Remarks on "Schwarz-type lemma, Landau-type theorem, and Lipschitz-type space of solutions to inhomogeneous biharmonic equations"
Shaolin Chen, Hidetaka Hamada
TL;DR
The paper investigates Hardy-Littlewood type Lipschitz regularity for solutions to the inhomogeneous biharmonic equation in the unit disk with mixed boundary data. By expressing solutions via the biharmonic Green function and Poisson kernel and decomposing into explicit boundary and interior terms, the authors derive tight Lipschitz bounds controlled by fast majorants ω. They show that a key limsup condition on ω is unnecessary and that certain boundary data restrictions can be weakened, broadening the applicability of Lipschitz regularity results to broader boundary data classes. These results extend classical Lipschitz estimates to inhomogeneous biharmonic problems with general boundary conditions, with potential implications for elasticity theory and related PDE models.
Abstract
Let $\varphi$, $ψ\in C(\mathbb{T})$, $g\in C(\overline{\mathbb{D}})$, where $\mathbb{D}$ and $\mathbb{T}$ denote the unit disk and the unit circle, respectively. Suppose that $f\in C^{4}(\mathbb{D})$ satisfies the following: (1) the inhomogeneous biharmonic equation $ Δ(Δf(z))=g(z)$ for $z\in\mathbb{D}$, (2) the Dirichlet boundary conditions $\partial_{\overline{z}}f(ζ)=\varphi(ζ)$ and $f(ζ)=ψ(ζ)$ for $ζ\in\mathbb{T}$. Recently, the authors in [J. Geom. Anal. 29: 2469-2491, 2019] showed that if $ω$ is a majorant with $\limsup_{t\rightarrow0^{+}}\left(ω(t)/t\right)<\infty$, $ψ=0$ and $\varphi_1 \in\mathscr{L}_ω(\mathbb{T})$, where $\varphi_1(e^{it})=\varphi(e^{it})e^{-it}$ for $t\in[0,2π]$, then $f\in\mathscr{L}_ω(\mathbb{D})$. The purpose of this paper is to improve and generalize this result. We not only prove that the condition "$\limsup_{t\rightarrow0^{+}}\left(ω(t)/t\right)<\infty$" is redundant, but also demonstrate that conditions "$ψ=0$" and "$\varphi_1\in\mathscr{L}_ω(\mathbb{T})$" can be replaced by weaker conditions.