We study the first Steklov-Dirichlet eigenvalue on eccentric spherical shells in $\mathbb{R}^{n+2}$ with $n\geq 1$, imposing the Steklov condition on the outer boundary sphere, denoted by $\Gamma_S$, and the Dirichlet …
We study the first Steklov-Dirichlet eigenvalue on eccentric spherical shells in $\mathbb{R}^{n+2}$ with $n\geq 1$, imposing the Steklov condition on the outer boundary sphere, denoted by $\Gamma_S$, and the Dirichlet condition on the inner boundary sphere. The first eigenfunction admits a Fourier--Gegenbauer series expansion via the bispherical coordinates, where the Dirichlet-to-Neumann operator on $\Gamma_S$ can be recursively expressed in terms of the expansion coefficients arXiv:2309.09587. In this paper, we develop a finite section approach for the Dirichlet-to-Neumann operator to approximate the first Steklov--Dirichlet eigenvalue on eccentric spherical shells. We prove the exponential convergence of this approach by using the variational characterization of the first eigenvalue. Furthermore, based on the convergence result, we propose a numerical computation scheme as an extension of the two-dimensional result in [Hong et al., Ann. Mat. Pura Appl., 2022] to general dimensions. We provide numerical examples of the first Steklov-Dirichlet eigenvalue on eccentric spherical shells with various geometric configurations.
We consider the Steklov-Dirichlet eigenvalue problem on eccentric annuli in Euclidean space of general dimensions. In recent work by the same authors of this paper [21], a limiting behavior of …
We consider the Steklov-Dirichlet eigenvalue problem on eccentric annuli in Euclidean space of general dimensions. In recent work by the same authors of this paper [21], a limiting behavior of the first eigenvalue, as the distance between the two boundary circles of an annulus approaches zero, was obtained in two dimensions. We extend this limiting behavior to general dimensions by employing bispherical coordinates and expressing the first eigenfunction as a Fourier-Gegenbauer series.
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 outer …
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 outer boundaries of an annulus. We first show the differentiability of the eigenvalue in $t$ and obtain an integral expression for the derivative value in two and higher dimensions. We then derive an upper bound of the eigenvalue for each $t$, in two dimensions, by the variational formulation. We also obtain a lower bound of the eigenvalue, given a restriction that the two boundaries of the annulus are sufficiently close. The key point of the proof of the lower bound is in analyzing the limit behavior of an infinite series expansion of the first eigenfunction in bipolar coordinates. We also perform numerical experiments that exhibit the monotonicity for two dimensions.
We consider mixed Steklov-Dirichlet eigenvalue problem on smooth bounded domains in Riemannian manifolds. Under certain symmetry assumptions on multiconnected domains in $\mathbb{R}^{n}$ with a spherical hole, we obtain isoperimetric inequalities …
We consider mixed Steklov-Dirichlet eigenvalue problem on smooth bounded domains in Riemannian manifolds. Under certain symmetry assumptions on multiconnected domains in $\mathbb{R}^{n}$ with a spherical hole, we obtain isoperimetric inequalities for $k$-th Steklov-Dirichlet eigenvalues for $2 \leq k \leq n+1$. We extend Theorem 3.1 of \cite{gavitone2023isoperimetric} from Euclidean domains to domains in space forms, that is, we obtain sharp lower and upper bounds of the first Steklov-Dirichlet eigenvalue on bounded star-shaped domains in the unit $n$-sphere and in the hyperbolic space.
We consider Steklov eigenvalues of three-dimensional, nearly spherical domains. In previous work, we have shown that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute …
We consider Steklov eigenvalues of three-dimensional, nearly spherical domains. In previous work, we have shown that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term of the asymptotic expansion, which can explicitly be written in terms of the Wigner 3-$j$ symbols. We analyze the asymptotic expansion and prove the isoperimetric result that, if $\ell$ is a square integer, the volume-normalized $\ell$th Steklov eigenvalue is stationary for a ball.
