Acta mathematica scientia,Series B ›› 2012, Vol. 32 ›› Issue (5): 1937-1958.doi: 10.1016/S0252-9602(12)60151-9

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REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS WITH DINI COEFFICIENTS UNDER NATURAL GROWTH CONDITION FOR THE CASE: 1 <m <2

 QIU Ya-Lin   

  1. School of Mathematics and Computing Science, Longyan University, Longyan 361005, China
  • Received:2011-01-24 Revised:2011-04-24 Online:2012-09-20 Published:2012-09-20
  • Supported by:

    Supported by National Natural Science Foundation of China (10976026) and the Education Department of Fujian Province (JK2009045).

Abstract:

In this article, we consider interior regularity for weak solutions to nonlinear elliptic systems of divergence type with Dini continuous coefficients under natural growth condition for the case 1 < m < 2. All estimates in the case of m ≥ 2 is no longer suitable, and we can’t obtain the Caccioppoli’s second inequality by using these techniques developed in the case of m ≥ 2. But the Caccioppoli’s second inequality is the key to use A-harmonic approximation method. Thus, we adopt another technique introduced by Acerbi and Fcsco to overcome the difficulty and we also overcome those difficulties due to Dini condition. And then we apply the A-harmonic approximation method to prove partial regularity of weak solutions.

Key words: nonlinear elliptic systems, natural growth condition, partial regularity, A-harmonic approximation technique

CLC Number: 

  • 35J45
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