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

Abstract

In this paper we study integral estimates of derivatives of conformal mappings $\phi :\mathbb{D}\to \Omega$ 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-Poincaré 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.

arXiv comment21 pages, 1 figure