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On the growth of positive entire solutions of elliptic PDEs and their gradients
1. | Department of Mathematics, University of Salerno, via Giovanni Paolo II n.132, 84084 Fisciano (SA), Italy |
References:
[1] |
M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second order elliptic equations in unbounded domains, Abstr. Appl. Anal., (2008), Art. ID 178534, 19 pp.
doi: 10.1155/2008/178534. |
[2] |
L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213.
doi: 10.2307/1971480. |
[3] |
L. A. Caffarelli and X. Cabrè, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995. |
[4] |
L. A. Caffarelli, M. G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49 (1996), 365-397.
doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. |
[5] |
I. Capuzzo Dolcetta, F. Leoni and A. Vitolo, The Alexandrov-Bakelman-Pucci weak maximum principle for fully nonlinear equations in unbounded domains, Commun. Partial Differ. Equations, 30 (2005), 1863-1881.
doi: 10.1080/03605300500300030. |
[6] |
I. Capuzzo Dolcetta and A. Vitolo, A qualitative Phragmèn-Lindelöf theorem for fully nonlinear elliptic equations, J. Differential Equations, 243 (2007), 578-592.
doi: 10.1016/j.jde.2007.08.001. |
[7] |
I. Capuzzo Dolcetta and A. Vitolo, Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations, Discrete Contin. Dyn. Syst., 28 (2010), 539-557.
doi: 10.3934/dcds.2010.28.539. |
[8] |
M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[9] |
M. G. Crandall, M. Kocan, P. L. Lions and A. Swiech, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electron. J. Differential Equations, 24 (1999), 1-22. |
[10] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin-New York, 1983.
doi: 10.1007/978-3-642-61798-0. |
[11] |
S. Koike, A Beginners Guide to the Theory of Viscosity Solutions, MSJ Memoirs 13, Math. Soc. Japan, Tokyo, 2004. |
[12] |
S. Koike and A. Swiech, Maximum principle for fully nonlinear equations via the iterated comparison function method, Math. Ann., 339 (2007), 461-484.
doi: 10.1007/s00208-007-0125-z. |
[13] |
S. Koike and T. Takahashi, Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients, Adv. Differential Equations, 7 (2002), 493-512. |
[14] |
E. Landau, Einige Ungleichungen für zweimal differenzierbare Funktionen, Proc. London Math. Soc., 13 (1913), 43-49.
doi: 10.1112/plms/s2-13.1.43. |
[15] |
Y. Y. Li and L. Nirenberg, Generalization of a well-known inequality, Progress in Nonlinear Differential Equations and Their Applications, 66 (2006), 365-370.
doi: 10.1007/3-7643-7401-2_24. |
[16] |
Y. Y. Li and L. Nirenberg, A miscellany, in Percorsi incrociati (in ricordo di Vittorio Cafagna) (eds I. Capuzzo Dolcetta, M. Transirico and A. Vitolo), Collana Scientifica di Ateneo, Università di Salerno, 2010, 193-208. |
[17] |
V. G. Maz'ya and T. O. Shaposhnikova, Sharp pointwise interpolation inequalities for derivatives, Funct. Anal. Appl., 36 (2002), 30-48.
doi: 10.1023/A:1014478100799. |
[18] |
A. Swiech, $W^{1,p}$-Interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027. |
[19] |
A. Vitolo, On the maximum principle for complete second-order elliptic operators in general domains, J. Differential Equations, 194 (2003), 166-184.
doi: 10.1016/S0022-0396(03)00193-1. |
[20] |
A. Vitolo, On the Phragmèn-Lindelöf principle for second-order elliptic equations, J. Math. Anal. Appl., 300 (2004), 244-259.
doi: 10.1016/j.jmaa.2004.04.067. |
show all references
References:
[1] |
M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second order elliptic equations in unbounded domains, Abstr. Appl. Anal., (2008), Art. ID 178534, 19 pp.
doi: 10.1155/2008/178534. |
[2] |
L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213.
doi: 10.2307/1971480. |
[3] |
L. A. Caffarelli and X. Cabrè, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995. |
[4] |
L. A. Caffarelli, M. G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49 (1996), 365-397.
doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. |
[5] |
I. Capuzzo Dolcetta, F. Leoni and A. Vitolo, The Alexandrov-Bakelman-Pucci weak maximum principle for fully nonlinear equations in unbounded domains, Commun. Partial Differ. Equations, 30 (2005), 1863-1881.
doi: 10.1080/03605300500300030. |
[6] |
I. Capuzzo Dolcetta and A. Vitolo, A qualitative Phragmèn-Lindelöf theorem for fully nonlinear elliptic equations, J. Differential Equations, 243 (2007), 578-592.
doi: 10.1016/j.jde.2007.08.001. |
[7] |
I. Capuzzo Dolcetta and A. Vitolo, Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations, Discrete Contin. Dyn. Syst., 28 (2010), 539-557.
doi: 10.3934/dcds.2010.28.539. |
[8] |
M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[9] |
M. G. Crandall, M. Kocan, P. L. Lions and A. Swiech, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electron. J. Differential Equations, 24 (1999), 1-22. |
[10] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin-New York, 1983.
doi: 10.1007/978-3-642-61798-0. |
[11] |
S. Koike, A Beginners Guide to the Theory of Viscosity Solutions, MSJ Memoirs 13, Math. Soc. Japan, Tokyo, 2004. |
[12] |
S. Koike and A. Swiech, Maximum principle for fully nonlinear equations via the iterated comparison function method, Math. Ann., 339 (2007), 461-484.
doi: 10.1007/s00208-007-0125-z. |
[13] |
S. Koike and T. Takahashi, Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients, Adv. Differential Equations, 7 (2002), 493-512. |
[14] |
E. Landau, Einige Ungleichungen für zweimal differenzierbare Funktionen, Proc. London Math. Soc., 13 (1913), 43-49.
doi: 10.1112/plms/s2-13.1.43. |
[15] |
Y. Y. Li and L. Nirenberg, Generalization of a well-known inequality, Progress in Nonlinear Differential Equations and Their Applications, 66 (2006), 365-370.
doi: 10.1007/3-7643-7401-2_24. |
[16] |
Y. Y. Li and L. Nirenberg, A miscellany, in Percorsi incrociati (in ricordo di Vittorio Cafagna) (eds I. Capuzzo Dolcetta, M. Transirico and A. Vitolo), Collana Scientifica di Ateneo, Università di Salerno, 2010, 193-208. |
[17] |
V. G. Maz'ya and T. O. Shaposhnikova, Sharp pointwise interpolation inequalities for derivatives, Funct. Anal. Appl., 36 (2002), 30-48.
doi: 10.1023/A:1014478100799. |
[18] |
A. Swiech, $W^{1,p}$-Interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027. |
[19] |
A. Vitolo, On the maximum principle for complete second-order elliptic operators in general domains, J. Differential Equations, 194 (2003), 166-184.
doi: 10.1016/S0022-0396(03)00193-1. |
[20] |
A. Vitolo, On the Phragmèn-Lindelöf principle for second-order elliptic equations, J. Math. Anal. Appl., 300 (2004), 244-259.
doi: 10.1016/j.jmaa.2004.04.067. |
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