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May  2019, 18(3): 1091-1115. doi: 10.3934/cpaa.2019053

## A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group

 Universidad Nacional de Cuyo-CONICET, Parque Gral. San Martin. M5502JMA Mendoza. Argentina

* Corresponding author

Received  December 2017 Revised  July 2018 Published  November 2018

Fund Project: The authors are supported by grants PICT 2015-1701 AGENCIA, and SECTyP UNCuyo B051

In this manuscript, we are interested in the study of existence, uniqueness and comparison of viscosity and weak solutions for quasilinear equations in the Heisenberg group. In particular, we highlight the limitation of applying the Euclidean theory of viscosity solutions to get comparison of solutions of sub-elliptic equations in the Heisenberg group. Moreover, we will be concerned with the equivalence of different notions of weak solutions under appropriate assumptions for the operators under analysis. We end the paper with an application to a Radó property.

Citation: Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053
##### References:
 [1] 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. doi: 10.3934/cpaa.2006.5.709. [2] T. Bieske, A sub-Riemannian maximum principle and its application to the p-Laplacian in Carnot groups, Annales Academiae Scientiarum Fennicae Mathematica, 37 (2012), 119-134. doi: 10.5186/aasfm.2012.3706. [3] T. Bieske, Equivalence of weak and viscosity for the p-Laplace equation in the Heisenberg group, Annales Academiae Scientiarum Fennicae Mathematica, 31 (2006), 363-379. [4] T. Bieske, On ∞-harmonic functions on the Heisenberg group, Communications in PDE, 27 (2002), 727-761. doi: 10.1081/PDE-120002872. [5] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, AMS, Colloquium Publications, Vol. 43, USA, 1995. doi: 10.1090/coll/043. [6] L. Capogna, D. Danielli, S. Pauls and J. Tyson, An introduction to the Heisenberg group and the sub-Riemannian Isoperimetric problem, Basel: Progress in Mathematics, Vol. 259, Birkhauser-Verlag, 2007. [7] L. Capogna, Regularity of quasi-linear equations in the Heisenberg group, Communications on Pure and Applied Analysis, 50 (1998), 867-889. doi: 10.1002/(SICI)1097-0312(199709)50:9<867::AID-CPA3>3.0.CO;2-3. [8] G. Citti, B. Franceschiello, G. Sanguinetti and A. Sarti, Sub-Riemannian mean curvature flow for image processing, SIAM Journal on Imaging Sciences, 9 (2016), 212-237. doi: 10.1137/15M1013572. [9] G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space, in Proceedings of the Workshop on Second Order Subelliptic Equations and Applications, 2003. [10] G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space, Journal of Mathematical Imaging and Vision, 24 (2006), 307-326. doi: 10.1007/s10851-005-3630-2. [11] M. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [12] Y. Dong, G. Lu and L. Sun, Global Poincaré Inequalities on the Heisenberg group and applications, Acta Mathematica Sinica, English Series, (2006). doi: 10.1007/s10114-005-0874-0. [13] L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton FL, 1992. [14] F. Ferrari, Q. Liu and J. Manfredi, On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group, Communications in Contemporary Mathematics, 16 (2014), 1350027. doi: 10.1142/S0219199713500272. [15] G. Folland and E. Stein, Hardy-spaces on Homogeneous Groups, Princeton University Press, 1982. [16] B. Franchi, R. Serapioni and F. Serra-Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated to Lipschitz continuous vector fields, Bollettino U.M.I., 11B (2001), 83-117. [17] N. Garofalo and E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana University Mathematics Journal, 41 (1992), 71-98. doi: 10.1512/iumj.1992.41.41005. [18] N. Garofalo and D. M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math., 49 (1996), 1081-1144. doi: 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A. [19] Y. Giga, Surface Evolution Equations, A Level Set Approach, Monographs in Mathematics 99, Birkhäuser Verlag, Basel, 2006. [20] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, New York, 2006. [21] R. Horn and C. Johnson, Matrix Analysis, Ed. 23. Cambridge University Press, New York, 2010. [22] H. Ishiii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac., 