arXiv subjects

Ahmad El Soufi

Publications and source records attributed to Ahmad El Soufi.

At least 19 recordsLinked to original sources

arXiv2018

Compact manifolds with fixed boundary and large Steklov eigenvalues

Let M be a compact Riemannian manifold with boundary. Let b>0 be the number of connected components of its boundary. For manifolds of dimension at least 3, we prove that it is possible to obtain an arbitrarily large (b+1)-th Steklov eigenvalue using a smooth conformal perturbation which is supported in a thin neighbourhood of the boundary, identically equal to 1 on the boundary. For j<b+1, it is also possible to obtain arbitrarily large j-th eigenvalue, but this require the conformal factor to spread throughout the interior of the manifold M. This is in stark contrast with the situation for the eigenvalues of the Laplace operator on a closed manifold, where a conformal factor that is large enough for the volume to become unbounded results in the spectrum collapsing to 0. We also prove that it is possible to obtain large Steklov eigenvalues while keeping different boundary components arbitrarily close to each other, by constructing a convenient Riemannian submersion.

math.SP↗

arXiv2017

Eigenvalue upper bounds for the magnetic Schroedinger operator

We study the eigenvalues of the magnetic Schroedinger operator associated with a magnetic potential A and a scalar potential q, on a compact Riemannian manifold M, with Neumann boundary conditions if the boundary is not empty. We obtain several bounds for the spectrum. Besides the dimension and the volume of the manifold, the geometric quantity which plays an important role in these estimates is the first eigenvalue of the Hodge-de Rham Laplacian acting on co-exact 1-forms. In the 2-dimensional case, this is nothing but the first positive eigenvalue of the Laplacian acting on functions. As for the dependence of the bounds on the potentials, it brings into play the mean value of the scalar potential q, the L^2-norm of the magnetic field B=dA, and the distance, taken in L^2, between the harmonic component of A and the subspace of all closed 1-forms whose cohomology class is integral (that is, having integral flux around any loop). In particular, this distance is zero when the first cohomology group is trivial.

math.DG↗

arXiv2016

Spectrum of the Laplacian with weights

Given a compact Riemannian manifold (M, g) and two positive functions $ρ$ and $σ$, we are interested in the eigenvalues of the Dirichlet energy functional weighted by $σ$, with respect to the L 2 inner product weighted by $ρ$. Under some regularity conditions on $ρ$ and $σ$, these eigenvalues are those of the operator $ρ$^{-1} div($σ$$\nabla$u) with Neumann conditions on the boundary if $\partial$M = $\emptyset$. We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is preserved.

math.DG↗

arXiv2016

The first positive eigenvalue of the sub-laplacian on cr spheres

We prove that the first positive eigenvalue, normalized by the volume, of the sub-Laplacian associated with a strictly pseudoconvex pseudo-Hermitian structure $\θ$ on the CR sphere S 2n+1 $\subset$ C n+1 , achieves its maximum when $\θ$ is the standard contact form.

math.DG↗

arXiv2016

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

We prove that among all doubly connected domains of $\R^n$ bounded by two spheres of given radii, $Z(t)$, the trace of the heat kernel with Dirichlet boundary conditions, achieves its minimum when the spheres are concentric (i.e., for the spherical shell). The supremum is attained when the interior sphere is in contact with the outer sphere.This is shown to be a special case of a more general theorem characterizing the optimal placement of a spherical obstacle inside a convex domain so as to maximize or minimize the trace of the Dirichlet heat kernel. In this case the minimizing position of the center of the obstacle belongs to the "heart" of the domain, while the maximizing situation occurs either in the interior of the heart or at a point where the obstacle is in contact with the outer boundary. Similar statements hold for the optimal positions of the obstaclefor any spectral property that can be obtained as a positivity-preserving or positivity-reversing transform of $Z(t)$,including the spectral zeta function and, through it, the regularized determinant.

math.MG↗

arXiv2015

On sums of eigenvalues of elliptic operators on manifolds

We use the averaged variational principle introduced in a recent article on graph spectra [7] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of Kr{ö}ger 's bound for Neumann spectra of Laplacians on Euclidean domains [12]. Among the operators we consider are the Laplace-Beltrami operator on compact subdomains of manifolds. These estimates become more explicit and asymptotically sharp when the manifold is conformal to homogeneous spaces (here extending a result of Strichartz [21] with a simplified proof). In addition we obtain results for the Witten Laplacian on the same sorts of domains and for Schr{ö}dinger operators with confining potentials on infinite Euclidean domains. Our bounds have the sharp asymptotic form expected from the Weyl law or classical phase-space analysis. Similarly sharp bounds for the trace of the heat kernel follow as corollaries.

math.MG↗

arXiv2014

Extremal Eigenvalues of the Laplacian on Euclidean domains and closed surfaces

We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part, we study sequences of extremal eigenvalues of the Laplace-Beltrami operator on closed surfaces of unit area.

math.MG↗

arXiv2013

Eigenvalues of the Laplacian on a compact manifold with density

In this paper, we study the spectrum of the weighted Laplacian (also called Bakry-Emery or Witten Laplacian) $L_σ$ on a compact, connected, smooth Riemannian manifold $(M,g)$ endowed with a measure $σdv_g$. First, we obtain upper bounds for the $k-$th eigenvalue of $L_σ$ which are consistent with the power of $k$ in Weyl's formula. These bounds depend on integral norms of the density $σ$, and in the second part of the article, we give examples showing that this dependence is, in some sense, sharp. As a corollary, we get bounds for the eigenvalues of Laplace type operators, such as the Schrödinger operator or the Hodge Laplacian on $p-$forms. In the special case of the weighted Laplacian on the sphere, we get a sharp inequality for the first nonzero eigenvalue which extends Hersch's inequality.

