# Spectral properties of the Neumann-Laplace operator in quasiconformal regular domains

@article{Goldshtein2019SpectralPO,
title={Spectral properties of the Neumann-Laplace
operator in quasiconformal regular domains},
author={Vladimir Gol'dshtein and V. A. Pchelintsev and A. Ukhlov},
journal={Contemporary Mathematics},
year={2019}
}
• Published 10 March 2017
• Mathematics
• Contemporary Mathematics
In this paper we study spectral properties of the Neumann-Laplace operator in planar quasiconformal regular domains $\Omega\subset\mathbb R^2$. This study is based on the quasiconformal theory of composition operators on Sobolev spaces. Using the composition operators theory we obtain estimates of constants in Poincare-Sobolev inequalities and as a consequence lower estimates of the first non-trivial eigenvalue of the Neumann-Laplace operator in planar quasiconformal regular domains.
7 Citations

## Figures from this paper

### Space quasiconformal composition operators with applications to Neumann eigenvalues

• Mathematics
• 2020
In this article we obtain estimates of Neumann eigenvalues of p -Laplace operators in a large class of space domains satisfying quasihyperbolic boundary conditions. The suggested method is based on

### Estimates of Dirichlet eigenvalues of divergent elliptic operators in non-Lipschitz domains

• Mathematics
• 2020
We study spectral estimates of the divergence form uniform elliptic operators $-\textrm{div}[A(z) \nabla f(z)]$ with the Dirichlet boundary condition in bounded non-Lipschitz simply connected domains

• Mathematics
• 2020

## References

SHOWING 1-10 OF 37 REFERENCES

### Conformal spectral stability for the Dirichlet-Laplace operator

• Mathematics
• 2014
We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains $\Omega\subset\mathbb{C}$ using conformal transformations of the original problem to the weighted

### On the First Eigenvalues of Free Vibrating Membranes in Conformal Regular Domains

• Mathematics
• 2016
AbstractIn 1961 G. Polya published a paper about the eigenvalues of vibrating membranes. The “free vibrating membrane” corresponds to the Neumann–Laplace operator in bounded plane domains. In this

### Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems

• Mathematics
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 2015
In this paper we prove a sharp lower bound for the first non-trivial Neumann eigenvalue μ1(Ω) for the p-Laplace operator (p > 1) in a Lipschitz bounded domain Ω in ℝn. Our estimate does not require

### Conformal Weights and Sobolev Embeddings

• Mathematics
• 2013
We study embeddings of the Sobolev space $${\mathop{W}\limits_{~}^{\circ}}{{~}_2^1}\left( \Omega \right)$$ into weighted Lebesgue spaces Lq(Ω, h) with the so-called universal conformal weight h

### Best constants in Poincaré inequalities for convex domains

• Mathematics
• 2011
We prove a Payne-Weinberger type inequality for the $p$-Laplacian Neumann eigenvalues ($p\ge 2$). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of

### Weighted Sobolev spaces and embedding theorems

• Mathematics
• 2007
In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A p -condition. Sufficient conditions for boundedness and compactness of

### Nonlinear Potential Theory of Degenerate Elliptic Equations

• Mathematics
• 1993
Introduction. 1: Weighted Sobolev spaces. 2: Capacity. 3: Supersolutions and the obstacle problem. 4: Refined Sobolev spaces. 5: Variational integrals. 6: A-harmonic functions. 7: A superharmonic

### Brennan's conjecture for composition operators on Sobolev spaces

• Mathematics
• 2012
We show that Brennan's conjecture is equivalent to boundedness of composition operators on homogeneous Sobolev spaces generated by conformal homeomorphisms of simply connected plane domains to the