Faber Krahn Type Inequality Research Articles

Stability of the Gaussian Faber–Krahn inequality

We prove a quantitative version of the Gaussian Faber–Krahn type inequality proved in (Betta et al. in Z. Angew. Math. Phys. 58:37–52, 2007) for the first Dirichlet eigenvalue of the Ornstein–Uhlenbeck operator, estimating the deficit in terms of the Gaussian Fraenkel asymmetry. As expected, the multiplicative constant only depends on the prescribed Gaussian measure.

Estimates for Robin p-Laplacian eigenvalues of convex sets with prescribed perimeter

In this paper, we prove an upper bound for the first Robin eigenvalue of the p-Laplacian with a positive boundary parameter and a quantitative version of the reverse Faber-Krahn type inequality for the first Robin eigenvalue of the p-Laplacian with negative boundary parameter, among convex sets with prescribed perimeter. The proofs are based on a comparison argument obtained by means of inner sets, introduced by Payne, Weinberger [32] and Pólya [33].

Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality

For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $\lambda _1(\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ (with mixed boundary conditions) under the polarizations. We apply this inequality to the obstacle problems on the domains of the form $\Omega \setminus \mathscr{O}$, where $\mathscr{O}\subset \subset \Omega $ is an obstacle. Under some geometric assumptions on $\Omega $ and $\mathscr{O}$, we prove the strict monotonicity of $\lambda _1 (\Omega \setminus \mathscr{O})$ with respect to certain translations and rotations of $\mathscr{O}$ in $\Omega $.

Weighted symmetrization results for a problem with variable Robin parameter

By means of a suitable weighted rearrangement, we obtain various apriori bounds for the solutions to a Robin problem. Among other things, we derive a family of Faber-Krahn type inequalities.

The logarithmic Schrödinger operator and associated Dirichlet problems

In this note, we study the integrodifferential operator (I−Δ)log corresponding to the logarithmic symbol log⁡(1+|ξ|2), which is a singular integral operator given by(I−Δ)logu(x)=dN∫RNu(x)−u(x+y)|y|Nω(|y|)dy, where dN=π−N2, ω(r)=21−N2rN2KN2(r) and Kν is the modified Bessel function of second kind with index ν. This operator is the Lévy generator of the variance gamma process and arises as derivative ∂s|s=0(I−Δ)s of fractional relativistic Schrödinger operators at s=0. In order to study associated Dirichlet problems in bounded domains, we first introduce the functional analytic framework and some properties related to (I−Δ)log, which allow to characterize the induced eigenvalue problem and Faber-Krahn type inequality. We also derive a decay estimate in RN of the Poisson problem and investigate small order asymptotics s→0+ of Dirichlet eigenvalues and eigenfunctions of (I−Δ)s in a bounded open Lipschitz set.

A Variational Formulation for Dirac Operators in Bounded Domains. Applications to Spectral Geometric Inequalities

We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szeg\"o type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences supporting the existence of a Faber-Krahn type inequality.

A quantitative reverse Faber-Krahn inequality for the first Robin eigenvalue with negative boundary parameter

The aim of this paper is to prove a quantitative form of a reverse Faber-Krahn type inequality for the first Robin Laplacian eigenvalueλβwith negative boundary parameter among convex sets of prescribed perimeter. In that framework, the ball is the only maximizer forλβand the distance from the optimal set is considered in terms of Hausdorff distance. The key point of our stategy is to prove a quantitative reverse Faber-Krahn inequality for the first eigenvalue of a Steklov-type problem related to the original Robin problem.

A Pólya–Szegö principle for general fractional Orlicz–Sobolev spaces

In this article, we prove modular and norm Pólya–Szegö inequalities in general fractional Orlicz–Sobolev spaces by using the polarization technique. We introduce a general framework which includes the different definitions of these spaces in the literature, and we establish some of its basic properties such as the density of smooth functions. As a corollary, we prove a Rayleigh–Faber–Krahn type inequality for Dirichlet eigenvalues under nonlocal nonstandard growth operators.

Faber-Krahn type inequalities and uniqueness of positive solutions on metric measure spaces

We consider a general class of metric measure spaces equipped with a strongly local regular Dirichlet form and provide a lower bound on the hitting time probabilities of the associated Hunt process. Using these estimates we establish (i) a generalization of the classical Lieb's inequality, and (ii) uniqueness of nonnegative super-solutions to semi-linear elliptic equations on metric measure spaces. Finally, using heat-kernel estimates we generalize the local Faber-Krahn inequality recently obtained in [28] to local and non-local Dirichlet spaces.

