Dirichlet Eigenfunctions Research Articles

On localisation of eigenfunctions of the Laplace operator

We prove (i) a simple sufficient geometric condition for localisation of a sequence of first Dirichlet eigenfunctions provided the corresponding Dirichlet Laplacians satisfy a uniform Hardy inequality and (ii) localisation of a sequence of first Dirichlet eigenfunctions for a wide class of elongating horn-shaped domains. We give examples of sequences of simply connected, planar, polygonal domains for which the corresponding sequence of first eigenfunctions with either Dirichlet or Neumann boundary conditions \kappa -localise in L^{2} .

On generalized eigenvalue problems of fractional (p, q)-Laplace operator with two parameters

For $s_1,\,s_2\in (0,\,1)$ and $p,\,q \in (1,\, \infty )$ , we study the following nonlinear Dirichlet eigenvalue problem with parameters $\alpha,\, \beta \in \mathbb {R}$ driven by the sum of two nonlocal operators: \[ (-\Delta)^{s_1}_p u+(-\Delta)^{s_2}_q u=\alpha|u|^{p-2}u+\beta|u|^{q-2}u\ \text{in }\Omega, \quad u=0\ \text{in } \mathbb{R}^d \setminus \Omega, \quad \mathrm{(P)} \] where $\Omega \subset \mathbb {R}^d$ is a bounded open set. Depending on the values of $\alpha,\,\beta$ , we completely describe the existence and non-existence of positive solutions to (P). We construct a continuous threshold curve in the two-dimensional $(\alpha,\, \beta )$ -plane, which separates the regions of the existence and non-existence of positive solutions. In addition, we prove that the first Dirichlet eigenfunctions of the fractional $p$ -Laplace and fractional $q$ -Laplace operators are linearly independent, which plays an essential role in the formation of the curve. Furthermore, we establish that every nonnegative solution of (P) is globally bounded.

A remark on quantitative unique continuation from subsets of the boundary of positive measure

The question of uniqueness of harmonic functions in a domain Ω ⊂ R d \Omega \subset \mathbb {R}^d with boundary ∂ Ω \partial \Omega , satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset of the boundary ω \omega of positive d − 1 d-1 dimensional measure has attracted a lot of attention (see e.g. V. Adolfsson and L. Escauriaza [Comm. Pure Appl. Math. 50 (1997), pp. 935–969]; F.-H. Lin [Comm. Pure Appl. Math. 44 (1991), pp. 287–308]; X. Tolsa [Comm. Pure Appl. Math. 76 (2023), pp. 305–336]). It is essentially a consequence of Carleman estimates when ω \omega contains an open subset of the boundary (uniqueness for second order elliptic operators across an hypersurface). The main open questions (about uniqueness) concern now Lipschitz domains and variable coefficients. Here, using results by Logunov and Malinnikova [Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimensions two and three, Birkhäuser/Springer, Cham, 2018, pp. 333-344; Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018, World Scientific Publishing, Hackensack, NJ, 2018, pp. 2391–2411], we consider the simpler case of W 2 , ∞ W^{2, \infty } domains but prove quantitative uniqueness both for Dirichlet and Neumann boundary conditions. As an application, we deduce quantitative estimates for the Dirichlet and Neumann Laplace eigenfunctions on a W 2 , ∞ W^{2, \infty } domain with boundary.

Uniqueness in Inverse Diffraction Grating Problems With Infinitely Many Plane Waves at a Fixed Frequency

This paper is concerned with uniqueness of solution of the inverse diffraction by problems by a periodic curve with Dirichlet boundary condition in two dimensions. It is proved that the periodic curve can be uniquely determined by the near-field measurement data corresponding to infinitely many incident plane waves with distinct directions at a fixed frequency. Our proof is based on Schiffer’s idea which consists of two ingredients: (i) the total fields for incident plane waves with distinct directions are linearly independent, and (ii) for a fixed wave number there exist only finitely many linearly independent Dirichlet eigenfunctions in a bounded domain or in a closed waveguide under additional assumptions on the waveguide boundary. Based on the Rayleigh expansion, we prove that the phased near-field data can be uniquely determined by the phaseless near-field data in a bounded domain, with the exception of a finite set of incident angles. Such a phase retrieval result leads to a new uniqueness result for the inverse grating diffraction problem with phaseless near-field data at a fixed frequency. Since the incident direction determines the quasi-periodicity of the boundary value problem, our inverse issues are different from the existing results of [F. Hettlich and A. Kirsch, Inverse Problems, 13 (1997), pp. 351–361], where fixed-direction plane waves at multiple frequencies were considered.

