TY - JOUR

T1 - A bilinear estimate for biharmonic functions in Lipschitz domains

AU - Kilty, Joel

AU - Shen, Zhongwei

N1 - Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2011/2

Y1 - 2011/2

N2 - 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 ℝd and 1 < q < ∞, the solvability of the Lq Dirichlet problem for Δ2u = 0 in Ω with boundary data in WA1,q(∂Ω) is equivalent to that of the Lp regularity problem for Δ2u = 0 in Ω with boundary data in WA2,p(∂Ω), where 1/p + 1/q = 1. This duality relation, together with known results on the Dirichlet problem, allows us to solve the Lp regularity problem for d ≥ 4 and p in certain ranges.

AB - 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 ℝd and 1 < q < ∞, the solvability of the Lq Dirichlet problem for Δ2u = 0 in Ω with boundary data in WA1,q(∂Ω) is equivalent to that of the Lp regularity problem for Δ2u = 0 in Ω with boundary data in WA2,p(∂Ω), where 1/p + 1/q = 1. This duality relation, together with known results on the Dirichlet problem, allows us to solve the Lp regularity problem for d ≥ 4 and p in certain ranges.

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U2 - 10.1007/s00208-010-0522-6

DO - 10.1007/s00208-010-0522-6

M3 - Article

AN - SCOPUS:78651253166

SN - 0025-5831

VL - 349

SP - 367

EP - 394

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 2

ER -