# Spectral estimates of the p-Laplace Neumann operator and Brennan’s conjecture

@article{Goldshtein2017SpectralEO,
title={Spectral estimates of the p-Laplace Neumann operator and Brennan’s conjecture},
author={Vladimir Gol'dshtein and V. A. Pchelintsev and Alexander Ukhlov},
journal={Bollettino dell'Unione Matematica Italiana},
year={2017},
volume={11},
pages={245-264}
}
• Published 18 January 2017
• Mathematics
• Bollettino dell'Unione Matematica Italiana
In this paper we obtain lower estimates for the first non-trivial eigenvalue of the p-Laplace Neumann operator in bounded simply connected planar domains $$\varOmega \subset {\mathbb {R}}^2$$Ω⊂R2. This study is based on a quasiconformal version of the universal two-weight Poincaré–Sobolev inequalities obtained in our previous papers for conformal weights and its non weighted version for so-called K-quasiconformal $$\alpha$$α-regular domains. The main technical tool is the geometric theory of…
10 Citations
• Mathematics
Integral Equations and Operator Theory
• 2018
In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate p-Laplace operator, $$p>2$$p>2, in a large class of non-convex domains. This study is based on
• Mathematics
• 2017
In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate $p$-Laplace operator, $p>2$, in a large class of non-convex domains. This study is based on applications
• Mathematics
Complex Analysis and Operator Theory
• 2018
In this paper we discuss applications of the geometric theory of composition operators on Sobolev spaces to the spectral theory of non-linear elliptic operators. Lower estimates of the first
We study variations of the first nontrivial eigenvalue of the two-dimensional p-Laplace operator, p > 2, generated by measure preserving quasiconformal mappings. The study is based on the geometric
• 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
• Mathematics
Journal of Spectral Theory
• 2020
In this paper we apply estimates of the norms of Sobolev extension operators to the spectral estimates of of the first nontrivial Neumann eigenvalue of the Laplace operator in non-convex extension
• Mathematics
• 2017
We study the variation of the Neumann eigenvalues of the $p$-Laplace operator under quasiconformal perturbations of space domains. This study allows to obtain lower estimates of the Neumann
• Mathematics
Georgian Mathematical Journal
• 2018
Abstract We study the variation of Neumann eigenvalues of the p-Laplace operator under quasiconformal perturbations of space domains. This study allows us to obtain the lower estimates of Neumann
• Materials Science, Mathematics
Complex Analysis and Operator Theory
• 2018
In this paper we discuss applications of the geometric theory of composition operators on Sobolev spaces to the spectral theory of non-linear elliptic operators. Lower estimates of the first
• Mathematics
Analysis and Mathematical Physics
• 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

## References

SHOWING 1-10 OF 41 REFERENCES

• 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
• Mathematics
• 2015
We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains Ω⊂C by reducing it, using conformal transformations, to the weighted eigenvalue problem for the
• 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
• Mathematics
• 2002
Abstract. We prove that a domain in ${\Bbb R}^n$ whose quasihyperbolic metric satisfies a logarithmic growth condition with coefficient $\beta\le 1$ is a (q,p)-\Poincare domain for all p and q
• 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
• Mathematics
• 2004
The classical theory of conformal mappings involves best possible pointwise estimates of the derivative, thus supplying a measure of the extremal expansion/contraction possible for a conformal
• 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