The Dirichlet problem for the logarithmic Laplacian

In this article, we study the logarithmic Laplacian operator which is a singular integral operator with symbol We show that this operator has the integral representationwith and where Γ is the Gamma function, is the Digamma function and is the Euler Mascheroni constant. This operator arises as formal derivative of fractional Laplacians at We develop the functional analytic framework for Dirichlet problems involving the logarithmic Laplacian on bounded domains and use it to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of as As a byproduct, we then derive a Faber-Krahn type inequality for the principal Dirichlet eigenvalue of Using this inequality, we also establish conditions on domains giving rise to the maximum principle in weak and strong forms. This allows us to also derive regularity up to the boundary of solutions to corresponding Poisson problems.

Universal constraints on the location of extrema of eigenfunctions of non-local Schrödinger operators

We derive a lower bound on the location of global extrema of eigenfunctions for a large class of non-local Schrödinger operators in convex domains under Dirichlet exterior conditions, featuring the symbol of the kinetic term, the strength of the potential, and the corresponding eigenvalue, and involving a new universal constant. We show a number of probabilistic and spectral geometric implications, and derive a Faber–Krahn type inequality for non-local operators. Our study also extends to potentials with compact support, and we establish bounds on the location of extrema relative to the boundary edge of the support or level sets around minima of the potential.

Some eigenvalue comparison theorems of Finsler p-Laplacian

We obtain Cheng type inequality, Cheeger type inequality, Faber–Krahn type inequality and McKean type inequality of [Formula: see text]-Laplacian on a Finsler manifold. These generalize the corresponding theorems in Riemannian geometry and sharpen some results in recent literatures. Moreover, for a complete noncompact Finsler manifold with negative constant flag curvature and vanishing [Formula: see text] curvature, the first eigenvalue is calculated.

The eigenvalue problem for the $$\infty $$-Bilaplacian

We consider the problem of finding and describing minimisers of the Rayleigh quotient $$\begin{aligned} \Lambda _\infty \, :=\, \inf _{u\in \mathcal {W}^{2,\infty }(\Omega )\setminus \{0\} }\frac{\Vert \Delta u\Vert _{L^\infty (\Omega )}}{\Vert u\Vert _{L^\infty (\Omega )}}, \end{aligned}$$where \(\Omega \subseteq \mathbb {R}^n\) is a bounded \(C^{1,1}\) domain and \(\mathcal {W}^{2,\infty }(\Omega )\) is a class of weakly twice differentiable functions satisfying either \(u=0\) on \(\partial \Omega \) or \(u=|\mathrm {D}u|=0\) on \(\partial \Omega \). Our first main result, obtained through approximation by \(L^p\)-problems as \(p\rightarrow \infty \), is the existence of a minimiser \(u_\infty \in \mathcal {W}^{2,\infty }(\Omega )\) satisfying $$\begin{aligned} \left\{ \begin{array}{ll} \Delta u_\infty \, \in \, \Lambda _\infty \mathrm {Sgn}(f_\infty ) &{} \text { a.e. in }\Omega , \\ \Delta f_\infty \, =\, \mu _\infty &{} \text { in }\mathcal {D}'(\Omega ), \end{array} \right. \end{aligned}$$for some \(f_\infty \in L^1(\Omega )\cap BV_{\text {loc}}(\Omega )\) and a measure \(\mu _\infty \in \mathcal {M}(\Omega )\), for either choice of boundary conditions. Here Sgn is the multi-valued sign function. We also study the dependence of the eigenvalue \(\Lambda _\infty \) on the domain, establishing the validity of a Faber–Krahn type inequality: among all \(C^{1,1}\) domains with fixed measure, the ball is a strict minimiser of \(\Omega \mapsto \Lambda _\infty (\Omega )\). This result is shown to hold true for either choice of boundary conditions and in every dimension.

Transmission eigenvalue problem for inhomogeneous absorbing media with mixed boundary condition

Consider the transmission eigenvalue problem for the wave scattering by a dielectric inhomogeneous absorbing obstacle lying on a perfect conducting surface. After excluding the purely imaginary transmission eigenvalues, we prove that the transmission eigenvalues exist and form a discrete set for inhomogeneous non-absorbing media, by using analytic Fredholm theory. Moreover, we derive the Faber-Krahn type inequalities revealing the lower bounds on real transmission eigenvalues in terms of the media parameters. Then, for inhomogeneous media with small absorption, we prove that the transmission eigenvalues also exist and form a discrete set by using perturbation theory. Finally, for homogeneous media, we present possible components of the eigenvalue-free zone quantitatively, giving the geometric understanding on this problem.

