Acta mathematica scientia,Series A ›› 2013, Vol. 33 ›› Issue (3): 424-430.

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Local C1(Ω) -minimizers Versus W1, p(x)(Ω)-minimizers of Nonsmooth Functionals

 DAI Guo-Wei, DA Ting   

  1. Department of Mathematics, Northwest Normal University, Gansu Lanzhou 730070
  • Received:2011-11-02 Revised:2012-11-08 Online:2013-06-25 Published:2013-06-25
  • Supported by:

    国家自然科学基金(11261052, 11101335)资助

Abstract:

In this paper, we study not necessarily differentiable functionals of the form
J(u)=∫Ω1/p(x)(|∨u|p(x)+|u|p(x))dx+∫Ω j1(x, u)dx+∫∂Ωj2(xγu)dσ,
where p(x)∈ C0, β(Ω), with β∈(0,1),  and 1<p-p+<+∞, j1: Ω×R→R as well as j2: ∂Ω×R→R are locally Lipschitz functions. We prove that local C1(Ω)-minimizers of J must be local W1, p(x)(Ω)-minimizers of J.

Key words: Nonsmooth functionals, Local minimizers, Variable exponent

CLC Number: 

  • 35R70
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