Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case.

*(English)*Zbl 1447.35125Aim of the authors is to establish two partial Hölder continuity results, Theorem 1.1 and Theorem 1.2. Firstly an appropriate Sobolev-Poincaré inequality, which plays an important part on proving Hölder regularity, is established. Then, an A-harmonic approximation lemma, and a prior estimate for weak solution \(h \in HW^{1,1}\) to the constant coefficient homogeneous sub-elliptic systems are given.

In the sequel the authors prove the first partial regularity result (Theorem 1.1) under sub-quadratic controllable structure assumptions by several steps. Step 1 is to gain a suitable Caccioppoli-type inequality which is an essential tool to get partial regularity. An appropriate linearization strategy is given in the second step. Then, one can achieve that solutions are approximately A-harmonic by the linearization procedure, and an excess improvement estimate for a functional, called \(\psi\), is obtained under two smallness condition assumptions, by combining with A-harmonic approximation lemma in the third steps. Once the excess improvement is established, the iteration for the \(\psi\)-excess and the \(C_y\)-excess can be acquired in Step 4. Finally, the authors show boundedness of the Campanato-type excess which leads immediately to desired Hölder continuity and Morrey regularity of Theorem 1.1. The last section shows the results of Theorem 1.2 under sub-quadratic natural structure assumptions. In such a case, the authors establish appropriate estimates just for the natural growth term, and the rest procedure is similar to the proof of Theorem 1.1.

In the sequel the authors prove the first partial regularity result (Theorem 1.1) under sub-quadratic controllable structure assumptions by several steps. Step 1 is to gain a suitable Caccioppoli-type inequality which is an essential tool to get partial regularity. An appropriate linearization strategy is given in the second step. Then, one can achieve that solutions are approximately A-harmonic by the linearization procedure, and an excess improvement estimate for a functional, called \(\psi\), is obtained under two smallness condition assumptions, by combining with A-harmonic approximation lemma in the third steps. Once the excess improvement is established, the iteration for the \(\psi\)-excess and the \(C_y\)-excess can be acquired in Step 4. Finally, the authors show boundedness of the Campanato-type excess which leads immediately to desired Hölder continuity and Morrey regularity of Theorem 1.1. The last section shows the results of Theorem 1.2 under sub-quadratic natural structure assumptions. In such a case, the authors establish appropriate estimates just for the natural growth term, and the rest procedure is similar to the proof of Theorem 1.1.

Reviewer: Maria Alessandra Ragusa (Catania)

##### MSC:

35H20 | Subelliptic equations |

35B65 | Smoothness and regularity of solutions to PDEs |

32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |

35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |

##### Keywords:

partial Hölder continuity; sub-quadratic controllable growth; sub-quadratic natural growth; \(p\)-Laplacian; Morrey regularity
PDF
BibTeX
XML
Cite

\textit{J. Wang} et al., Adv. Nonlinear Anal. 10, 420--449 (2021; Zbl 1447.35125)

Full Text:
DOI

##### References:

[1] | V. Bögelein, F. Duzaar, J. Habermann, C. Scheven, Partial Hölder continuity for discontinuous elliptic problems with VMO-coefficients, Proc. Lond. Math. Soc. 103 (2011), 371-404. · Zbl 1241.35034 |

[2] | M. Bramanti, An Invitation to Hypoelliptic Operators and Hörmander’s Vector Fields, Springer, 2014. · Zbl 1298.47001 |

[3] | L. Capogna, D. Danielli, N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear sub-elliptic equations, Comm. Partial Differential Equations 18 (1993), 1765-1794. · Zbl 0802.35024 |

[4] | M. Carozza, N. Fusco, G. Mingione, Partial regularity of minimizers of quasiconvex integrals with sub-quadratic growth, Ann. Mat. Pura Appl. 175 (1998), 141-164. · Zbl 0960.49025 |

[5] | L. Capogna, N. Garofalo, Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type, J. European Math. Society 5 (2003), 1-40. · Zbl 1064.49026 |

[6] | S. Chen, Z. Tan, The method of A-harmonic approximation and optimal interior partial regularity for nonlinear elliptic systems under the controllable growth condition, J. Math. Anal. Appl. 335 (2007), 20-42. · Zbl 1387.35210 |

[7] | G. Di Fazio, M. Fanciullo, Gradient estimates for elliptic systems in Carnot-Carathéodory spaces. Comment. Math. Univ. Caroline. 43 (2002), 605-618. · Zbl 1090.35058 |

[8] | F. Duzaar, A. Gastel, Nonlinear elliptic systems with Dini continuous coefficients, Arch. Math. 78 (2002), 58-73. · Zbl 1013.35028 |

[9] | F. Duzaar, J. F. Grotowski, Partial regularity for nonlinear elliptic systems: The method of A-harmonic approximation, Manuscripta Math. 103 (2000), 267-298. · Zbl 0971.35025 |

[10] | F. Duzaar, J. F. Grotowski, M. Kronz, Regularity of almost minimizers of quasi-convex variational integrals with sub-quadratic growth, Ann. Mat. Pura Appl. 184 (2005), 421-448. · Zbl 1223.49040 |

