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Regularity and weak comparison principles for double phase quasilinear elliptic equations
Dipartimento di Matematica e Informatica, Università della Calabria, Via P. Bucci, 87036 Rende (CS), Italy |
$ \int_{\Omega}(| \nabla u|^p+a(x)| \nabla u|^q) \,{ {\rm{d}}} x, $ |
$ 1<p<q<p+1 $ |
$ a(x)\geq 0 $ |
References:
[1] |
P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), Art. 62, 48 pp.
doi: 10.1007/s00526-018-1332-z. |
[2] |
D. Castorina, G. Riey and B. Sciunzi, Hopf Lemma and Regularity Results for Quasilinear Anisotropic Elliptic Equations, Calc. Var. Partial Differential Equations, To appear. |
[3] |
M. Colombo and G. Mingione,
Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 215 (2015), 443-496.
|
[4] |
M. Colombo and G. Mingione,
Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273.
doi: 10.1007/s00205-015-0859-9. |
[5] |
M. Colombo and G. Mingione,
Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal., 270 (2016), 1416-1478.
doi: 10.1016/j.jfa.2015.06.022. |
[6] |
G. Cupini, F. Leonetti and E. Mascolo,
Existence of weak solutions for elliptic systems with $p, q$-growth conditions, Ann. Acad. Sci. Fenn. Ser A I Math., 40 (2015), 645-658.
doi: 10.5186/aasfm.2015.4035. |
[7] |
G. Cupini, P. Marcellini and E. Mascolo,
Existence for elliptic equations under $p, q$-growth, Adv. Differential Equations, 19 (2014), 693-724.
|
[8] |
L. Damascelli,
Comparison theorems for some quesilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré. Anal. non linéaire, 15 (1998), 493-516.
doi: 10.1016/S0294-1449(98)80032-2. |
[9] |
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. |
[10] |
L. Damascelli and B. Sciunzi,
Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations, Calc. Var. Partial Differential Equations, 25 (2006), 139-159.
doi: 10.1007/s00526-005-0337-6. |
[11] |
E. Di Benedetto,
$C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.
doi: 10.1016/0362-546X(83)90061-5. |
[12] |
L. Esposito, F. Leonetti and G. Mingione,
Sharp regularity for functionals with (p, q) growth, J. Differential Equations, 204 (2004), 5-55.
doi: 10.1016/j.jde.2003.11.007. |
[13] |
A. Farina, L. Montoro, G. Riey and B. Sciunzi,
Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 32 (2015), 1-22.
doi: 10.1016/j.anihpc.2013.09.005. |
[14] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983.
doi: 10.1007/978-3-642-61798-0. |
[15] |
T. Leonori, A. Porretta and G. Riey,
Comparison principles for p-Laplace equations with lower order terms, Ann. Mat. Pura Appl., 196 (2017), 877-903.
doi: 10.1007/s10231-016-0600-9. |
[16] |
P. Marcellini,
Regularity of minimizers of integrals of the calculus of variations with non standard growth cconditions, Arch. Ration. Mech. Anal., 105 (1989), 267-284.
doi: 10.1007/BF00251503. |
[17] |
P. Marcellini,
Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions, J. Differential Equations, 90 (1991), 1-30.
doi: 10.1016/0022-0396(91)90158-6. |
[18] |
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. |
[19] |
N. G. Meyers and J. Serrin, H = W, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 1055-1056.
doi: 10.1073/pnas.51.6.1055. |
[20] |
G. Montoro, G. Riey and B. Sciunzi,
Qualitative properties of positive solutions to systems of quasilinear elliptic equations, Adv. Differential Equations, 20 (2015), 717-740.
|
[21] |
M. K. V. Murthy and G. Stampacchia,
Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 1-122.
doi: 10.1007/BF02413623. |
[22] |
P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007. |
[23] |
G. Riey,
Boundary regularity for quasi-linear elliptic equations with lower order term, Electron. J. Differential Equations, 283 (2017), 1-9.
|
[24] |
G. Riey and B. Sciunzi,
A note on the boundary regularity of solutions to quasilinear elliptic equations, ESAIM Control Optim. Calc. Var., 24 (2018), 849-858.
doi: 10.1051/cocv/2017040. |
[25] |
B. Sciunzi,
Some results on the qualitative properties of positive solutions of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 315-334.
doi: 10.1007/s00030-007-5047-7. |
[26] |
B. Sciunzi, Regularity and comparison principles for p-Laplace equations with vanishing source term, Commum. Contemp. Math., 16 (2014), 1450013, 20pp.
doi: 10.1142/S0219199714500138. |
[27] |
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. |
[28] |
N. S. Trudinger,
Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 265-308.
