Schwarz-Pick and Landau type theorems for solutions to the Dirichlet-Neumann problem in the unit disk

TL;DR

This paper establishes Schwarz-Pick and Landau type inequalities for solutions to a specific Dirichlet-Neumann boundary value problem in the unit disk, extending classical complex analysis results to this PDE context.

Contribution

It introduces new inequalities and theorems for solutions of the Dirichlet-Neumann problem, generalizing classical results to a PDE setting in the unit disk.

Findings

Derived Schwarz-Pick type inequalities for solutions.

Established Landau type theorems for the problem.

Extended classical complex analysis results to PDE solutions.

Abstract

The aim of this paper is to establish some properties of solutions to the Dirichlet-Neumann problem: $(\partial_z\partial_{\overline{z}})^2 w=g$ in the unit disc $\ID$, $w=\gamma_0$ and $\partial_{\nu}\partial_z\partial_{\overline{z}}w=\gamma$ on $\mathbb{T}$ (the unit circle), $\frac{1}{2\pi i}\int_{\mathbb{T}}w_{\zeta\overline{\zeta}}(\zeta)\frac{d\zeta}{\zeta}=c$, where $\partial_\nu$ denotes differentiation in the outward normal direction. More precisely, we obtain Schwarz-Pick type inequalities and Landau type theorem for solutions to the Dirichlet-Neumann problem.