38 (1995), 101-120. [23] V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation, Communications in PDE, 37 (2012), 934-946. doi: 10.1080/03605302.2011.615878. [24] P. Juutinen and P. Lindqvist, A theorem of Radó's type for the solutions of a quasi-linear equation, Mathematical Research Letters, 11 (2004), 31-34. doi: 10.4310/MRL.2004.v11.n1.a4. [25] P. Juutinen, P. Lindqvist and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear equation, SIAM Journal on Mathematical Analysis, 33 (2001), 699-717. doi: 10.1137/S0036141000372179. [26] D. Kinderleher and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics in Applied Mathematics, SIAM, 2000. doi: 10.1137/1.9780898719451. [27] P-L. Lions, Optimal control of diffusion process and Hamilton-Jacobi-Bellman equations, Part 2: viscosity solutions and uniqueness, Comm. in P.D.E., 8 (1983), 1229-1276. doi: 10.1080/03605308308820301. [28] J. Manfredi, Non-linear subelliptic equations on Carnot groups: Analysis and geometry in metric spaces, Notes of a course given at the Third School on Analysis and Geometry in Metric Spaces, Trento, 2003. [29] P. Mannucci, The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction, Commun. Pure Appl. Anal., 13 (2014), 119-133. doi: 10.3934/cpaa.2014.13.119. [30] V. Martino and A. Montanari, Nonsmooth solutions for a class of fully nonlinear PDE's on Lie groups, Nonlinear Anal., 126 (2015), 115-130. doi: 10.1016/j.na.2015.02.009. [31] V. Martino and A. Montanari, Lipschitz continuous viscosity solutions for a class of fully nonlinear equations on Lie groups, J. Geom. Anal., 24 (2014), 169-189. doi: 10.1007/s12220-012-9332-2. [32] M. Medina and P. Ochoa, On viscosity and weak solutions for non-homogeneous p-Laplace equations, Advances in Nonlinear Analysis, to appear, arXiv: math/161009216. [33] R. Monti and D. Morbidelli, Regular domains in homogeneous groups, Transactions of the AMS, 357 (2005), 2975-3011. doi: 10.1090/S0002-9947-05-03799-2. [34] R. Monti, Distances, Boundaries and Surface Measures in Carnot-Carathéodory Spaces, PhD Thesis in Mathematics, Università di Trento, 2001. [35] P. Ochoa and J. A. Ruiz, Existence and uniqueness results for linear second-order equations in the Heisenberg group, Annales Academiae Scientiarum Fennicae Mathematica, 42 (2017), 1063-1085. doi: 10.5186/aasfm.2017.4264. [36] P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Vol. 73, Berlin, 2007. [37] G. Wang, Viscosity convex functions on Carnot groups, Proceedings of the American Mathematical Society, 133 (2004), 1247-1253. doi: 10.1090/S0002-9939-04-07836-0. [38] G. Yuan and Z. Yuan, Dirichlet problems for linear and semilinear sub-Laplace equations on Carnot groups, Journal of Inequalities and Applications, 2012 (2012), 1-12. doi: 10.1186/1029-242X-2012-136. [39] J. Wang, P. Hong, D. Liao and Z. Yu, Partial regularity for non-linear elliptic systems with Dini continuous coefficients in the Heisenberg group, Abstract and Applied Analysis, (2013), 1-12. doi: 10.1155/2013/950134. [40] C. Xu, The Dirichlet problem for a class of semi-linear subelliptic equations, Nonlinear Analysis, 37 (1999), 1039-1049. doi: 10.1016/S0362-546X(97)00722-0. [41] C. Xu and C. Zuily, Higher interior regularity for quasilinear subelliptic systems, Calc. Var., 5 (1997), 323-343. doi: 10.1007/s005260050069. [42] C. Xu, An existence result for a class of semilinear degenerate elliptic equations, Adv. in Math., 22 (1996), 492-499. [43] C. Xu, Dirichlet problems for the quasilinear second order subelliptic equations, Acta Mathematica Sinica, New Series, 12 (1996), 18-32. doi: 10.1007/BF02109387. [44] C. Xu, Existence of bounded solutions for quasilinear subelliptic Dirichlet problems, J. Partial Diff. Eqs., 8 (1995), 97-107. [45] C. Xu, Semilinear subelliptic equations and Sobolev inequality for vector fields satisfying Hörmandar's condition, Chinise Journal of Contemporary Mathematics, 15 (1994), 183-192. [46] C. Xu, Subelliptic variational problems, Bulletin de la S. M. F., 118 (1990), 147-169. [47] J. Zhang and P. Niu, Existence results for positive solutions of semilinear equations on the Heisenberg group, Nonlinear Analysis, Theory, Methods and Applications, 31 (1998), 181-189. doi: 10.1016/S0362-546X(96)00303-3.

