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

@article{Goldshtein2015OnTF, title={On the First Eigenvalues of Free Vibrating Membranes in Conformal Regular Domains}, author={Vladimir Gol'dshtein and A. Ukhlov}, journal={Archive for Rational Mechanics and Analysis}, year={2015}, volume={221}, pages={893-915} }

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 paper we obtain estimates for the first
non-trivial eigenvalue of this operator in a large class of domains that we call conformal regular domains. This class includes convex domains, John domains etc. On the basis of our estimates we conjecture that the eigenvalues of the Neumann–
Laplace operator…

## 34 Citations

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

- MathematicsContemporary Mathematics
- 2019

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…

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

- Mathematics
- 2017

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.…

### Principal frequencies of free non-homogeneous membranes and Sobolev extension operators

- Mathematics
- 2021

Using the quasiconformal mappings theory and Sobolev extension operators, we obtain estimates of principal frequencies of free non-homogeneous membranes. The suggested approach is based on…

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

- Mathematics
- 2020

In this paper we study variations of the first non-trivial eigenvalues of the two-dimensional $p$-Laplace operator, $p>2$, generated by measure preserving quasiconformal mappings $\varphi : \mathbb…

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

- MathematicsJournal of Mathematical Analysis and Applications
- 2018

### Conformal spectral stability estimates for the Neumann Laplacian

- Mathematics
- 2016

We study the eigenvalue problem for the Neumann–Laplace operator in conformal regular planar domains Ω⊂C . Conformal regular domains support the Poincaré–Sobolev inequality and this allows us to…

### Asymptotic analysis and numerical computation of the Laplacian eigenvalues using the conformal mapping

- MathematicsArXiv
- 2021

An asymptotic formula for the Laplace eigenvalues with respect to the perturbation of the domain is obtained and a fully computable a priori error estimate with no assumption on the domain’s convexity is derived.

### 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…

### On conformal spectral gap estimates of the Dirichlet–Laplacian

- Mathematics
- 2018

We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains $\Omega\subset\mathbb R^2$. With the help of these estimates we obtain asymptotically sharp…

## References

SHOWING 1-10 OF 48 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…

### 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…

### 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…

### Sharp Estimates for Eigenfunctions of a Neumann Problem

- Mathematics
- 2009

In this paper we provide some bounds for eigenfunctions of the Laplacian with homogeneous Neumann boundary conditions in a bounded domain Ω of ℝ n . To this aim we use the so-called symmetrization…

### Poincaré inequalities in quasihyperbolic boundary condition domains

- Mathematics
- 2015

We study the validity of (q, p)-Poincaré inequalities, q < p, on domains in $${\mathbb{R}^n}$$Rn which satisfy a quasihyperbolic boundary condition, i.e. domains whose quasihyperbolic metric…

### 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…

### 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…

### Applications of change of variables operators for exact embedding theorems

- Mathematics
- 1994

We propose here a new method for the investigation of embedding operators. It is based on an exact description of classes of homeomorphisms that induce change of variables operators on the Sobolev…