A bilinear estimate for biharmonic functions in Lipschitz domains

作者：Kilty Joel^*; Shen Zhongwei

来源：Mathematische Annalen, 2011, 349(2): 367-394.

DOI：10.1007/s00208-010-0522-6

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain Omega is equivalent to the solvability of the Dirichlet problem for the biharmonic equation in Omega. As a result, we prove that for any given bounded Lipschitz domain Omega in R-d and 1 < q < infinity, the solvability of the L-q Dirichlet problem for Delta (2) u = 0 in Omega with boundary data in WA (1,q) (partial derivative,Omega) is equivalent to that of the L (p) regularity problem for Delta (2) u = 0 in Omega with boundary data in WA (2,p) (partial derivative,Omega), where 1/p + 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.

• 出版日期2011-2