# American Institute of Mathematical Sciences

August  2010, 27(3): 981-993. doi: 10.3934/dcds.2010.27.981

## Optimal interior partial regularity for nonlinear elliptic systems

 1 Department of Information and Mathematics Sciences, China Jiliang University, Hangzhou 310018, Zhejiang, China 2 School of Mathematical Science, Xiamen University, Xiamen 361005, Fujian, China

Received  December 2008 Revised  February 2010 Published  March 2010

We consider interior regularity for weak solutions of nonlinear elliptic systems with subquadratic under controllable growth condition. By $\mathcal{A}$-harmonic approximation technique, we obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly, the regular result is optimal.
Citation: Shuhong Chen, Zhong Tan. Optimal interior partial regularity for nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 981-993. doi: 10.3934/dcds.2010.27.981
 [1] Shuhong Chen, Zhong Tan. Optimal partial regularity results for nonlinear elliptic systems in Carnot groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3391-3405. doi: 10.3934/dcds.2013.33.3391 [2] Shenzhou Zheng, Xueliang Zheng, Zhaosheng Feng. Optimal regularity for $A$-harmonic type equations under the natural growth. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 669-685. doi: 10.3934/dcdsb.2011.16.669 [3] Shenzhou Zheng, Laping Zhang, Zhaosheng Feng. Everywhere regularity for P-harmonic type systems under the subcritical growth. Communications on Pure & Applied Analysis, 2008, 7 (1) : 107-117. doi: 10.3934/cpaa.2008.7.107 [4] Guji Tian, Xu-Jia Wang. Partial regularity for elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 899-913. doi: 10.3934/dcds.2010.28.899 [5] Luigi C. Berselli, Carlo R. Grisanti. On the regularity up to the boundary for certain nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 53-71. doi: 10.3934/dcdss.2016.9.53 [6] Luisa Fattorusso, Antonio Tarsia. Regularity in Campanato spaces for solutions of fully nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1307-1323. doi: 10.3934/dcds.2011.31.1307 [7] R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497 [8] Jinglai Qiao, Li Yang, Jiawei Yao. Passive control for a class of Nonlinear systems by using the technique of Adding a power integrator. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2020009 [9] G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279 [10] Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020002 [11] Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184 [12] Paulo Rabelo. Elliptic systems involving critical growth in dimension two. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2013-2035. doi: 10.3934/cpaa.2009.8.2013 [13] Dung Le. Partial regularity of solutions to a class of strongly coupled degenerate parabolic systems. Conference Publications, 2005, 2005 (Special) : 576-586. doi: 10.3934/proc.2005.2005.576 [14] Sun-Sig Byun, Hongbin Chen, Mijoung Kim, Lihe Wang. Lp regularity theory for linear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 121-134. doi: 10.3934/dcds.2007.18.121 [15] Rong Dong, Dongsheng Li, Lihe Wang. Regularity of elliptic systems in divergence form with directional homogenization. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 75-90. doi: 10.3934/dcds.2018004 [16] Mostafa Fazly. Regularity of extremal solutions of nonlocal elliptic systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 107-131. doi: 10.3934/dcds.2020005 [17] Ellina Grigorieva, Evgenii Khailov. Optimal control of a nonlinear model of economic growth. Conference Publications, 2007, 2007 (Special) : 456-466. doi: 10.3934/proc.2007.2007.456 [18] Lin Du, Yun Zhang. $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method. Journal of Industrial & Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035 [19] Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105 [20] Patrick Winkert. Multiplicity results for a class of elliptic problems with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (2) : 785-802. doi: 10.3934/cpaa.2013.12.785

2018 Impact Factor: 1.143