Boundary value problems and integral operators for the bi-Laplacian in non-smooth domains (Q374113)

The Dirichlet and Neumann boundary problems for the bi-Laplacian \(\Delta^2\) in a bounded Lipschitz domain \(\Omega\subset\mathbb{R}^n\) with boundary \(\partial\Omega\) were studied by \textit{G. C. Verchota} [Acta Math. 194, No. 2, 217--279 (2005; Zbl 1216.35021)] and by \textit{Z. Shen} [Adv. Math. 216, No. 1, 212--254 (2007; Zbl 1210.35080)]. The paper under review refines and extends the cited work. Existence and regularity of the boundary problems are analysed for various spaces, also for the Besov and Triebel-Lizorkin scales, \(B_s^{p,q}\) and \(F_s^{p,q}\). Invertibility of multi-layer operators, which is basic to the proofs of the main results, holds only for a restricted range for the exponents \(p\). It is shown that the range becomes larger when the mean oscillation of the exterior normal \(\nu\) becomes sufficiently small. In fact, the range is \(1<p<\infty\) when \(\nu\) belongs to the Sarason space \(VMO(\partial\Omega)\), in particular, when \(\Omega\) is a \(C^1\) domain. The proof is based on a result about a class of singular integral operators \(T\) on \(\partial\Omega\), Theorem 4.36 of NEWLINE\textit{S. Hofmann} et al. [Int. Math. Res. Not. 2010, No. 14, 2567--2865 (2010; Zbl 1221.31010)], which says roughly that the distance from \(T\) to the space of compact operators tends to zero if the distance in \(BMO(\partial\Omega)\) from \(\nu\) to \(VMO(\Omega)\) does.