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Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities
The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction
1. | Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, Via Belzoni, 7, 35131, Padova |
References:
[1] |
M. Bardi and S. Bottacin, On the Dirichlet problem for nonlinear degenerate elliptic equations and applications to optimal control,, Rend. Sem. Mat. Univ. Pol. Torino, 56 (1998), 13. Google Scholar |
[2] |
M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi Bellman Equations,", Systems and Control: Foundations and Applications. Birkhauser, (1997). Google Scholar |
[3] |
M. Bardi and P. Mannucci, On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations,, Commun. Pure Appl. Anal., 5 (2006), 709.
|
[4] |
M. Bardi and P. Mannucci, Comparison principles for subelliptic equations of Monge-Ampère type,, Boll. Unione Mat. Ital., 9 (2008), 489. Google Scholar |
[5] |
M. Bardi and P. Mannucci, Comparison principles for equations of Monge-Ampère type in Carnot groups: a direct proof,, Lecture Notes of Seminario Interdisciplinare di Matematica, 7 (2008), 41. Google Scholar |
[6] |
M. Bardi and P. Mannucci, Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of Monge-Ampère type,, to appear on Forum Math., (2013), 2013.
doi: DOI: 10.1515/forum-2013-0067. |
[7] |
F. H. Beatrous, T. J. Bieske and J. J. Manfredi, The maximum principle for vector fields, in, Amer. Math. Soc., Providence, RI, 2005, 1-9., 370 (2005), 1. Google Scholar |
[8] |
T. Bieske, On infinite harmonic functions on the Heisenberg group,, Comm. Partial Differential Equations, 27 (2002), 727. Google Scholar |
[9] |
T. Bieske, Viscosity solutions on Grushin-type planes,, Illinois J. Math., 46 (2002), 893. Google Scholar |
[10] |
T. Bieske and L. Capogna, The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathodory metrics,, Trans. Amer. Math. Soc., 357 (2005), 795. Google Scholar |
[11] |
I. Birindelli, I. Capuzzo Dolcetta and A. Cutrì, Indefinite semi-linear equations on the Heisenberg group: a priori bounds and existence,, Comm. Partial Differential Equations, 23 (1998), 1123. Google Scholar |
[12] |
I. Birindelli and B. Stroffolini, Existence theorems for fully nonlinear equations in the Heisenberg group,, Subelliptic PDE's and applications to geometry and finance, 6 (2007), 49. Google Scholar |
[13] |
A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory For Their Sub-Laplacians,", Springer, (2007). Google Scholar |
[14] |
M. G. Crandall, Viscosity solutions: a primer,, In, (1660). Google Scholar |
[15] |
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. Google Scholar |
[16] |
A. Cutrì and N. Tchou, Barrier functions for Pucci-Heisenberg operators and applications,, Int. J. Dyn. Syst. Differ. Equ., 1, 2 (2007), 117. Google Scholar |
[17] |
G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups,", Princeton University Press, (1982). Google Scholar |
[18] |
C. E. Gutierrez and A. Montanari, Maximum and comparison principles for convex functions on the Heisenberg group,, Comm. Partial Differential Equations, 29 (2004), 1305. Google Scholar |
[19] |
L. Hörmander, Hypoelliptic Second Order Differential Equations,, Acta Math. Uppsala, 119 (1967), 147. Google Scholar |
[20] |
M. A. Katsoulakis, A representation formula and regularizing properties for viscosity solutions of second-order fully nonlinear degenerate parabolic equations,, Nonlinear Analysis, 24 (1995), 147. Google Scholar |
[21] |
H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations,, J. Diff. Eq., 83 (1990), 26. Google Scholar |
[22] |
R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations,, Arch. Rational Mech., 101 (1988), 1. Google Scholar |
[23] |
J. J. Manfredi, Nonlinear subelliptic equations on Carnot groups,, Notes of a course given at the Third School on Analysis and Geometry in Metric Spaces, (2003). Google Scholar |
[24] |
C. Y. Wang, The Aronsson equation for absolute minimizers of $L^\infty$-functionals associated with vector fields satisfying Hörmander's condition,, Trans. Amer. Math. Soc. 359, 1 (2007), 91. Google Scholar |
show all references
References:
[1] |
M. Bardi and S. Bottacin, On the Dirichlet problem for nonlinear degenerate elliptic equations and applications to optimal control,, Rend. Sem. Mat. Univ. Pol. Torino, 56 (1998), 13. Google Scholar |
[2] |
M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi Bellman Equations,", Systems and Control: Foundations and Applications. Birkhauser, (1997). Google Scholar |
[3] |
M. Bardi and P. Mannucci, On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations,, Commun. Pure Appl. Anal., 5 (2006), 709.
