For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $λ_1(Ω)$ of the $p$-Laplace operator on a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$ (with mixed boundary conditions) under the polarizations. We apply this inequality to the obstacle problems on the domains of the form $Ω\setminus \mathscr{O}$, where $\mathscr{O}\subset \subset Ω$ is an obstacle. Under some geometric assumptions on $Ω$ and $\mathscr{O}$, we prove the strict monotonicity of $λ_1 (Ω\setminus \mathscr{O})$ with respect to certain translations and rotations of $\mathscr{O}$ in $Ω$.
We prove a quantitative version of the Faber-Krahn inequality for the first eigenvalue of the fractional Dirichlet-Laplacian of order s. This is done by using the so-called Caffarelli-Silvestre extension and adapting to the nonlocal setting a trick by Hansen and Nadirashvili. The relevant stability estimate comes with an explicit constant, which is stable as the fractional order of differentiability goes to 1.
We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for a class of very degenerate elliptic operators, with the aim to show that, at least for square type domains having fixed volume, the symmetry of the domain maximize the principal eigenvalue, contrary to what happens for the Laplacian.
For a given bounded Lipschitz set $Ω$, we consider a Steklov--type eigenvalue problem for the Laplacian operator whose solutions provide extremal functions for the compact embedding $H^1(Ω)\hookrightarrow L^2(\partial Ω)$. We prove that a conjectured reverse Faber--Krahn inequality holds true at least in the class of Lipschitz sets which are "close" to a ball in a Hausdorff metric sense. The result implies that among sets of prescribed measure, balls are local minimizers of the embedding constant.
In this paper we study the main properties of the first eigenvalue and its eigenfunctions of a class of highly nonlinear elliptic operators in a bounded Lipschitz domain, assuming a Robin boundary condition. Moreover, we prove a Faber-Krahn inequality.
In this paper we prove a Faber-Krahn type inequality for the first eigenvalue of the Hermite operator with Robin boundary condition. We prove that the optimal set is an half-space and we also address the equality case in such inequality.
The eigenvalue problem for the p-Laplace operator with Robin boundary condition is considered in this paper. A Faber-Krahn type inequality is proved. More precisely, it is shown that amongst all the domains of fixed volume, the ball has the smallest first eigenvalue.
We consider a general class of metric measure spaces equipped with a regular Dirichlet form and then 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 on metric measure spaces and (ii) uniqueness of nonnegative super-solutions on metric measure spaces. Finally, using heat-kernel estimates we generalize the local Faber-Krahn inequality recently obtained in [LS18].
We prove a quantitative Faber-Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. The asymmetry term involves the square power of the Fraenkel asymmetry, multiplied by a constant depending on the Robin parameter, the dimension of the space and the measure of the set.
We consider the first Dirichlet eigenvalue problem for a mixed local/nonlocal elliptic operator and we establish a quantitative Faber-Krahn inequality. More precisely, we show that balls minimize the first eigenvalue among sets of given volume and we provide a stability result for sets that almost attain the minimum.
When revisiting the Faber-Krahn inequality for the principal $p$-Laplacian eigenvalue of a bounded open set in $\mathbb R^n$ with smooth boundary, we simply rename it as the $p$-Faber-Krahn inequality and interestingly find that this inequality may be improved but also characterized through Maz'ya's capacity method, the Euclidean volume, the Sobolev type inequality and Moser-Trudinger's inequality.
The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman. More generally, the result applies to every optimal Poincaré-Sobolev constant for the embeddings $W^{1,2}_0(Ω)\hookrightarrow L^q(Ω)$.