show all references

##### References:
 [1] 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. doi: 10.3934/cpaa.2006.5.709. [2] T. Bieske, A sub-Riemannian maximum principle and its application to the p-Laplacian in Carnot groups, Annales Academiae Scientiarum Fennicae Mathematica, 37 (2012), 119-134. doi: 10.5186/aasfm.2012.3706. [3] T. Bieske, Equivalence of weak and viscosity for the p-Laplace equation in the Heisenberg group, Annales Academiae Scientiarum Fennicae Mathematica, 31 (2006), 363-379. [4] T. Bieske, On ∞-harmonic functions on the Heisenberg group, Communications in PDE, 27 (2002), 727-761. doi: 10.1081/PDE-120002872. [5] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, AMS, Colloquium Publications, Vol. 43, USA, 1995. doi: 10.1090/coll/043. [6] L. Capogna, D. Danielli, S. Pauls and J. Tyson, An introduction to the Heisenberg group and the sub-Riemannian Isoperimetric problem, Basel: Progress in Mathematics, Vol. 259, Birkhauser-Verlag, 2007. [7] L. Capogna, Regularity of quasi-linear equations in the Heisenberg group, Communications on Pure and Applied Analysis, 50 (1998), 867-889. doi: 10.1002/(SICI)1097-0312(199709)50:9<867::AID-CPA3>3.0.CO;2-3. [8] G. Citti, B. Franceschiello, G. Sanguinetti and A. Sarti, Sub-Riemannian mean curvature flow for image processing, SIAM Journal on Imaging Sciences, 9 (2016), 212-237. doi: 10.1137/15M1013572. [9] G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space, in Proceedings of the Workshop on Second Order Subelliptic Equations and Applications, 2003. [10] G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space, Journal of Mathematical Imaging and Vision, 24 (2006), 307-326. doi: 10.1007/s10851-005-3630-2. [11] M. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [12] Y. Dong, G. Lu and L. Sun, Global Poincaré Inequalities on the Heisenberg group and applications, Acta Mathematica Sinica, English Series, (2006). doi: 10.1007/s10114-005-0874-0. [13] L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton FL, 1992. [14] F. Ferrari, Q. Liu and J. Manfredi, On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group, Communications in Contemporary Mathematics, 16 (2014), 1350027. doi: 10.1142/S0219199713500272. [15] G. Folland and E. Stein, Hardy-spaces on Homogeneous Groups, Princeton University Press, 1982. [16] B. Franchi, R. Serapioni and F. Serra-Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated to Lipschitz continuous vector fields, Bollettino U.M.I., 11B (2001), 83-117. [17] N. Garofalo and E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana University Mathematics Journal, 41 (1992), 71-98. doi: 10.1512/iumj.1992.41.41005. [18] N. Garofalo and D. M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math., 49 (1996), 1081-1144. doi: 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A. [19] Y. Giga, Surface Evolution Equations, A Level Set Approach, Monographs in Mathematics 99, Birkhäuser Verlag, Basel, 2006. [20] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, New York, 2006. [21] R. Horn and C. Johnson, Matrix Analysis, Ed. 23. Cambridge University Press, New York, 2010. [22] H. Ishiii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac., 38 (1995), 101-120. [23] V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation, Communications in PDE, 37 (2012), 934-946. doi: 10.1080/03605302.2011.615878. [24] P. Juutinen and P. Lindqvist, A theorem of Radó's type for the solutions of a quasi-linear equation, Mathematical Research Letters, 11 (2004), 31-34. doi: 10.4310/MRL.2004.v11.n1.a4. [25] P. Juutinen, P. Lindqvist and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear equation, SIAM Journal on Mathematical Analysis, 33 (2001), 699-717. doi: 10.1137/S0036141000372179. [26] D. Kinderleher