We consider Steklov eigenvalues of three-dimensional, nearly-spherical domains. In previous work, we have shown that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the …
We consider Steklov eigenvalues of three-dimensional, nearly-spherical domains. In previous work, we have shown that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term of the asymptotic expansion, which can explicitly be written in terms of the Wigner 3-jsymbols. We analyze the asymptotic expansion and prove the isoperimetric result that, if l is a square integer, the volume-normalized l-th Steklov eigenvalue is stationary for a ball.
In this paper we find lower bounds for the first Steklov eigenvalue in Riemannian n-manifolds, n = 2, 3, with non-positive sectional curvature.
In this paper we find lower bounds for the first Steklov eigenvalue in Riemannian n-manifolds, n = 2, 3, with non-positive sectional curvature.
Let ( M n , g ) be a complete simply connected n -dimensional Riemannian manifold with curvature bounds Sect g ≤ κ for κ ≤ 0 and Ric g …
Let ( M n , g ) be a complete simply connected n -dimensional Riemannian manifold with curvature bounds Sect g ≤ κ for κ ≤ 0 and Ric g ≥ ( n − 1) Kg for K ≤ 0. We prove that for any bounded domain Ω ⊂ M n with diameter d and Lipschitz boundary, if Ω* is a geodesic ball in the simply connected space form with constant sectional curvature κ enclosing the same volume as Ω, then σ 1 (Ω) ≤ C σ 1 (Ω*), where σ 1 (Ω) and σ 1 (Ω*) denote the first nonzero Steklov eigenvalues of Ω and Ω* respectively, and C = C ( n , κ, K , d ) is an explicit constant. When κ = K , we have C = 1 and recover the Brock–Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space.
Let $(M^n,g)$ be a complete simply connected $n$-dimensional Riemannian manifold with curvature bounds $\operatorname{Sect}_g\leq \kappa$ for $\kappa\leq 0$ and $\operatorname{Ric}_g\geq(n-1)Kg$ for $K\leq 0$. We prove that for any bounded domain …
Let $(M^n,g)$ be a complete simply connected $n$-dimensional Riemannian manifold with curvature bounds $\operatorname{Sect}_g\leq \kappa$ for $\kappa\leq 0$ and $\operatorname{Ric}_g\geq(n-1)Kg$ for $K\leq 0$. We prove that for any bounded domain $\Omega \subset M^n$ with diameter $d$ and Lipschitz boundary, if $\Omega^*$ is a geodesic ball in the simply connected space form with constant sectional curvature $\kappa$ enclosing the same volume as $\Omega$, then $\sigma_1(\Omega) \leq C \sigma_1(\Omega^*)$, where $\sigma_1(\Omega)$ and $ \sigma_1(\Omega^*)$ denote the first nonzero Steklov eigenvalues of $\Omega$ and $\Omega^*$ respectively, and $C=C(n,\kappa, K, d)$ is an explicit constant. When $\kappa=K$, we have $C=1$ and recover the Brock-Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space.
We study the problem of maximizing the first nontrivial Steklov eigenvalue of the Laplace-Beltrami Operator among subdomains of fixed volume of a Riemannian manifold. More precisely, we study the expansion …
We study the problem of maximizing the first nontrivial Steklov eigenvalue of the Laplace-Beltrami Operator among subdomains of fixed volume of a Riemannian manifold. More precisely, we study the expansion of the corresponding profile of this isoperimetric (or isochoric) problem as the volume tends to zero. The main difficulty encountered in our study is the lack of existence results for maximizing domains and the possible degeneracy of the first nontrivial Steklov eigenvalue, which makes it difficult to tackle the problem with domain variation techniques. As a corollary of our results, we deduce local comparison principles for the profile in terms of the scalar curvature on $\mathcal{M}$. In the case where the underlying manifold is a closed surface, we obtain a global expansion and thus a global comparison principle.