math.MG↗

arXiv2013

Inequalities and bounds for the eigenvalues of the sub-Laplacian on a strictly pseudoconvex CR manifold

We establish inequalities for the eigenvalues of the sub-Laplace operator associated with a pseudo-Hermitian structure on a strictly pseudoconvex CR manifold. Our inequalities extend those obtained by Niu and Zhang \cite{NiuZhang} for the Dirichlet eigenvalues of the sub-Laplacian on a bounded domain in the Heisenberg group and are in the spirit of the well known Payne-Pólya-Weinberger and Yang universal inequalities.

math.MG↗

arXiv2011

Isoperimetric control of the spectrum of a compact hypersurface

Upper bounds for the eigenvalues of the Laplace-Beltrami operator on a hypersurface bounding a domain in some ambient Riemannian manifold are given in terms of the isoperimetric ratio of the domain. These results are applied to the extrinsic geometry of isometric embeddings.

math.MG↗

arXiv2011

Isoperimetric control of the Steklov spectrum

Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, the normalized Steklov eigenvalues are uniformly bounded above in terms of the genus. We also obtain a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its bounding hypersurface.

math.SP↗

arXiv2009

Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds

We give upper bounds for the eigenvalues of the La-place-Beltrami operator of a compact $m$-dimensional submanifold $M$ of $\R^{m+p}$. Besides the dimension and the volume of the submanifold and the order of the eigenvalue, these bounds depend on either the maximal number of intersection points of $M$ with a $p$-plane in a generic position (transverse to $M$), or an invariant which measures the concentration of the volume of $M$ in $\R^{m+p}$. These bounds are asymptotically optimal in the sense of the Weyl law. On the other hand, we show that even for hypersurfaces (i.e., when $p=1$), the first positive eigenvalue cannot be controlled only in terms of the volume, the dimension and (for $m\ge 3$) the differential structure.

math.MG↗

arXiv2009

Isoperimetric inequalities for the eigenvalues of natural Schrödinger operators on surfaces

This paper deals with eigenvalue optimization problems for a family of natural Schrödinger operators arising in some geometrical or physical contexts. These operators, whose potentials are quadratic in curvature, are considered on closed surfaces immersed in space forms and we look for geometries that maximize the eigenvalues. We show that under suitable assumptions on the potential, the first and the second eigenvalues are maximized by (round) spheres.

math.DG↗

arXiv2009

Universal inequalities for the eigenvalues of Schrodinger operators on submanifolds

We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Weinberger and to Yang, but which depend in an explicit way on the mean curvature. In later sections, we prove similar results for Schrödinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as for the Kohn Laplacian on subdomains of the Heisenberg group. Among the consequences of this analysis are an extension of Reilly's inequality, bounding any eigenvalue of the Laplacian in terms of the mean curvature, and spectral criteria for the immersibility of manifolds in homogeneous spaces.

math.MG↗

arXiv2008

Pseudo-radial solutions of semi-linear elliptic equations on symmetric domains

In this paper we investigate existence and characterization of non-radial pseudo-radial (or separable) solutions of some semi-linear elliptic equations on symmetric 2-dimensional domains. The problem reduces to the phase plane analysis of a dynamical system. In particular, we give a full description of the set of pseudo-radial solutions of equations of the form $Δu = \pm a^2(|x|) u|u|^{q-1}$, with $q>0$, $q\neq 1$. We also study such equations over spherical or hyperbolic symmetric domains.

math.AP↗

arXiv2007

Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue

We prove that among all doubly connected domains of $\mathbb{R}^n$ bounded by two spheres of given radii, the second eigenvalue of the Dirichlet Laplacian achieves its maximum when the spheres are concentric (spherical shell). The corresponding result for the first eigenvalue has been established by Hersch in dimension 2, and by Harrell, Kröger and Kurata and Kesavan in any dimension. We also prove that the same result remains valid when the ambient space $\mathbb{R}^n$ is replaced by the standard sphere $\mathbb{S}^n$ or the hyperbolic space $\mathbb{H}^n$ .

math.MG↗

arXiv2007

Extremal first Dirichlet eigenvalue of doubly connected plane domains and dihedral symmetry

We deal with the following eigenvalue optimization problem: Given a bounded domain $D\subset \R^2$, how to place an obstacle $B$ of fixed shape within $D$ so as to maximize or minimize the fundamental eigenvalue $λ_1$ of the Dirichlet Laplacian on $D\setminus B$. This means that we want to extremize the function $ρ\mapsto λ_1(D\setminus ρ(B))$, where $ρ$ runs over the set of rigid motions such that $ρ(B)\subset D$. We answer this problem in the case where both $D$ and $B$ are invariant under the action of a dihedral group $\mathbb{D}_n$, $n\ge2$, and where the distance from the origin to the boundary is monotonous as a function of the argument between two axes of symmetry. The extremal configurations correspond to the cases where the axes of symmetry of $B$ coincide with those of $D$.

math.SP↗

arXiv2007

Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold

For any bounded regular domain $Ω$ of a real analytic Riemannian manifold $M$, we denote by $λ_{k}(Ω)$ the $k$-th eigenvalue of the Dirichlet Laplacian of $Ω$. In this paper, we consider $λ_k$ and as a functional upon the set of domains of fixed volume in $M$. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for $λ_k$. These results rely on Hadamard type variational formulae that we establish in this general setting.

math.MG↗