|
[29] |
V. V. Zhykov,
Averaging of functional of the calculus of variations and elasticity theory, Izk. Akad. Nauk. SSSR Ser. Mat., 50 (1986), 675-710.
|
[30] |
V. V. Zhykov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.
doi: 10.1007/978-3-642-84659-5. |
show all references
References:
[1] |
P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), Art. 62, 48 pp.
doi: 10.1007/s00526-018-1332-z. |
[2] |
D. Castorina, G. Riey and B. Sciunzi, Hopf Lemma and Regularity Results for Quasilinear Anisotropic Elliptic Equations, Calc. Var. Partial Differential Equations, To appear. |
[3] |
M. Colombo and G. Mingione,
Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 215 (2015), 443-496.
|
[4] |
M. Colombo and G. Mingione,
Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273.
doi: 10.1007/s00205-015-0859-9. |
[5] |
M. Colombo and G. Mingione,
Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal., 270 (2016), 1416-1478.
doi: 10.1016/j.jfa.2015.06.022. |
[6] |
G. Cupini, F. Leonetti and E. Mascolo,
Existence of weak solutions for elliptic systems with $p, q$-growth conditions, Ann. Acad. Sci. Fenn. Ser A I Math., 40 (2015), 645-658.
doi: 10.5186/aasfm.2015.4035. |
[7] |
G. Cupini, P. Marcellini and E. Mascolo,
Existence for elliptic equations under $p, q$-growth, Adv. Differential Equations, 19 (2014), 693-724.
|
[8] |
L. Damascelli,
Comparison theorems for some quesilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré. Anal. non linéaire, 15 (1998), 493-516.
doi: 10.1016/S0294-1449(98)80032-2. |
[9] |
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. |
[10] |
L. Damascelli and B. Sciunzi,
Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations, Calc. Var. Partial Differential Equations, 25 (2006), 139-159.
doi: 10.1007/s00526-005-0337-6. |
[11] |
E. Di Benedetto,
$C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.
doi: 10.1016/0362-546X(83)90061-5. |
[12] |
L. Esposito, F. Leonetti and G. Mingione,
Sharp regularity for functionals with (p, q) growth, J. Differential Equations, 204 (2004), 5-55.
doi: 10.1016/j.jde.2003.11.007. |
[13] |
A. Farina, L. Montoro, G. Riey and B. Sciunzi,
Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 32 (2015), 1-22.
doi: 10.1016/j.anihpc.2013.09.005. |
[14] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983.
doi: 10.1007/978-3-642-61798-0. |
[15] |
T. Leonori, A. Porretta and G. Riey,
Comparison principles for p-Laplace equations with lower order terms, Ann. Mat. Pura Appl., 196 (2017), 877-903.
doi: 10.1007/s10231-016-0600-9. |
[16] |
P. Marcellini,
Regularity of minimizers of integrals of the calculus of variations with non standard growth cconditions, Arch. Ration. Mech. Anal., 105 (1989), 267-284.
doi: 10.1007/BF00251503. |
[17] |
P. Marcellini,
Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions, J. Differential Equations, 90 (1991), 1-30.
doi: 10.1016/0022-0396(91)90158-6. |
[18] |
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. |
[19] |
N. G. Meyers and J. Serrin, H = W, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 1055-1056.
doi: 10.1073/pnas.51.6.1055. |
[20] |
G. Montoro, G. Riey and B. Sciunzi,
Qualitative properties of positive solutions to systems of quasilinear elliptic equations, Adv. Differential Equations, 20 (2015), 717-740.
|
[21] |
M. K. V. Murthy and G. Stampacchia,
Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 1-122.
doi: 10.1007/BF02413623. |
[22] |
P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007. |
[23] |
G. Riey,
Boundary regularity for quasi-linear elliptic equations with lower order term, Electron. J. Differential Equations, 283 (2017), 1-9.
|
[24] |
G. Riey and B. Sciunzi,
A note on the boundary regularity of solutions to quasilinear elliptic equations, ESAIM Control Optim. Calc. Var., 24 (2018), 849-858.
doi: 10.1051/cocv/2017040. |
[25] |
B. Sciunzi,
Some results on the qualitative properties of positive solutions of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 315-334.
doi: 10.1007/s00030-007-5047-7. |
[26] |
B. Sciunzi, Regularity and comparison principles for p-Laplace equations with vanishing source term, Commum. Contemp. Math., 16 (2014), 1450013, 20pp.
doi: 10.1142/S0219199714500138. |
[27] |
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. |
[28] |
N. S. Trudinger,
Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 265-308.
|
[29] |
V. V. Zhykov,
Averaging of functional of the calculus of variations and elasticity theory, Izk. Akad. Nauk. SSSR Ser. Mat., 50 (1986), 675-710.
|
[30] |
V. V. Zhykov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.
doi: 10.1007/978-3-642-84659-5. |
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