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Hölder continuity for sub-elliptic systems under the sub-quadratic controllable growth in Carnot groups. (English) Zbl 1286.35080

Summary: This paper is devoted to optimal partial regularity of weak solutions to nonlinear sub-elliptic systems for the case \(1<m<2\) under the controllable growth condition in Carnot groups. We begin with establishing a Sobolev-Poincaré type inequality for the function \(u\in HW^{1,m}(\varOmega , \mathbb R^n)\) with \(m\in (1,2)\), and the partial regularity with optimal local Hölder exponent for horizontal gradients of weak solutions to the system is established by using \(\mathcal{A}\)-harmonic approximation technique.

MSC:

35H20 Subelliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
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[1] E. DE GIORGI, Un esempio di estremali discontinue per un problema va- riazionale di tipo ellitico. Boll. Unione Mat. Italiana 4 (1968), pp. 135-137. · Zbl 0155.17603
[2] M. GIAQUINTA, Multiple Integrals in the Calculus of Variations and Non- linear Elliptic Systems. Princeton Univ. Press, Princeton, 1983. · Zbl 0516.49003
[3] M. GIAQUINTA, Introduction to regularity theory for nonlinear elliptic systems. BirkhaÈuser, Berlin, 1993. · Zbl 0786.35001
[4] Y. CHEN - L. WU, Second order elliptic equations and elliptic systems. Science Press, Beijing, 2003.
[5] M. GIAQUINTA - G. MODICA, Almost-everywhere regularity results for solu- tions of non linear elliptic systems. Manuscripta Math. 28 (1979), pp. 109-158, · Zbl 0411.35018 · doi:10.1007/BF01647969
[6] E. GIUSTI - M. MIRANDA, Sulla regolaritaÁ delle soluzioni deboli di una classe di sistemi ellittici quasilineari. Arch. Rat. Mech. Anal. 31 (1968), pp. 173-184.
[7] F. DUZAAR - J. F. GROTOWSKI, Partial regularity for nonlinear elliptic systems: The method of A-harmonic approximation, Manuscripta Math. 103 (2000), pp. 267-298. · Zbl 0971.35025 · doi:10.1007/s002290070007
[8] L. SIMON, Lectures on Geometric Measure Theory. Australian National University Press, Canberra, 1983. · Zbl 0546.49019
[9] F. DUZAAR - J. F. GROTOWSKI - M. KRONZ, Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth. Annali Mat. Pura Appl. (4) 184 (2005), pp. 421-448. · Zbl 1223.49040 · doi:10.1007/s10231-004-0117-5
[10] F. DUZAAR - G. MINGIONE, The p-harmonic approximation and the regularity of p-harmonic maps. Calc. Var. Partial Differential Equations 20 (2004), pp. 235-256. · Zbl 1142.35433 · doi:10.1007/s00526-003-0233-x
[11] F. DUZAAR - G. MINGIONE, Regularity for degenerate elliptic problems via p-harmonic approximation. Ann. Inst. H. Poincare Anal. Non LineÁaire 21 (2004), pp. 735-766. · Zbl 1112.35078 · doi:10.1016/j.anihpc.2003.09.003
[12] M. CAROZZA - N. FUSCO - G. MINGIONE, Partial regularity of minimizers of quasiconvex integrals with subquadratic growth, Annali Mat. Pura Appl. (4) 175 (1998), pp. 141-164. · Zbl 0960.49025 · doi:10.1007/BF01783679
[13] S. CHEN - Z. TAN, The method of A-harmonic approximation and optimal interior partial regularity for nonlinear elliptic systems under the control- lable growth condition. J. Math. Anal. Appl. 335 (2007), pp. 20-42. · Zbl 1387.35210
[14] S. CHEN - Z. TAN, Optimal interior partial regularity for nonlinear elliptic systems. Discrete Contin. Dyn. Syst. 27 (2010), pp. 981-993. · Zbl 1191.35115 · doi:10.3934/dcds.2010.27.981
[15] L. CAPOGNA - N. GAROFALO, Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of HoÈrmander type. J. European Math. Society 5 (2003), pp. 1-40. · Zbl 1064.49026 · doi:10.1007/s100970200043
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