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January  2014, 13(1): 119-133. doi: 10.3934/cpaa.2014.13.119

The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction

Paola Mannucci 1,

1. 

Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, Via Belzoni, 7, 35131, Padova

Received  May 2012 Revised  August 2012 Published  July 2013

We prove a comparison principle for viscosity solutions of a fully nonlinear equation satisfying a condition of non-degeneracy in a fixed direction. We apply these results to prove that a continuous solution of the corresponding Dirichlet problem exists. To obtain the existence of barrier functions and well-posedness, we find suitable explicit assumptions on the domain and on the ellipticity constants of the operator.
Keywords: degenerate elliptic equation, Viscosity solution, Pucci's operators, comparison principle, Dirichlet problem, subelliptic equation, Carnot groups..
Mathematics Subject Classification: 35J70, 35H20, 35J6.
Citation: Paola Mannucci. The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction. Communications on Pure & Applied Analysis, 2014, 13 (1) : 119-133. doi: 10.3934/cpaa.2014.13.119
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), 13-39. 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, Boston, MA, 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), 709-731.  Google Scholar

[4]

M. Bardi and P. Mannucci, Comparison principles for subelliptic equations of Monge-Ampère type, Boll. Unione Mat. Ital., 9 (2008), 489-495. 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), 41-51. 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., published online May 2013, 2013-0067. doi: DOI: 10.1515/forum-2013-0067.  Google Scholar

[7]

F. H. Beatrous, T. J. Bieske and J. J. Manfredi, The maximum principle for vector fields, in "The $p$-harmonic Equation and Recent Advances in Analysis," Contemp. Math., 370, Google Scholar

[8]

T. Bieske, On infinite harmonic functions on the Heisenberg group, Comm. Partial Differential Equations, 27 (2002), 727-761. Google Scholar

[9]

T. Bieske, Viscosity solutions on Grushin-type planes, Illinois J. Math., 46 (2002), 893-911. 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), 795-823. 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), 1123-1157. 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, Lect. Notes Semin. Interdiscip. Mat., Potenza, 6 (2007), 49-55. Google Scholar

[13]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory For Their Sub-Laplacians," Springer, Berlin, 2007. Google Scholar

[14]

M. G. Crandall, Viscosity solutions: a primer, In "Viscosity Solutions and Applications," Lecture Notes in Mathematics, 1660. Springer, Berlin; C.I.M.E., Florence, 1997. 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), 1-67. 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), 117-131. Google Scholar

[17]

G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups," Princeton University Press, Princeton, 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), 1305-1334. Google Scholar

[19]

L. Hörmander, Hypoelliptic Second Order Differential Equations, Acta Math. Uppsala, 119 (1967), 147-17 . 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, Theory, Methods & Appl., 24 (1995), 147-158. 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), 26-78. Google Scholar

[22]

R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech., 101 (1988), 1-27. 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, Trento, May 2003, available at http://www.pitt.edu/ manfredi/. 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), 91-113. 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), 13-39. 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, Boston, MA, 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), 709-731.  Google Scholar

[4]

M. Bardi and P. Mannucci, Comparison principles for subelliptic equations of Monge-Ampère type, Boll. Unione Mat. Ital., 9 (2008), 489-495. 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), 41-51. 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., published online May 2013, 2013-0067. doi: DOI: 10.1515/forum-2013-0067.  Google Scholar

[7]

F. H. Beatrous, T. J. Bieske and J. J. Manfredi, The maximum principle for vector fields, in "The $p$-harmonic Equation and Recent Advances in Analysis," Contemp. Math., 370, Google Scholar

[8]

T. Bieske, On infinite harmonic functions on the Heisenberg group, Comm. Partial Differential Equations, 27 (2002), 727-761. Google Scholar

[9]

T. Bieske, Viscosity solutions on Grushin-type planes, Illinois J. Math., 46 (2002), 893-911. 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), 795-823. 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), 1123-1157. 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, Lect. Notes Semin. Interdiscip. Mat., Potenza, 6 (2007), 49-55. Google Scholar

[13]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory For Their Sub-Laplacians," Springer, Berlin, 2007. Google Scholar

[14]

M. G. Crandall, Viscosity solutions: a primer, In "Viscosity Solutions and Applications," Lecture Notes in Mathematics, 1660. Springer, Berlin; C.I.M.E., Florence, 1997. 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), 1-67. 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), 117-131. Google Scholar

[17]

G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups," Princeton University Press, Princeton, 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), 1305-1334. Google Scholar

[19]

L. Hörmander, Hypoelliptic Second Order Differential Equations, Acta Math. Uppsala, 119 (1967), 147-17 . 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, Theory, Methods & Appl., 24 (1995), 147-158. 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), 26-78. Google Scholar

[22]

R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech., 101 (1988), 1-27. 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, Trento, May 2003, available at http://www.pitt.edu/ manfredi/. 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), 91-113. Google Scholar

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