数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (4): 1301-1317.doi: 10.1016/S0252-9602(10)60126-9

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THE REGULARITY OF QUASI-MINIMA AND ω-MINIMA OF INTEGRAL FUNCTIONALS

 宁正元, 王秀丽   

  1. College of Computer and Information, Fujian Agriculture and Forestry University, Fuzhou 350002, China
  • 收稿日期:2008-01-15 修回日期:2008-05-20 出版日期:2010-07-20 发布日期:2010-07-20
  • 基金资助:

    Supported by the Program  of Fujian Province-Hong Kong

THE REGULARITY OF QUASI-MINIMA AND ω-MINIMA OF INTEGRAL FUNCTIONALS

 NING Zheng-Yuan, WANG Xiu-Li   

  1. College of Computer and Information, Fujian Agriculture and Forestry University, Fuzhou 350002, China
  • Received:2008-01-15 Revised:2008-05-20 Online:2010-07-20 Published:2010-07-20
  • Supported by:

    Supported by the Program  of Fujian Province-Hong Kong

摘要:

In this article, we have two parts. In the first part, we are concerned with the locally Holder continuity of quasi-minima of the following integral functional
 ∫Ω f(x, u, Du) dx,                     (1)
 where Ω is an open subset of Euclidean N-space (N ≥3), u: Ω→R, the Carath\'eodory function f satisfies the critical Sobolev exponent growth condition

|Du|p-|u|p*-a(x)l≤ f(x, u, Du)≤ L(|Du|p+|u|p*+a(x)),              (2)
where L≤1, 1<p<N, p*=Np/N-p, and a(x) is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally H\"{o}lder continuity of ω-minima of (1). Our method is to compare the ω-minima of (1) with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland's variational principal.

关键词: Integral functional, Q-minima,  ω-minima, Ekelands variational principle, Holder continuous

Abstract:

In this article, we have two parts. In the first part, we are concerned with the locally Holder continuity of quasi-minima of the following integral functional
 ∫Ω f(x, u, Du) dx,                     (1)
 where Ω is an open subset of Euclidean N-space (N ≥3), u: Ω→R, the Carath\'eodory function f satisfies the critical Sobolev exponent growth condition

|Du|p-|u|p*-a(x)l≤ f(x, u, Du)≤ L(|Du|p+|u|p*+a(x)),              (2)
where L≤1, 1<p<N, p*=Np/N-p, and a(x) is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally H\"{o}lder continuity of ω-minima of (1). Our method is to compare the ω-minima of (1) with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland's variational principal.

Key words: Integral functional, Q-minima,  ω-minima, Ekelands variational principle, Holder continuous

中图分类号: 

  • 35B65