- 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] |
A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 |
[2] |
Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024 |
[3] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 |
[4] |
Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 |
[5] |
Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637 |
[6] |
Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 |
[7] |
Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 |
[8] |
Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451 |
[9] |
Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 |
[10] |
W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349 |
[11] |
Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341 |
[12] |
John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021004 |
[13] |
Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533 |
[14] |
Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021004 |
[15] |
Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021009 |
[16] |
Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021 |
[17] |
Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243 |
[18] |
Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 |
[19] |
Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184 |
[20] |
Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]