On Faber-Krahn inequality for Dirichlet Fractional-Logarithmic Laplacian
Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）；Brandenburg University of Technology Cottbus–Senftenberg（勃兰登堡工业大学科特布斯-森夫滕贝格分校）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）
In this paper, we study the minimization of $λ_{1}(Ω)$, the first Dirichlet eigenvalue of the Laplace-Beltrami operator, within the class of open sets $Ω$ of fixed volume in a Riemmanian manifold $(M,g)$. In the Euclidian setting (when $(M,g)=(\mathbb{R}^n,e)$), the well-known Faber-Krahn inequality asserts that the solution of such problem is any ball of suitable volume. Even if similar results are known or may be expected for Riemannian manifolds with symmetries, we cannot expect to find explicit solutions for general manifolds $(M,g)$. In this paper we study existence and regularity propert…
Let $Ω$ be a bounded $C^{2,α}$ domain in $\R^n$ ($n\geq 1$, $0<α<1$), $Ω^{\ast}$ be the open Euclidean ball centered at 0 having the same Lebesgue measure as $Ω$, $τ\geq 0$ and $v\in L^{\infty}(Ω,\R^n)$ with $\left\Vert v\right\Vert\_{\infty}\leq τ$. If $λ\_{1}(Ω,τ)$ denotes the principal eigenvalue of the operator $-Δ+v\cdot\nabla$ in $Ω$ with Dirichlet boundary condition, we establish that $λ\_{1}(Ω,v)\geq λ\_{1}(Ω^{\ast},τe\_{r})$ where $e\_{r}(x)=x/| x|$. Moreover, equality holds only when, up to translation, $Ω=Ω^{\ast}$ and $v=τe\_{r}$. This result can be viewed as an isoperimetric inequ…
In this paper we prove a reverse Faber-Krahn inequality for the principal eigenvalue $μ_1(Ω)$ of the fully nonlinear eigenvalue problem \[ \label{eq} \left\{\begin{array}{r c l l} -λ_N(D^2 u) & = & μu & \text{in }Ω, \\ u & = & 0 & \text{on }\partial Ω. \end{array}\right. \] Here $ λ_N(D^2 u)$ stands for the largest eigenvalue of the Hessian matrix of $u$. More precisely, we prove that, for an open, bounded, convex domain $Ω\subset \mathbb{R}^N$, the inequality \[ μ_1(Ω) \leq \frac{π^2}{[\text{diam}(Ω)]^2} = μ_1(B_{\text{diam}(Ω)/2}),\] where $\text{diam}(Ω)$ is the diameter of $Ω$, holds true.…
We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators. In particular, we show that there exists a compactly supported nonnegative Sobolev function $u_0$ that attains the infimum (which will be a positive real number) of the set \[ \left\{ \int\int_{\{u > 0\}\times\{u>0\}} \frac{|u(x) - u(y)|^2}{|x - y|^{n + 2 σ}}d x d y : u \in \mathring H^σ(\mathbb{R}^n), \int_{\mathbb{R}^n} u^2 = 1, |\{u > 0 \}| \leq 1\right\}. \] Unlike the corresponding problem for the usual fractional Laplacian, where the domain of the integration is $\mathbb{R}^n \times \mathbb{R}^n$, sym…
We prove a local Faber-Krahn inequality for solutions $u$ to the Dirichlet problem for $Δ+ V$ on an arbitrary domain $Ω$ in $\mathbb{R}^n$. Suppose a solution $u$ assumes a global maximum at some point $x_0 \in Ω$ and $u(x_0)>0$. Let $T(x_0)$ be the smallest time at which a Brownian motion, started at $x_0$, has exited the domain $Ω$ with probability $\ge 1/2$. For nice (e.g., convex) domains, $T(x_0) \asymp d(x_0,\partialΩ)^2$ but we make no assumption on the geometry of the domain. Our main result is that there exists a ball $B$ of radius $\asymp T(x_0)^{1/2}$ such that $$ \| V \|_{L^{\frac{…
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 annulus $Ω^\#$ has the highest fundamental frequency,'} where $Ω^\#$ is a concentric annulus with the same area as $Ω$ and the same outer boundary length as $L_0$. We extend this result for the higher dimensional domains and $p$-Laplacian with $p\in (1,\infty),$ under the additional assumption that the outer boundary is a sphere. As an application, we prove that the nodal set of the second eigenfunctions of $p$-…
For a domain $Ω\subset \mathbb{R}^n$ and a small number $\frak{T} > 0$, let \[ \mathcal{E}_0(Ω) = λ_1(Ω) + {\frak{T}} {\text{tor}}(Ω) = \inf_{u, w \in H^1_0(Ω)\setminus \{0\}} \frac{\int |\nabla u|^2}{\int u^2} + {\frak{T}} \int \frac{1}{2} |\nabla w|^2 - w \] be a modification of the first Dirichlet eigenvalue of $Ω$. It is well-known that over all $Ω$ with a given volume, the only sets attaining the infimum of $\mathcal{E}_0$ are balls $B_R$; this is the Faber-Krahn inequality. The main result of this paper is that, if for all $Ω$ with the same volume and barycenter as $B_R$ and whose bounda…
The objective of this paper is two-fold. First, we establish new sharp quantitative estimates for Faber-Krahn inequalities on simply connected space forms. We prove that the gap between the first eigenvalue of a given set $Ω$ and that of the ball quantitatively controls both the $L^1$ distance of this set from a ball {\it and} the $L^2$ distance between the corresponding eigenfunctions: \[ λ_1(Ω) - λ_1(B) \gtrsim |ΩΔB|^2 + \int |u_Ω - u_B|^2, \] where $B$ denotes the nearest geodesic ball to $Ω$ with $|B|=|Ω|$ and $u_Ω$ denotes the first eigenfunction with suitable normalization. On Euclidean …
Reverse Faber-Krahn and Szego-Weinberger type inequalities for annular domains under Robin-Neumann boundary conditions
Indian Institute of Technology Madras（印度马德拉斯理工学院）；Institute of Mathematics, Ufa Federal Research Centre, RAS（俄罗斯科学院乌法联邦研究中心数学研究所）；Ufa University of Science and Technology（乌法科学技术大学）；Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia（西波希米亚大学应用科学学院数学与NTIS系）
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 Szegö 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.
