# Integral estimates of conformal derivatives and spectral properties of the Neumann-Laplacian

## TL;DR

This paper derives integral estimates for conformal derivatives of mappings onto domains satisfying the Ahlfors condition, leading to spectral bounds for the Neumann-Laplace operator in complex, non-Lipschitz domains like quasidiscs and fractals.

## Contribution

It introduces new integral estimates for conformal derivatives that enable spectral analysis of the Neumann-Laplace operator in non-Lipschitz, fractal-like domains.

## Key findings

- Lower bounds for the first non-trivial eigenvalues in fractal domains
- Estimates of Sobolev-Poincaré constants in quasidiscs
- Spectral estimates linked to conformal geometry

## Abstract

In this paper we study integral estimates of derivatives of conformal mappings $\varphi:\mathbb D\to\Omega$ of the unit disc $\mathbb D\subset\mathbb C$ onto bounded domains $\Omega$ that satisfy the Ahlfors condition. These integral estimates lead to estimates of constants in Sobolev-Poincar\'e inequalities, and by the Rayleigh quotient we obtain spectral estimates of the Neumann-Laplace operator in non-Lipschitz domains (quasidiscs) in terms of the (quasi)conformal geometry of the domains. Specifically, the lower estimates of the first non-trivial eigenvalues of the Neumann-Laplace operator in some fractal type domains (snowflakes) were obtained.