# On the strict monotonicity of the first eigenvalue of the π-Laplacian on annuli

@article{Anoop2018OnTS, title={On the strict monotonicity of the first eigenvalue of the π-Laplacian on annuli}, author={T. V. Anoop and Vladimir Bobkov and Sarath Sasi}, journal={Transactions of the American Mathematical Society}, year={2018} }

<p>Let <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B 1">
<mml:semantics>
<mml:msub>
<mml:mi>B</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:annotation encoding="application/x-tex">B_1</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> be a ball in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Rβ¦Β

## Figures from this paper

## 10 Citations

On reverse Faber-Krahn inequalities

- Mathematics, Physics
- 2018

Payne-Weinberger showed that \textit{`among the class of membranes with given area $A$, free along the interior boundaries and fixed along the outer boundary of given length $L_0$, the annulusβ¦

Optimal Shapes for the First Dirichlet Eigenvalue of the $p$-Laplacian and Dihedral symmetry

- Mathematics
- 2021

In this paper, we consider the optimization problem for the first Dirichlet eigenvalue Ξ»1(Ξ©) of the p-Laplacian βp, 1 < p <β, over a family of doubly connected planar domains Ξ© = B\P , where B is anβ¦

On the first SteklovβDirichlet eigenvalue for eccentric annuli

- MathematicsAnnali di Matematica Pura ed Applicata (1923 -)
- 2021

In this paper, we investigate the first SteklovβDirichlet eigenvalue on eccentric annuli. The main geometric parameter is the distance t between the centers of the inner and outer boundaries of anβ¦

On qualitative properties of solutions for elliptic problems with the p-Laplacian through domain perturbations

- Mathematics, PhysicsCommunications in Partial Differential Equations
- 2019

Abstract We study the dependence of least nontrivial critical levels of the energy functional corresponding to the zero Dirichlet problem in a bounded domain upon domain perturbations. Assuming thatβ¦

The Cheeger constant of curved tubes

- Mathematics, PhysicsArchiv der Mathematik
- 2019

We compute the Cheeger constant of tubular neighbourhoods of complete curves in an arbitrary dimensional Euclidean space and raise a question about curved spherical shells.

On the placement of an obstacle so as to optimize the Dirichlet heat content

- Mathematics
- 2021

We prove that among all doubly connected domains of Rn (n β₯ 2) bounded by two spheres of given radii, the Dirichlet heat content at any fixed time achieves its minimum when the spheres areβ¦

A shape variation result via the geometry of eigenfunctions

- Mathematics
- 2020

We discuss some of the geometric properties, such as the foliated Schwarz symmetry, the monotonicity along the axial and the affine-radial directions, of the first eigenfunctions of the Zarembaβ¦

Second-order derivative of domain-dependent functionals along Nehari manifold trajectories

- Physics, MathematicsESAIM: Control, Optimisation and Calculus of Variations
- 2020

Assume that a family of domain-dependent functionals EΞ©t possesses a corresponding family of least energy critical points ut which can be found as (possibly nonunique) minimizers of EΞ©t over theβ¦

Shape monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli

- Mathematics, Physics
- 2020

In this paper, we investigate the monotonicity of the first Steklov--Dirichlet eigenvalue on eccentric annuli with respect to the distance, namely $t$, between the centers of the inner and outerβ¦

Second-order shape derivative along Nehari manifold trajectories

- Physics
- 2018

Assume that a family of domain-dependent functionals $E_{\Omega_t}$ possesses a corresponding family of least energy critical points $u_t$ which can be found as (possibly nonunique) minimizers ofβ¦

## References

SHOWING 1-10 OF 31 REFERENCES

On the p-torsion functions of an annulus

- Mathematics, Computer ScienceAsymptot. Anal.
- 2015

A new proof of the calibrabiliy of a;b is given by combining Pohozaevβs identity for the p-torsional creep problem with a kind of LβHospital rule for monotonicity based on estimates for m p ; the radius of the sphere on which u p assumes its maximum value.

THE BEGINNING OF THE FUCIK SPECTRUM FOR THE P-LAPLACIAN

- Mathematics
- 1999

has a nontrivial solution u. Here 1<p< , 2p u= div( |{u| p&2 {u), u= max[u, 0], u=uu in (1.1). Denoting by *1<*2 the first two eigenvalues of &2p on W 1, p 0 (0) (cf. the end of this introduction),β¦

An eigenvalue optimization problem for the p-Laplacian

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2015

It has been shown by Kesavan (Proc. R. Soc. Edinb. A (133) (2003), 617β624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with the puncture having the shape of a ball, isβ¦

ON THE FABER-KRAHN INEQUALITY FOR THE DIRICHLET p-LAPLACIAN β

- Mathematics
- 2015

A famous conjecture made by Lord Rayleigh is the following: βThe first eigenvalue of the Laplacian on an open domain of given measure with Dirichlet boundary conditions is minimum when the domain isβ¦

On the Placement of an Obstacle or a Well so as to Optimize the Fundamental Eigenvalue

- Computer Science, MathematicsSIAM J. Math. Anal.
- 2001

We investigate how to place an obstacle B within a domain $\Omega$ in Euclidean space so as to maximize or minimize the principal Dirichlet eigenvalue for the Laplacian on $\Omega \setminus B$. Theβ¦

The β-Eigenvalue Problem

- Mathematics
- 1999

Abstract. The EulerβLagrange equation of the nonlinear Rayleigh quotient
$$ \left(\int_{\Omega}|\nabla u|^{p}\,dx\right) \bigg/ \left(\int_{\Omega}|u|^{p}\,dx\right)$$ is
$$ -\div\left( |\nablaβ¦

On the structure of the second eigenfunctions of the -Laplacian on a ball

- Mathematics
- 2015

In this paper, we prove that the second eigenfunctions of the $p$-Laplacian, $p>1$, are not radial on the unit ball in $\mathbb{R}^N,$ for any $N\ge 2.$ Our proof relies on the variationalβ¦

Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant

- Mathematics
- 2003

First we recall a Faber-Krahn type inequality and an estimate forp() in terms of the so-called Cheeger constant. Then we prove that the eigenvaluep() converges to the Cheeger constant h() as p β 1.β¦

A Strong Maximum Principle for some quasilinear elliptic equations

- Mathematics
- 1984

In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain Ξ© β βn,n β©Ύ 1, is in fact positive everywhere. Here we prove that the sameβ¦

Inequalities for the minimal eigenvalue of the Laplacian in an annulus

- Mathematics
- 1998

We discuss the behavior of the minimal eigenvalue of the Dirichlet Laplacian in the domain D1\D2 := D (an annulus) where D1 is a circular disc and D2 D1 is a smaller circular disc. It is conjecturedβ¦