|
[4] |
M. Bardi and P. Mannucci, Comparison principles for subelliptic equations of Monge-Ampère type,, Boll. Unione Mat. Ital., 9 (2008), 489. Google Scholar |
[5] |
M. Bardi and P. Mannucci, Comparison principles for equations of Monge-Ampère type in Carnot groups: a direct proof,, Lecture Notes of Seminario Interdisciplinare di Matematica, 7 (2008), 41. Google Scholar |
[6] |
M. Bardi and P. Mannucci, Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of Monge-Ampère type,, to appear on Forum Math., (2013), 2013.
doi: DOI: 10.1515/forum-2013-0067. |
[7] |
F. H. Beatrous, T. J. Bieske and J. J. Manfredi, The maximum principle for vector fields, in, Amer. Math. Soc., Providence, RI, 2005, 1-9., 370 (2005), 1. Google Scholar |
[8] |
T. Bieske, On infinite harmonic functions on the Heisenberg group,, Comm. Partial Differential Equations, 27 (2002), 727. Google Scholar |
[9] |
T. Bieske, Viscosity solutions on Grushin-type planes,, Illinois J. Math., 46 (2002), 893. Google Scholar |
[10] |
T. Bieske and L. Capogna, The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathodory metrics,, Trans. Amer. Math. Soc., 357 (2005), 795. Google Scholar |
[11] |
I. Birindelli, I. Capuzzo Dolcetta and A. Cutrì, Indefinite semi-linear equations on the Heisenberg group: a priori bounds and existence,, Comm. Partial Differential Equations, 23 (1998), 1123. Google Scholar |
[12] |
I. Birindelli and B. Stroffolini, Existence theorems for fully nonlinear equations in the Heisenberg group,, Subelliptic PDE's and applications to geometry and finance, 6 (2007), 49. Google Scholar |
[13] |
A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory For Their Sub-Laplacians,", Springer, (2007). Google Scholar |
[14] |
M. G. Crandall, Viscosity solutions: a primer,, In, (1660). Google Scholar |
[15] |
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. Google Scholar |
[16] |
A. Cutrì and N. Tchou, Barrier functions for Pucci-Heisenberg operators and applications,, Int. J. Dyn. Syst. Differ. Equ., 1, 2 (2007), 117. Google Scholar |
[17] |
G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups,", Princeton University Press, (1982). Google Scholar |
[18] |
C. E. Gutierrez and A. Montanari, Maximum and comparison principles for convex functions on the Heisenberg group,, Comm. Partial Differential Equations, 29 (2004), 1305. Google Scholar |
[19] |
L. Hörmander, Hypoelliptic Second Order Differential Equations,, Acta Math. Uppsala, 119 (1967), 147. Google Scholar |
[20] |
M. A. Katsoulakis, A representation formula and regularizing properties for viscosity solutions of second-order fully nonlinear degenerate parabolic equations,, Nonlinear Analysis, 24 (1995), 147. Google Scholar |
[21] |
H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations,, J. Diff. Eq., 83 (1990), 26. Google Scholar |
[22] |
R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations,, Arch. Rational Mech., 101 (1988), 1. Google Scholar |
[23] |
J. J. Manfredi, Nonlinear subelliptic equations on Carnot groups,, Notes of a course given at the Third School on Analysis and Geometry in Metric Spaces, (2003). Google Scholar |
[24] |
C. Y. Wang, The Aronsson equation for absolute minimizers of $L^\infty$-functionals associated with vector fields satisfying Hörmander's condition,, Trans. Amer. Math. Soc. 359, 1 (2007), 91. Google Scholar |
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