We study the problem of maximizing the first nontrivial Steklov eigenvalue of the Laplace-Beltrami Operator among subdomains of fixed volume of a Riemannian manifold. More precisely, we study the expansion …
We study the problem of maximizing the first nontrivial Steklov eigenvalue of the Laplace-Beltrami Operator among subdomains of fixed volume of a Riemannian manifold. More precisely, we study the expansion of the corresponding profile of this isoperimetric (or isochoric) problem as the volume tends to zero. The main difficulty encountered in our study is the lack of existence results for maximizing domains and the possible degeneracy of the first nontrivial Steklov eigenvalue, which makes it difficult to tackle the problem with domain variation techniques. As a corollary of our results, we deduce local comparison principles for the profile in terms of the scalar curvature on $\mathcal{M}$. In the case where the underlying manifold is a closed surface, we obtain a global expansion and thus a global comparison principle.
We study the problem of maximizing the first nontrivial Steklov eigenvalue of the Laplace-Beltrami Operator among subdomains of fixed volume of a Riemannian manifold. More precisely, we study the expansion …
We study the problem of maximizing the first nontrivial Steklov eigenvalue of the Laplace-Beltrami Operator among subdomains of fixed volume of a Riemannian manifold. More precisely, we study the expansion of the corresponding profile of this isoperimetric (or isochoric) problem as the volume tends to zero. The main difficulty encountered in our study is the lack of existence results for maximizing domains and the possible degeneracy of the first nontrivial Steklov eigenvalue, which makes it difficult to tackle the problem with domain variation techniques. As a corollary of our results, we deduce local comparison principles for the profile in terms of the scalar curvature on $\mathcal{M}$. In the case where the underlying manifold is a closed surface, we obtain a global expansion and thus a global comparison principle.
Let $U\subset \mathbb{R}^n$ ($n\geq 3$) be an exterior Euclidean domain with smooth boundary $\partial U$. We consider the Steklov eigenvalue problem on $U$. First we derive a sharp lower bound …
Let $U\subset \mathbb{R}^n$ ($n\geq 3$) be an exterior Euclidean domain with smooth boundary $\partial U$. We consider the Steklov eigenvalue problem on $U$. First we derive a sharp lower bound for the first eigenvalue in terms of the support function and the distance function to the origin of $\partial U$. Second under various geometric conditions on $\partial U$ we obtain sharp upper bounds for the first eigenvalue. Along the proof, we get a sharp upper bound for the capacity of $\partial U$ when $n=3$ and $\partial U$ is connected. Last we also discuss an upper bound for the second eigenvalue.
En este trabajo proveemos una cota inferior para el primer valor propio del problema de Steklov en un dominio estrellado acotado en Rn. Este resultado extiende a dimensiones altas un …
En este trabajo proveemos una cota inferior para el primer valor propio del problema de Steklov en un dominio estrellado acotado en Rn. Este resultado extiende a dimensiones altas un estimativo inferior de Kuttler-Sigillito en un dominio estrellado acotado dos dimensional.
We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated …
We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem
The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the …
The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces.
We discuss the behavior of the minimal eigenvalue λ of the Dirichlet Laplacian in the domain D 1 \D 2 := D (an annulus) where D 1 is a circular …
We discuss the behavior of the minimal eigenvalue λ of the Dirichlet Laplacian in the domain D 1 \D 2 := D (an annulus) where D 1 is a circular disc and D 2 ⊂ D 1 is a smaller circular disc.It is conjectured that the minimal eigenvalue λ has a maximum value when D 2 is a concentric disc.If h is a displacement of the center of the disc D 2 and λ (h) is the corresponding minimal eigenvalue, then dλ (h) dh < 0 so that λ (h) is minimal when ∂D 2 touches ∂D 1 , where ∂D is the boundary of D .Numerical results are given to back the conjecture.Upper and lower bounds are given for λ (h) .