Limiting Sobolev inequalities and the 1-biharmonic operator

Abstract In this article we present recent results on optimal embeddings, and associated PDEs, of the space of functions whose distributional Laplacian belongs to L1 . We discuss sharp embedding inequalities which allow to improve the optimal summability results for solutions of Poisson equations with L1 -data by Maz'ya (N ≥ 3) and Brezis–Merle (N = 2). Then, we consider optimal embeddings of the mentioned space into L1 , for the simply supported and the clamped case, which yield corresponding eigenvalue problems for the 1-biharmonic operator (a higher order analogue of the 1-Laplacian). We derive some properties of the corresponding eigenfunctions, and prove some Faber–Krahn type inequalities.

The interior transmission problem for regions on a conducting surface

We consider the interior transmission problem corresponding to inverse scattering for a bounded isotropic dielectric medium lying on an infinite conducting surface. In particular, we investigate the 2D scalar case of this problem where, in the corresponding scattering problem, the dielectric medium is illuminated by time harmonic transverse-electric (TE) or transverse-magnetic (TM) polarized electromagnetic waves, respectively. In both cases we establish the Fredholm property for this problem and show that transmission eigenvalues exist and form a discrete set. We also derive Faber–Krahn type inequalities for the transmission eigenvalues. Numerical results for the TE and TM cases are given showing that real transmission eigenvalues can be found from near field data, although in some cases the accuracy requirements on the data is very stringent.

A numerical method to compute interior transmission eigenvalues

In this paper the numerical calculation of eigenvalues of the interior transmission problem arising in acoustic scattering for constant contrast in three dimensions is considered. From the computational point of view existing methods are very expensive, and are only able to show the existence of such transmission eigenvalues. Furthermore, they have trouble finding them if two or more eigenvalues are situated closely together. We present a new method based on complex-valued contour integrals and the boundary integral equation method which is able to calculate highly accurate transmission eigenvalues. So far, this is the first paper providing such accurate values for various surfaces different from a sphere in three dimensions. Additionally, the computational cost is even lower than those of existing methods. Furthermore, the algorithm is capable of finding complex-valued eigenvalues for which no numerical results have been reported yet. Until now, the proof of existence of such eigenvalues is still open. Finally, highly accurate eigenvalues of the interior Dirichlet problem are provided and might serve as test cases to check newly derived Faber–Krahn type inequalities for larger transmission eigenvalues that are not yet available.

On eigenvalue problems of the one-speed neutron transport equation for isotropic scattering

In this paper, we consider eigenvalue problems of the one-speed neutron transport equation with isotropic scattering in a steady state and prove the Rayleigh–Faber–Krahn type inequality for the first eigenvalue.

Faber–Krahn type inequality for unicyclic graphs

The Faber–Krahn inequality is related to the smallest nonzero eigenvalue of the Laplacian operator with Dirichlet boundary condition on a bounded domain in ℝ n . In this article, we investigate some properties of the first Dirichlet eigenvalue of unicyclic graphs. These results are used to show that the Faber–Krahn type inequality also holds for unicyclic graphs with a given graphic unicyclic degree sequence with minor conditions.

Algorithm 922

Transmission eigenvalue problem has important applications in inverse scattering. Since the problem is non-self-adjoint, the computation of transmission eigenvalues needs special treatment. Based on a fourth-order reformulation of the transmission eigenvalue problem, a mixed finite element method is applied. The method has two major advantages: 1) the formulation leads to a generalized eigenvalue problem naturally without the need to invert a related linear system, and 2) the nonphysical zero transmission eigenvalue, which has an infinitely dimensional eigenspace, is eliminated. To solve the resulting non-Hermitian eigenvalue problem, an iterative algorithm using restarted Arnoldi method is proposed. To make the computation efficient, the search interval is decided using a Faber-Krahn type inequality for transmission eignevalues and the interval is updated at each iteration. The algorithm is implemented using Matlab. The code can be easily used in the qualitative methods in inverse scattering and modified to compute transmission eigenvalues for other models such as elasticity problem.