and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics in Applied Mathematics, SIAM, 2000. doi: 10.1137/1.9780898719451. [27] P-L. Lions, Optimal control of diffusion process and Hamilton-Jacobi-Bellman equations, Part 2: viscosity solutions and uniqueness, Comm. in P.D.E., 8 (1983), 1229-1276. doi: 10.1080/03605308308820301. [28] J. Manfredi, Non-linear subelliptic equations on Carnot groups: Analysis and geometry in metric spaces, Notes of a course given at the Third School on Analysis and Geometry in Metric Spaces, Trento, 2003. [29] P. Mannucci, The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction, Commun. Pure Appl. Anal., 13 (2014), 119-133. doi: 10.3934/cpaa.2014.13.119. [30] V. Martino and A. Montanari, Nonsmooth solutions for a class of fully nonlinear PDE's on Lie groups, Nonlinear Anal., 126 (2015), 115-130. doi: 10.1016/j.na.2015.02.009. [31] V. Martino and A. Montanari, Lipschitz continuous viscosity solutions for a class of fully nonlinear equations on Lie groups, J. Geom. Anal., 24 (2014), 169-189. doi: 10.1007/s12220-012-9332-2. [32] M. Medina and P. Ochoa, On viscosity and weak solutions for non-homogeneous p-Laplace equations, Advances in Nonlinear Analysis, to appear, arXiv: math/161009216. [33] R. Monti and D. Morbidelli, Regular domains in homogeneous groups, Transactions of the AMS, 357 (2005), 2975-3011. doi: 10.1090/S0002-9947-05-03799-2. [34] R. Monti, Distances, Boundaries and Surface Measures in Carnot-Carathéodory Spaces, PhD Thesis in Mathematics, Università di Trento, 2001. [35] P. Ochoa and J. A. Ruiz, Existence and uniqueness results for linear second-order equations in the Heisenberg group, Annales Academiae Scientiarum Fennicae Mathematica, 42 (2017), 1063-1085. doi: 10.5186/aasfm.2017.4264. [36] P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Vol. 73, Berlin, 2007. [37] G. Wang, Viscosity convex functions on Carnot groups, Proceedings of the American Mathematical Society, 133 (2004), 1247-1253. doi: 10.1090/S0002-9939-04-07836-0. [38] G. Yuan and Z. Yuan, Dirichlet problems for linear and semilinear sub-Laplace equations on Carnot groups, Journal of Inequalities and Applications, 2012 (2012), 1-12. doi: 10.1186/1029-242X-2012-136. [39] J. Wang, P. Hong, D. Liao and Z. Yu, Partial regularity for non-linear elliptic systems with Dini continuous coefficients in the Heisenberg group, Abstract and Applied Analysis, (2013), 1-12. doi: 10.1155/2013/950134. [40] C. Xu, The Dirichlet problem for a class of semi-linear subelliptic equations, Nonlinear Analysis, 37 (1999), 1039-1049. doi: 10.1016/S0362-546X(97)00722-0. [41] C. Xu and C. Zuily, Higher interior regularity for quasilinear subelliptic systems, Calc. Var., 5 (1997), 323-343. doi: 10.1007/s005260050069. [42] C. Xu, An existence result for a class of semilinear degenerate elliptic equations, Adv. in Math., 22 (1996), 492-499. [43] C. Xu, Dirichlet problems for the quasilinear second order subelliptic equations, Acta Mathematica Sinica, New Series, 12 (1996), 18-32. doi: 10.1007/BF02109387. [44] C. Xu, Existence of bounded solutions for quasilinear subelliptic Dirichlet problems, J. Partial Diff. Eqs., 8 (1995), 97-107. [45] C. Xu, Semilinear subelliptic equations and Sobolev inequality for vector fields satisfying Hörmandar's condition, Chinise Journal of Contemporary Mathematics, 15 (1994), 183-192. [46] C. Xu, Subelliptic variational problems, Bulletin de la S. M. F., 118 (1990), 147-169. [47] J. Zhang and P. Niu, Existence results for positive solutions of semilinear equations on the Heisenberg group, Nonlinear Analysis, Theory, Methods and Applications, 31 (1998), 181-189. doi: 10.1016/S0362-546X(96)00303-3.
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