4.56 W. STEKLOFF.satisfaisant aux conditions ^=^V, sur (S).Un an après, M. H. Poincaré a découvert l'existence de ses fonctions remarquables, qu'il a appelées fonctions fondamentales, pour toute surface (S) admettant …
4.56 W. STEKLOFF.satisfaisant aux conditions ^=^V, sur (S).Un an après, M. H. Poincaré a découvert l'existence de ses fonctions remarquables, qu'il a appelées fonctions fondamentales, pour toute surface (S) admettant une certaine transformation ponctuelle (transformation de M. Poincaré).Ce fait important a été démontré, en i8()7-i8<)8, par'M.Éd.Le Roy, qui a généralisé le problème.Les fonctions fondamentales de M. Ed.Le Roy dépendent d'une (onction arbitraire y, positive et ne s'annulant pas sur (S).En employant la même fonction y, j'ai généralisé mes recherches mentionnées plus haut; j'ai remplacé la condition (i)par la suivante : ^=Ay(.^-cp/sur (S), et, par la même méthode de M. Poincaré?fai démontré l'existence d'une infinité de nombres positifs Ai,.Àg, . .,, Àê t de fonctions correspondantes l? ^2» * ' • 1 T Si harmoniques à ^intérieur de (S) et satisfaisant aux conditions ^==À,9V, sur (S) (r=ï,2,..p ourvu que la surface (S) admette la transformation de M. Poincaré, Fai exposé mon analyse dans une petite Note Sur l'existence des fonctions fondamentales, insérée aux Comptes rendus du 27 mars 1899 C 1 ) ( 1 ) Voir aussi ma Note Sur la théorie des fonctions fondamentales (Comptes rendus, 17 avril 1899).SUR LES PROBLÈMES FONDAMENTAUX DE LA PHYSIQUE MATHÉMATIQUE.45ê t: puis dans mon Ouvrage Les mélhodes générales pour résoudre les problèmes fondamentaux de la Physique mathématique, paru récemment en russe.Enfin, M. S. Zaremba, dans son Mémoire récent, a généralisé de plus le problème dont il s'agit.Il a démontré l'existence des fonctions V,y(^ = i, '2, ...) satisfaisant aux conditions AV,,-!-^^ o à l'intérieur de (S), àV,/ , , ,., ,, ^x =À,9'V.<;sur (S), an pour toute surface (S) jouissant des propriétés du n° 1 du Chapitre 1 dans le cas particulier de a == i.Je pourrais maintenant, en m'appuyant sur les recherches des deux Chapitres précédents, étendre ma méthode au cas général des surfaces tout a l'heure mentionnées, mais je n'insisterai pas sur ce point.J'adopterai dans ce Chapitre les notations les plus simples de M. Ed.Le Roy et je démontrerai l'existence de ses fonctions, sans employer aucune transformation, pour toute surface (S) satisfaisant aux trois conditions ï:°, 2° et 3° (/i° 1, Chap, î), quel que soit le nombre a, plus petit ou é^al à l'unité.
This book is a collection of articles devoted to the theory of linear operators in Hilbert spaces and its applications. The subjects covered range from the abstract theory of Toeplitz …
This book is a collection of articles devoted to the theory of linear operators in Hilbert spaces and its applications. The subjects covered range from the abstract theory of Toeplitz operators to the analysis of very specific differential operators arising in quantum mechanics, electromagnetism, and the theory of elasticity; the stability of numerical methods is also discussed. Many of the articles deal with spectral problems for not necessarily selfadjoint operators. Some of the articles are surveys outlining the current state of the subject and presenting open problems.
