Abstract and Applied Analysis

Partial Regularity for Nonlinear Subelliptic Systems with Dini Continuous Coefficients in Heisenberg Groups

Abstract

This paper is concerned with partial regularity to nonlinear subelliptic systems with Dini continuous coefficients under quadratic controllable growth conditions in the Heisenberg group ${ℍ}^{n}$. Based on a generalization of the technique of $\mathrm{𝒜}$-harmonic approximation introduced by Duzaar and Steffen, partial regularity to the sub-elliptic system is established in the Heisenberg group. Our result is optimal in the sense that in the case of Hölder continuous coefficients we establish the optimal Hölder exponent for the horizontal gradients of the weak solution on its regular set.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 950134, 12 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512072

Digital Object Identifier
doi:10.1155/2013/950134

Mathematical Reviews number (MathSciNet)
MR3108667

Zentralblatt MATH identifier
07095529

Citation

Wang, Jialin; Hong, Pingzhou; Liao, Dongni; Yu, Zefeng. Partial Regularity for Nonlinear Subelliptic Systems with Dini Continuous Coefficients in Heisenberg Groups. Abstr. Appl. Anal. 2013 (2013), Article ID 950134, 12 pages. doi:10.1155/2013/950134. https://projecteuclid.org/euclid.aaa/1393512072

References

• J. Wang and P. Niu, “Optimal partial regularity for weak solutions of nonlinear sub-elliptic systems in Carnot groups,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 11, pp. 4162–4187, 2010.
• J. Wang and D. Liao, “Optimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groups,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2499–2519, 2012.
• E. de Giorgi, “Un esempio di estremali discontinue per un problema variazionale di tipo ellittico,” Bollettino della Unione Matematica Italiana, vol. 4, pp. 135–137, 1968.
• M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, NJ, USA, 1983.
• Y. Chen and L. Wu, Second Order Elliptic Equations and Elliptic Systems, Science Press, Beijing, China, 2003.
• F. Duzaar and K. Steffen, “Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals,” Journal für die Reine und Angewandte Mathematik, vol. 546, pp. 73–138, 2002.
• F. Duzaar and J. F. Grotowski, “Optimal interior partial regularity for nonlinear elliptic systems: the method of \emphA-harmonic approximation,” Manuscripta Mathematica, vol. 103, no. 3, pp. 267–298, 2000.
• F. Duzaar, J. F. Grotowski, and M. Kronz, “Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth,” Annali di Matematica Pura ed Applicata IV, vol. 184, no. 4, pp. 421–448, 2005.
• F. Duzaar and G. Mingione, “The p-harmonic approximation and the regularity of p-harmonic maps,” Calculus of Variations and Partial Differential Equations, vol. 20, no. 3, pp. 235–256, 2004.
• F. Duzaar and G. Mingione, “Regularity for degenerate elliptic problems via p-harmonic approximation,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 21, no. 5, pp. 735–766, 2004.
• S. Chen and Z. Tan, “The method of \emphA-harmonic approximation and optimal interior partial regularity for nonlinear elliptic systems under the controllable growth condition,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 20–42, 2007.
• F. Duzaar and A. Gastel, “Nonlinear elliptic systems with Dini continuous coefficients,” Archiv der Mathematik, vol. 78, no. 1, pp. 58–73, 2002.
• F. Duzaar, A. Gastel, and G. Mingione, “Elliptic systems, singular sets and Dini continuity,” Communications in Partial Differential Equations, vol. 29, no. 7-8, pp. 1215–1240, 2004.
• Y. Qiu and Z. Tan, “Optimal interior partial regularity for nonlinear elliptic systems with Dini continuous coefficients,” Acta Mathematica Scientia B, vol. 30, no. 5, pp. 1541–1554, 2010.
• Y. Qiu, “Optimal partial regularity of second order nonlinear elliptic systems with Dini continuous coefficients for the superquadratic case,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 8, pp. 3574–3590, 2012.
• L. Capogna, “Regularity of quasi-linear equations in the Heisenberg group,” Communications on Pure and Applied Mathematics, vol. 50, no. 9, pp. 867–889, 1997.
• L. Capogna, “Regularity for quasilinear equations and 1-quasiconformal maps in Carnot groups,” Mathematische Annalen, vol. 313, no. 2, pp. 263–295, 1999.
• S. Marchi, “${C}^{1,\alpha }$ local regularity for the solutions of the p-Laplacian on the Heisenberg group for $2<p<1+\sqrt{5}$,” Zeitschrift für Analysis und ihre Anwendungen, vol. 20, no. 3, pp. 617–636, 2001.
• S. Marchi, “${C}^{1,\alpha }$ local regularity for the solutions of the p-Laplacian on the Heisenberg group. The case $1+1/\sqrt{5}<p\leq2$,” Commentationes Mathematicae Universitatis Carolinae, vol. 44, no. 1, pp. 33–56, 2003.
• S. Marchi, “\emphL$^{p}$ regularity of the derivative in the second commutator direction for nonlinear elliptic equations on the Heisenberg group,” Accademia Nazionale delle Scienze detta dei XL. Rendiconti. Serie V. Memorie di Matematica e Applicazioni. Parte I, vol. 26, pp. 1–15, 2002.
• A. Domokos, “Differentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg group,” Journal of Differential Equations, vol. 204, no. 2, pp. 439–470, 2004.
• A. Domokos, On the regularity of p-harmonic functions in the Heisenberg group [Ph.D. thesis], University of Pittsburgh, Pittsburgh, Pa, USA, 2004.
• J. J. Manfredi and G. Mingione, “Regularity results for quasilinear elliptic equations in the Heisenberg group,” Mathematische Annalen, vol. 339, no. 3, pp. 485–544, 2007.
• G. Mingione, A. Zatorska-Goldstein, and X. Zhong, “Gradient regularity for elliptic equations in the Heisenberg group,” Advances in Mathematics, vol. 222, no. 1, pp. 62–129, 2009.
• L. Capogna and N. Garofalo, “Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type,” Journal of the European Mathematical Society, vol. 5, no. 1, pp. 1–40, 2003.
• E. Shores, “Hypoellipticity for čommentComment on ref. [26?]: Please update the information of this reference, if possible.linear degenerate elliptic systems in Carnot groups and applications,” http://arxiv.org/abs/math/ 0502569.
• A. Föglein, “Partial regularity results for subelliptic systems in the Heisenberg group,” Calculus of Variations and Partial Differential Equations, vol. 32, no. 1, pp. 25–51, 2008.
• G. Lu, “The sharp Poincaré inequality for free vector fields: an endpoint result,” Revista Matemática Iberoamericana, vol. 10, no. 2, pp. 453–466, 1994.