{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T22:17:49Z","timestamp":1761171469644,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:00:00Z","timestamp":1760227200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12061010"],"award-info":[{"award-number":["12061010"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004479","name":"Jiangxi Provincial Natural Science Foundation","doi-asserted-by":"crossref","award":["20242BAB26003"],"award-info":[{"award-number":["20242BAB26003"]}],"id":[{"id":"10.13039\/501100004479","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Science and Technology Planning Project of Jiangxi Province","award":["GJJ2201204"],"award-info":[{"award-number":["GJJ2201204"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper studies discontinuous quasilinear sub-elliptic systems associated with H\u00f6rmander\u2019s vector fields under controllable and natural growth conditions. By a new A-harmonic approximation reformulation for bilinear forms A\u2208Bil(RkN,RkN), we obtain optimal partial H\u00f6lder continuity with exact exponents for weak solutions with vanishing mean oscillation coefficients.<\/jats:p>","DOI":"10.3390\/axioms14100761","type":"journal-article","created":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T11:57:42Z","timestamp":1760529462000},"page":"761","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimal H\u00f6lder Regularity for Discontinuous Sub-Elliptic Systems Structured on H\u00f6rmander\u2019s Vector Fields"],"prefix":"10.3390","volume":"14","author":[{"given":"Dongni","family":"Liao","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0992-2952","authenticated-orcid":false,"given":"Jialin","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/BF02392081","article-title":"Hypoelliptic second order differential equation","volume":"119","year":"1967","journal-title":"Acta Math. Upps."},{"key":"ref_2","first-page":"135","article-title":"Un esempio di estremali discontinue per un problem variazionale di upo ellitico","volume":"4","year":"1968","journal-title":"Boll. Unione Mat. Ital."},{"unstructured":"Chen, Y., and Wu, L. (2003). Second Order Elliptic Equations and Elliptic Systems, Science Press.","key":"ref_3"},{"doi-asserted-by":"crossref","unstructured":"Giaquinta, M. (1983). Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press.","key":"ref_4","DOI":"10.1515\/9781400881628"},{"key":"ref_5","first-page":"73","article-title":"Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals","volume":"546","author":"Duzaar","year":"2002","journal-title":"J. Reine Angew. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s002290070007","article-title":"Partial regularity for nonlinear elliptic systems: The method of A-harmonic approximation","volume":"103","author":"Duzaar","year":"2000","journal-title":"Manuscripta Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1016\/j.anihpc.2003.09.003","article-title":"Regularity for degenerate elliptic problems via p-harmonic approximation","volume":"21","author":"Duzaar","year":"2004","journal-title":"Ann. Inst. H. Poincar\u00e9 Anal. Non Lin\u00e8aire"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"113691","DOI":"10.1016\/j.na.2024.113691","article-title":"Partial gradient regularity for parabolic systems with degenerate diffusion and H\u00f6lder continuous coefficients","volume":"251","year":"2025","journal-title":"Nonlinear Anal. TMA"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1112\/plms\/pdr009","article-title":"Partial H\u00f6lder continuity for discontinuous elliptic problems with VMO-coefficients","volume":"103","author":"Duzaar","year":"2011","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1215","DOI":"10.1081\/PDE-200033734","article-title":"Elliptic systems, singular sets and Dini continuity","volume":"29","author":"Duzaar","year":"2004","journal-title":"Commun. Part. Diff. Equ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1007\/s10231-010-0169-7","article-title":"Morrey space regularity for weak solutions of Stokes systems with VMO coefficients","volume":"190","author":"John","year":"2011","journal-title":"Ann. Mat. Pura Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1016\/j.anihpc.2007.02.003","article-title":"Partial continuity for elliptic problems","volume":"25","author":"Foss","year":"2008","journal-title":"Ann. Inst. H. Poincar\u00e9 Anal. Non Lin\u00e9ire"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/s10231-016-0564-9","article-title":"Partial regularity in the interior for discontinuous inhomogeneous elliptic system with VMO-coefficients","volume":"196","author":"Tan","year":"2017","journal-title":"Ann. Mat. Pura Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s00526-022-02324-2","article-title":"A-caloric approximation and partial regularity for parabolic systems with Orlicz growth","volume":"62","author":"Foss","year":"2023","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"4271","DOI":"10.3934\/cpaa.2021160","article-title":"Partial regularity result for non-autonomous elliptic systems with general growth","volume":"20","author":"Isernia","year":"2021","journal-title":"Commun. Pure Appl. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00526-024-02687-8","article-title":"Partial regularity for degenerate parabolic systems with general growth via caloric approximations","volume":"63","author":"Ok","year":"2024","journal-title":"Calc. Var. Partial Diff. Equ."},{"key":"ref_17","first-page":"271","article-title":"Partial regularity for elliptic systems with critical growth","volume":"33","author":"Isernia","year":"2022","journal-title":"Rend. Lincei Mat. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s100970200043","article-title":"Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of H\u00f6rmander type","volume":"5","author":"Capogna","year":"2003","journal-title":"J. Eur. Math. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s00526-007-0127-4","article-title":"Partial regularity results for sub-elliptic systems in the Heisenberg group","volume":"32","year":"2008","journal-title":"Calc. Var. Partial Differ. Equ."},{"unstructured":"Shores, E. (2005). Regularity Theory for Weak Solutions of Systems in Carnot Groups. [Ph.D. Thesis, University of Arkansas].","key":"ref_20"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1515\/anona-2020-0145","article-title":"Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: The sub-quadratic structure case","volume":"10","author":"Wang","year":"2021","journal-title":"Adv. Nonlinear Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s005260050069","article-title":"Higher interior regularity for quasilinear sub-elliptic systems","volume":"5","author":"Xu","year":"1997","journal-title":"Calc. Var. Partial. Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1080\/17476933.2022.2152444","article-title":"Partial regularity for a nonlinear discontinuous sub-elliptic system with drift on the Heisenberg group: The superquadratic case","volume":"69","author":"Zhang","year":"2024","journal-title":"Complex Var. Elliptic Equ."},{"unstructured":"Domokos, A. (2004). On the Regularity of p-Harmonic Functions in the Heisenberg Group. [Ph.D. Thesis, University of Pittsburgh].","key":"ref_24"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"567","DOI":"10.2140\/apde.2021.14.567","article-title":"C1,\u03b1 regularity for variational problems in the Heisenberg group","volume":"14","author":"Mukherjee","year":"2021","journal-title":"Anal. PDE"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.aim.2009.03.016","article-title":"Gradient regularity for elliptic equations in the Heisenberg group","volume":"222","author":"Mingione","year":"2009","journal-title":"Adv. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"108593","DOI":"10.1016\/j.aim.2022.108593","article-title":"Regularity of quasi-linear equations with H\u00f6rmander vector fields of Step two","volume":"408","author":"Citti","year":"2022","journal-title":"Adv. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1007\/s00229-011-0498-x","article-title":"Estimates in Morrey spaces and H\u00f6lder continuity for weak solutions to degenerate elliptic systems","volume":"138","author":"Dong","year":"2012","journal-title":"Manuscr. Math."},{"key":"ref_29","first-page":"605","article-title":"Gradient estimates for elliptic systems in Carnot-Carath\u00e9odory spaces","volume":"43","author":"Fanciullo","year":"2002","journal-title":"Comment. Math. Univ. Carol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3209","DOI":"10.1016\/j.na.2010.07.001","article-title":"Partial regularity for degenerate sub-elliptic systems associated with H\u00f6rmander\u2019s vector fields","volume":"73","author":"Gao","year":"2010","journal-title":"Nonlinear Anal."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1407","DOI":"10.3934\/math.2025065","article-title":"Gradient regularity for nonlinear sub-elliptic systems with the drift term: Sub-quadratic growth case","volume":"10","author":"Chen","year":"2025","journal-title":"AIMS Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"251","DOI":"10.2206\/kyushujm.65.251","article-title":"Optimal partial regularity for nonlinear sub-elliptic systems related to H\u00f6rmander\u2019s vector fields","volume":"65","author":"Wang","year":"2011","journal-title":"Kyushu J. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1007\/BF02392539","article-title":"Balls and metrics defined by vector fields I. Basic properties","volume":"155","author":"Nagel","year":"1985","journal-title":"Acta Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"453","DOI":"10.4171\/rmi\/158","article-title":"The sharp Poincar\u00e9 inequality for free vector fields: An endpoint result","volume":"10","author":"Lu","year":"1994","journal-title":"Rev. Mat. Iberoam."},{"unstructured":"Evans, L., and Gariepy, R. (1992). 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