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arXiv · 1701.02616

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

Vladimir Gol'dshtein·Valerii Pchelintsev·Alexander Ukhlov
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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.

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Vladimir Gol'dshtein, Valerii Pchelintsev, Alexander Ukhlov. 2017-01-10. Integral estimates of conformal derivatives and spectral properties of the Neumann-Laplacian. https://arxiv.org/abs/1701.02616

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