Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Regularity for Very Weak Solutions of A-Harmonic Equation with Weight

  • Gao, Hong-Ya (College of Mathematics and Computer Science, Hebei University) ;
  • Zhang, Yu (College of Mathematics and Computer, Hebei University) ;
  • Chu, Yu-Ming (Faculty of Science, Huzhou Teachers College)
  • Received : 2007.01.18
  • Accepted : 2008.12.26
  • Published : 2009.06.30
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Abstract

This paper deals with very weak solutions of the A-harmonic equation $divA(x,{\nabla}u)$ = 0 (*) with the operator $A:{\Omega}{\times}R^n{\rightarrow}R^n$ satisfies some coercivity and controllable growth conditions with Muckenhoupt weight. By using the Hodge decomposition with weight, a regularity property is proved: There exists an integrable exponent $r_1=r_1({\lambda},n,p)$ < p, such that every very weak solution $u{\in}W_{loc}^{1,r}({\Omega},{\omega})$ with $r_1$ < r < p belongs to $W_{loc}^{1,p}({\Omega},{\omega})$. That is, u is a weak solution to (*) in the usual sense.

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References

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