It has been shown by Kesavan ( Proc. R. Soc. Edinb. A ( 133 ) (2003), 617–624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with …
It has been shown by Kesavan ( Proc. R. Soc. Edinb. A ( 133 ) (2003), 617–624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with the puncture having the shape of a ball, is maximum if and only if the balls are concentric. Recently, Emamizadeh and Zivari-Rezapour ( Proc. Am. Math. Soc. 136 (2007), 1325–1331) have tried to generalize this result to the case of the p -Laplacian but could succeed only in proving a domain monotonicity result for a weighted eigenvalue problem in which the weights need to satisfy some artificial conditions. In this paper we generalize the result of Kesavan to the case of the p -Laplacian (1 < p < ∞) without any artificial restrictions, and in the process we simplify greatly the proof, even in the case of the Laplacian. The uniqueness of the maximizing domain in the nonlinear case is still an open question.
The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the …
The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometry. While this problem shares some common properties with its more familiar Dirichlet and Neumann cousins, its eigenvalues and eigenfunctions have a number of distinctive geometric features, which makes the subject especially appealing. In this survey we discuss some recent advances and open questions, particularly in the study of spectral asymptotics, spectral invariants, eigenvalue estimates, and nodal geometry.
Progress in Partial Differential Equations, presents some of the latest research in this important field. Both volumes contain the lectures and papers of top international researchers contributed at the Third …
Progress in Partial Differential Equations, presents some of the latest research in this important field. Both volumes contain the lectures and papers of top international researchers contributed at the Third European Conference on Elliptic and Parabolic Problems.
We study bounds on the Riesz means of the mixed Steklov–Neumann and Steklov–Dirichlet eigenvalue problem on a bounded domain [Formula: see text] in [Formula: see text]. The Steklov–Neumann eigenvalue problem …
We study bounds on the Riesz means of the mixed Steklov–Neumann and Steklov–Dirichlet eigenvalue problem on a bounded domain [Formula: see text] in [Formula: see text]. The Steklov–Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov–Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov–Dirichlet eigenvalue problem, the proof is based on a well-known bound on the Riesz means of the Dirichlet fractional Laplacian, and an inequality between the Dirichlet and Navier fractional Laplacian. The two-term asymptotic results for the Riesz means of mixed Steklov eigenvalue problems are discussed in the Appendix which in particular show the asymptotic sharpness of the bounds we obtain.
Let $B_1$ be a ball in $\mathbb{R}^N$ centred at the origin and $B_0$ be a smaller ball compactly contained in $B_1$. For $p\in(1, \infty)$, using the shape derivative method, we …
Let $B_1$ be a ball in $\mathbb{R}^N$ centred at the origin and $B_0$ be a smaller ball compactly contained in $B_1$. For $p\in(1, \infty)$, using the shape derivative method, we show that the first eigenvalue of the $p$-Laplacian in annulus $B_1\setminus \overline{B_0}$ strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as $p \to 1$ and $p \to \infty$ are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fu\v{c}ik spectrum of the $p$-Laplacian on bounded radial domains.
<p style='text-indent:20px;'>In this paper we study the first Steklov-Laplacian eigenvalue with an internal fixed spherical obstacle. We prove that the spherical shell locally maximizes the first eigenvalue among nearly spherical …
<p style='text-indent:20px;'>In this paper we study the first Steklov-Laplacian eigenvalue with an internal fixed spherical obstacle. We prove that the spherical shell locally maximizes the first eigenvalue among nearly spherical sets when both the internal ball and the volume are fixed.
In this paper we prove the existence of a maximum for the first Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole under volume constraint.More precisely, …
In this paper we prove the existence of a maximum for the first Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole under volume constraint.More precisely, if Ω " Ω 0 zB R1 , where B R1 is the ball centered at the origin with radius R 1 ą 0 and Ω 0 Ă R n , n ě 2, is an open bounded and convex set such that B R1 Ť Ω 0 , then the first Steklov-Dirichlet eigenvalue σ 1 pΩq has a maximum when R 1 and the measure of Ω are fixed.Moreover, if Ω 0 is contained in a suitable ball, we prove that the spherical shell is the maximum.