[11] | F. Duzaar, A. Gastel, G. Mingione, Elliptic systems, singular sets and Dini continuity, Comm. Partial Differential Equations 29 (2004), 1215-1240. · Zbl 1140.35415 |

[12] | F. Duzaar, G. Mingione, The p-harmonic approximation and the regularity of p-harmonic maps, Calc. Var. Partial Differention Equations 20 (2004), 235-256. · Zbl 1142.35433 |

[13] | F. Duzaar, G. Mingione, Regularity for degenerate elliptic problems via p-harmonic approximation, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 735-766. · Zbl 1112.35078 |

[14] | Y. Dong, P. Niu, Estimates in Morrey spaces and Hölder continuity for weak solutions to degenerate elliptic systems, Manuscripta Math. 138 (2012), 419-437. · Zbl 1253.35052 |

[15] | F. Duzaar, K. Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math. 546 (2002), 73-138. · Zbl 0999.49024 |

[16] | A. Föglein, Partial regularity results for sub-elliptic systems in the Heisenberg group, Calc. Var. Partial Differential Equations 32 (2008), 25-51. · Zbl 1145.35059 |

[17] | M. Foss, G. Mingione, Partial continuity for elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 471-503. · Zbl 1153.35017 |

[18] | M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, NJ, 1983. · Zbl 0516.49003 |

[19] | D. Gao, P. Niu, J. Wang, Partial regularity for degenerate sub-elliptic systems associated with Hörmander’s vector fields, Nonlinear Anal. 73 (2010), 3209-3223. · Zbl 1208.35021 |

[20] | C. S. Goodrich, M. A. Ragusa, A. Scapellato, Partial regularity of solutions to p(x)-Laplacian PDEs with discontinuous coefficients, J. Differential Equations 268 (2020), 5440-5468. · Zbl 1436.35206 |

[21] | G. Lu, Embedding theorems on Campanato-Morrey space for vector fields on Hörmander type, Approx. Theory Appl. 14 (1998) 69-80. · Zbl 0916.46026 |

[22] | S. Polidoro, M. A. Ragusa, Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term, Revista Matematica Iberoamericana 24 (2008), 1011-1046. · Zbl 1175.35081 |

[23] | M. A. Ragusa, A. Tachikawa, Regularity of minimizers of some variational integrals with discontinuity, Z. Anal. Anwend. 27 (2008), 469-482. · Zbl 1153.49036 |

[24] | A. Scapellato, New perspectives in the theory of some function spaces and their applications, AIP Conference Proceedings 1978, 140002(2018); doi:10.1063/1.5043782. |

[25] | E. Shores, Regularity theory for weak solutions of systems in Carnot groups, Ph. D. Thesis, University of Arkansas. 2005. |

[26] | T. Kanazawa, Partial regularity for elliptic systems with VMO-coefficients, Riv. Math. Univ. Parma (N.S.) 5 (2014), 311-333. · Zbl 1325.35037 |

[27] | Z. Tan, Y. Wang, S. Chen, Partial regularity in the interior for discontinuous inhomogeneous elliptic system with VMO-coefficients, Ann. Mat. Pura Appl. 196 (2017), 85-105. · Zbl 1356.74073 |

[28] | Z. Tan, Y. Wang, S. Chen, Partial regularity up to the boundary for solutions of sub-quadratic elliptic systems, Adv. Nonlinear Anal. 7 (2018), 469-483. · Zbl 1404.35180 |

[29] | G. Lu, The sharp Poincaré inequality for free vector fields: an endpoint result, Revista Matematica Iberoamericana 10 (1994), 453-466. · Zbl 0860.35006 |

[30] | J. Wang, D. Liao, Optimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groups, Nonlinear Anal. 75 (2012), 2499-2519. · Zbl 1236.35020 |

[31] | J. Wang, D. Liao, S. Gao, Z. Yu, Optimal partial regularity for sub-elliptic systems with Dini continuous coefficients under the superquadratic natural growth, Nonlinear Anal. 114 (2015), 13-25. · Zbl 1317.35274 |

[32] | J. Wang, Juan J. Manfredi, Partial Hölder continuity for nonlinear sub-elliptic systems with VMO-coefficients in the Heisenberg group, Adv. Nonlinear Anal. 7 (2018), 96-114. · Zbl 1381.35020 |

[33] | J. Wang, S. Zhang, Q. Yang, Partial regularity for discontinuous sub-elliptic systems with sub-quadratic growth in the Heisenberg group, Nonlinear Anal. 195 (2020), 111719; doi:10.1016/j.na.2019.111719. · Zbl 1440.35040 |

[34] | C. Xu, C. Zuily, Higher interior regularity for quasilinear sub-elliptic systems, Calc. Var. Partial Differential Equations 5 (1997), 323-343. · Zbl 0902.35019 |

[35] | S. Zheng, Partial regularity for quasi-linear elliptic systems with VMO coefficients under the natural growth, Chinese Ann. Math. Ser. A 29 (2008), 49-58. · Zbl 1164.35034 |

[36] | S. Zheng, Z. Feng, Regularity of sub-elliptic p-harmonic systems with subcritical growth in Carnot group, J. Differential Equations 258 (2015), 2471-2494. · Zbl 1322.35005 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.