# 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} }

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

### On the First Eigenvalue of the Degenerate $$\varvec{p}$$p-Laplace Operator in Non-convex Domains

- MathematicsIntegral 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…

### On the First Eigenvalue of the Degenerate $p$-Laplace Operator in Non-Convex Domains

- 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…

### Composition Operators on Sobolev Spaces and Neumann Eigenvalues

- MathematicsComplex 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…

### On Variations of the Neumann Eigenvalues of p-Laplacian Generated by Measure Preserving Quasiconformal Mappings

- MathematicsJournal of Mathematical Sciences
- 2021

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…

### 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…

### Sobolev extension operators and Neumann eigenvalues

- MathematicsJournal 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…

### Space Quasiconformal Mappings and Neumann Eigenvalues in Fractal Domains

- 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…

### Space quasiconformal mappings and Neumann eigenvalues in fractal type domains

- MathematicsGeorgian 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…

### Composition Operators on Sobolev Spaces and Neumann Eigenvalues

- Materials Science, MathematicsComplex 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…

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

- MathematicsAnalysis 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

### 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…

### Conformal spectral stability estimates for the Dirichlet Laplacian

- 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…

### 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…

### Quasihyperbolic boundary conditions and Poincaré domains

- 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…

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

- MathematicsProceedings 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…

### Weighted Bergman spaces and the integral means spectrum of conformal mappings

- 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…

### 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…