Abstract We derive a shape derivative formula for the family of principal Dirichlet eigenvalues $$\lambda _s(\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> …
Abstract We derive a shape derivative formula for the family of principal Dirichlet eigenvalues $$\lambda _s(\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of the fractional Laplacian $$(-\Delta )^s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> </mml:math> associated with bounded open sets $$\Omega \subset \mathbb {R}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> </mml:math> of class $$C^{1,1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . This extends, with a help of a new approach, a result in Dalibard and Gérard-Varet (Calc. Var. 19(4):976–1013, 2013) which was restricted to the case $$s=\frac{1}{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> . As an application, we consider the maximization problem for $$\lambda _s(\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> among annular-shaped domains of fixed volume of the type $$B\setminus \overline{B}'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>\</mml:mo> <mml:msup> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:math> , where B is a fixed ball and $$B'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>B</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> is ball whose position is varied within B . We prove that $$\lambda _s(B\setminus \overline{B}')$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>B</mml:mi> <mml:mo>\</mml:mo> <mml:msup> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is maximal when the two balls are concentric. Our approach also allows to derive similar results for the fractional torsional rigidity. More generally, we will characterize one-sided shape derivatives for best constants of a family of subcritical fractional Sobolev embeddings.
These are the notes for two lectures I gave at the Belfast Functional Analysis Day 1999.The purpose of these notes is to give an idea of how C * -algebra …
These are the notes for two lectures I gave at the Belfast Functional Analysis Day 1999.The purpose of these notes is to give an idea of how C * -algebra techniques can be successfully employed in order to solve some concrete problems of Numerical Analysis.I focus my attention on several questions concerning the asymptotic behavior of large Toeplitz matrices.This limitation ignores the potential and the triumphs of C * -algebra methods in connection with large classes of other operators and plenty of different approximation methods, but it allows me to demonstrate the essence of the C *algebra approach and to illustrate it with nevertheless nontrivial examples.
We prove that among all doubly connected domains of ℝ n of the form B 1 \ B ̅ 2 , where B 1 and B 2 are open balls …
We prove that among all doubly connected domains of ℝ n of the form B 1 \ B ̅ 2 , where B 1 and B 2 are open balls of fixed radii such that B ̅ 2 ⊂ B 1 , the first nonzero Steklov eigenvalue achieves its maximal value uniquely when the balls are concentric. Furthermore, we show that the ideas of our proof also apply to a mixed boundary conditions eigenvalue problem found in literature.
In the present paper, we study properties of the second Dirichlet eigenvalue of the fractional Laplacian of annuli-like domains and the corresponding eigenfunctions. In the first part, we consider an …
In the present paper, we study properties of the second Dirichlet eigenvalue of the fractional Laplacian of annuli-like domains and the corresponding eigenfunctions. In the first part, we consider an annulus with inner radius $ R $ and outer radius $ R+1 $. We show that for $ R $ sufficiently large any corresponding second eigenfunction of this annulus is nonradial. In the second part, we investigate the second eigenvalue in domains of the form $ B_1(0)\setminus \overline{B_{\tau}(a)} $, where $ a $ is in the unitary ball and $ 0<\tau<1-|a| $. We show that this value is maximized for $ a = 0 $, if the set $ B_1(0)\setminus \overline{B_{\tau}(0)} $ has no radial second eigenfunction. We emphasize that the first part of our paper implies that this assumption is indeed nonempty.
Abstract The Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of …
Abstract The Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of the recent developments linking the Steklov eigenvalues and eigenfunctions of compact Riemannian manifolds to the geometry of the manifolds. Topics include isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the case of surfaces and then in higher dimensions), stability and instability of eigenvalues under deformations of the Riemannian metric, optimisation of eigenvalues and connections to free boundary minimal surfaces in balls, inverse problems and isospectrality, discretisation, and the geometry of eigenfunctions. We begin with background material and motivating examples for readers that are new to the subject. Throughout the tour, we frequently compare and contrast the behavior of the Steklov spectrum with that of the Laplace spectrum. We include many open problems in this rapidly expanding area.