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

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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

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

Abstract
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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.
[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).
[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.
[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.
[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.
[8]	T. Bieske, On infinite harmonic functions on the Heisenberg group,, Comm. Partial Differential Equations, 27 (2002), 727.
[9]	T. Bieske, Viscosity solutions on Grushin-type planes,, Illinois J. Math., 46 (2002), 893.
[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.
[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.
[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.
[13]	A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory For Their Sub-Laplacians,", Springer, (2007).
[14]	M. G. Crandall, Viscosity solutions: a primer,, In, (1660).
[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.
[16]	A. Cutrì and N. Tchou, Barrier functions for Pucci-Heisenberg operators and applications,, Int. J. Dyn. Syst. Differ. Equ., 1, 2 (2007), 117.
[17]	G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups,", Princeton University Press, (1982).
[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.
[19]	L. Hörmander, Hypoelliptic Second Order Differential Equations,, Acta Math. Uppsala, 119 (1967), 147.
[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.
[21]	H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations,, J. Diff. Eq., 83 (1990), 26.
[22]	R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations,, Arch. Rational Mech., 101 (1988), 1.
[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).
[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.

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.
[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).
[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.
[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.
[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.
[8]	T. Bieske, On infinite harmonic functions on the Heisenberg group,, Comm. Partial Differential Equations, 27 (2002), 727.
[9]	T. Bieske, Viscosity solutions on Grushin-type planes,, Illinois J. Math., 46 (2002), 893.
[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.
[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.
[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.
[13]	A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory For Their Sub-Laplacians,", Springer, (2007).
[14]	M. G. Crandall, Viscosity solutions: a primer,, In, (1660).
[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.
[16]	A. Cutrì and N. Tchou, Barrier functions for Pucci-Heisenberg operators and applications,, Int. J. Dyn. Syst. Differ. Equ., 1, 2 (2007), 117.
[17]	G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups,", Princeton University Press, (1982).
[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.
[19]	L. Hörmander, Hypoelliptic Second Order Differential Equations,, Acta Math. Uppsala, 119 (1967), 147.
[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.
[21]	H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations,, J. Diff. Eq., 83 (1990), 26.
[22]	R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations,, Arch. Rational Mech., 101 (1988), 1.
[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).
[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.

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