, Shulin Zhou. Regularity theory in Orlicz spaces for the poisson and heat equations. Communications on Pure and Applied Analysis, 2008, 7 (2) : 407-416. doi: 10.3934/cpaa.2008.7.407 |
| [6] |
Junjie Zhang, Shenzhou Zheng, Chunyan Zuo. $ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3305-3318. doi: 10.3934/dcdss.2021080 |
| [7] |
Shuhong Chen, Zhong Tan. Optimal interior partial regularity for nonlinear elliptic systems. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 981-993. doi: 10.3934/dcds.2010.27.981 |
| [8] |
Luigi C. Berselli, Carlo R. Grisanti. On the regularity up to the boundary for certain nonlinear elliptic systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 53-71. doi: 10.3934/dcdss.2016.9.53 |
| [9] |
Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure and Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425 |
| [10] |
Luisa Fattorusso, Antonio Tarsia. Regularity in Campanato spaces for solutions of fully nonlinear elliptic systems. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1307-1323. doi: 10.3934/dcds.2011.31.1307 |
| [11] |
Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685 |
| [12] |
Shuhong Chen, Zhong Tan. Optimal partial regularity results for nonlinear elliptic systems in Carnot groups. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3391-3405. doi: 10.3934/dcds.2013.33.3391 |
| [13] |
Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703 |
| [14] |
Geng Chen, Yannan Shen. Existence and regularity of solutions in nonlinear wave equations. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3327-3342. doi: 10.3934/dcds.2015.35.3327 |
| [15] |
Kunquan Lan, Wei Lin. Uniqueness of nonzero positive solutions of Laplacian elliptic equations arising in combustion theory. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 849-861. doi: 10.3934/dcdsb.2016.21.849 |
| [16] |
Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121 |
| [17] |
Sunra J. N. Mosconi. Optimal elliptic regularity: A comparison between local and nonlocal equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 547-559. doi: 10.3934/dcdss.2018030 |
| [18] |
Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure and Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631 |
| [19] |
Giuseppe Riey. Regularity and weak comparison principles for double phase quasilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4863-4873. doi: 10.3934/dcds.2019198 |
| [20] |
Ugur G. Abdulla. Regularity of $\infty$ for elliptic equations with measurable coefficients and its consequences. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3379-3397. doi: 10.3934/dcds.2012.32.3379 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]