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Mathematische Annalen

, Volume 349, Issue 2, pp 367–394 | Cite as

A bilinear estimate for biharmonic functions in Lipschitz domains

  • Joel KiltyEmail author
  • Zhongwei ShenEmail author
Article

Abstract

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain Ω is equivalent to the solvability of the Dirichlet problem for the biharmonic equation in Ω. As a result, we prove that for any given bounded Lipschitz domain Ω in \({\mathbb{R}^{d}}\) and 1 < q < ∞, the solvability of the L q Dirichlet problem for Δ 2 u = 0 in Ω with boundary data in WA 1,q (∂Ω) is equivalent to that of the L p regularity problem for Δ 2 u = 0 in Ω with boundary data in WA 2,p (∂Ω), where \({\frac{1}{p} + \frac{1}{q}=1}\). This duality relation, together with known results on the Dirichlet problem, allows us to solve the L p regularity problem for d ≥ 4 and p in certain ranges.

Mathematics Subject Classification (2000)

35J40 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsCentre CollegeDanvilleUSA
  2. 2.Department of MathematicsUniversity of KentuckyLexingtonUSA

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