In this note we prove that the ball is a maximiser of some Schatten $p$-norms of the Riesz potential operators among all domains of a given measure in $\mathbb R^{d}$. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain and isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh-Faber-Krahn and Hong-Krahn-Szegö inequalities.
We prove an isoperimetric inequality of the Rayleigh-Faber-Krahn type for a nonlinear generalization of the first twisted Dirichlet eigenvalue. More precisely, we show that the minimizer among sets of given volume is the union of two equal balls.
A Faber-Krahn type argument gives a sharp lower estimate for the first Dirichlet eigenvalue for subdomains of wedge domains in spheres, generalizing the inequality in the plane, found by Payne and Weinberger. An application is an alternative proof to the finiteness of a Brownian motion capture time estimate.
In this paper, we determine, in the case of the Laplacian on the flat two-dimensional torus (R/Z) 2 , all the eigenvalues having an eigenfunction which satisfies Courant's theorem with equality (Courant-sharp situation). Following the strategy o A. Pleijel (1956), the proof is a combination of a lower bound a la Weyl) of the counting function, with an explicit remainder term, and of a Faber--Krahn inequality for domains on the torus (deduced as in B{é}rard-Meyer from an isoperimetric inequality), with an explicit upper bound on the area.
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, Weimberger and Polya.
Eigenfunctions of the fractional Schrödinger operators in a domain $\mathcal{D}$ are considered, and a relation between the supremum of the potential and the distance of a maximizer of the eigenfunction from $\partial\mathcal{D}$ is established. This, in particular, extends a recent result of Rachh and Steinerberger to the fractional Schrödinger operators. We also propose a fractional version of the Barta's inequality and also generalize a celebrated Lieb's theorem for fractional Schrödinger operators. As applications of above results we obtain a Faber-Krahn inequality for non-local Schrödinge…
In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus $(\mathbb{R}/\mathbb{Z})^3$, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domains theorem with equality (Courant-sharp situation). Following the strategy of Å. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber-Krahn-type inequality for domains on the torus, deduced as, in the work of P. Bérard and D. Meyer (1982), from an isoperimetric inequality. This inequality relies on the work of L. Hauswirth, J. Perez, P. R…
We prove that, if $Ω$ is an open bounded domain with smooth and connected boundary, for every $p \in (1, + \infty)$ the first Dirichlet eigenvalue of the normalized $p$-Laplacian is simple in the sense that two positive eigenfunctions are necessarily multiple of each other. We also give a (non-optimal) lower bound for the eigenvalue in terms of the measure of $Ω$, and we address the open problem of proving a Faber-Krahn type inequality with balls as optimal domains.
Given a bounded open set $Ω\subseteq{\mathbb{R}}^n$, we consider the eigenvalue problem of a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $Ω$. We prove that the second eigenvalue $λ_2(Ω)$ is always strictly larger than the first eigenvalue $λ_1(B)$ of a ball $B$ with volume half of that of $Ω$. This bound is proven to be sharp, by comparing to the limit case in which $Ω$ consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing se…