Deterministic-statistical approach for an inverse acoustic source problem using multiple frequency limited aperture data

We propose a deterministic-statistical method for an inverse source problem using multiple frequency limited aperture far field data. The direct sampling method is used to obtain a disc such that it contains the compact support of the source. The Dirichlet eigenfunctions of the disc are used to expand the source function. Then the inverse problem is recast as a statistical inference problem and the Bayesian inversion is employed to reconstruct the coefficients of the eigen-expansion for the source function. The stability of the statistical inverse problem with respect to the measured data is justified in the sense of Hellinger distance. A preconditioned Crank-Nicolson (pCN) Metropolis-Hastings (MH) algorithm is implemented to explore the posterior density function. Numerical examples show that the proposed method is effective for both smooth and non-smooth sources.

A variational approach to the optimal locations of the nodes of the second Dirichlet eigenfunctions

By a node of a Sturm–Liouville problem, it means an interior zero of an eigenfunction. In this paper, by considering the unique node of the second Dirichlet eigenfunction as a nonlinear functional of potential from the Lebesgue space , we will study the optimization problems to minimize or to maximize subject to the constraint . By applying the recent results on the differentiability and complete continuity of in , it will be proved that for the case , these optimization problems are attained by some potentials. Moreover, a critical equation for optimizers will be derived. Finally, by considering the limit case , it will be found that the optimizers for the optimization problems for the case are certain Dirac measures. These results are then used to deduce the optimal locations of nodes .

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.

Dirichlet spectral-Galerkin approximation method for the simply supported vibrating plate eigenvalues

In this paper, we analyze and implement the Dirichlet spectral-Galerkin method for approximating simply supported vibrating plate eigenvalues with variable coefficients. This is a Galerkin approximation that uses the approximation space that is the span of finitely many Dirichlet eigenfunctions for the Laplacian. Convergence and error analysis for this method is presented for two and three dimensions. Here we will assume that the domain has either a smooth or Lipschitz boundary with no reentrant corners. An important component of the error analysis is Weyl’s law for the Dirichlet eigenvalues. Numerical examples for computing the simply supported vibrating plate eigenvalues for the unit disk and square are presented. In order to test the accuracy of the approximation, we compare the spectral-Galerkin method to the separation of variables for the unit disk. Whereas for the unit square we will numerically test the convergence rate for a variable coefficient problem.

Small Order Asymptotics of the Dirichlet Eigenvalue Problem for the Fractional Laplacian

We study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian (-Delta )^s in bounded open Lipschitz sets in the small order limit s rightarrow 0^+. While it is easy to see that all eigenvalues converge to 1 as s rightarrow 0^+, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with Fourier symbol 2log |xi |. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that L^2-normalized Dirichlet eigenfunctions of (-Delta )^s corresponding to the k-th eigenvalue are uniformly bounded and converge to the set of L^2-normalized eigenfunctions of the logarithmic Laplacian. In order to derive these spectral asymptotics, we establish new uniform regularity and boundary decay estimates for Dirichlet eigenfunctions for the fractional Laplacian. As a byproduct, we also obtain corresponding regularity properties of eigenfunctions of the logarithmic Laplacian.

Efficiency and localisation for the first Dirichlet eigenfunction

Bounds are obtained for the efficiency or mean to peak ratio $E(\Omega)$ for the first Dirichlet eigenfunction (positive) for open, connected sets $\Omega$ with finite measure in Euclidean space $\R^m$. It is shown that (i) localisation implies vanishing efficiency, (ii) a vanishing upper bound for the efficiency implies localisation, (iii) localisation occurs for the first Dirichlet eigenfunctions for a wide class of elongating bounded, open, convex and planar sets, (iv) if $\Omega_n$ is any quadrilateral with perpendicular diagonals of lengths $1$ and $n$ respectively, then the sequence of first Dirichlet eigenfunctions localises, and $E(\Omega_n)=O\big(n^{-2/3}\log n\big)$. This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.

Doubling inequalities and critical sets of Dirichlet eigenfunctions

We study the sharp doubling inequalities for the gradients and upper bounds for the critical sets of Dirichlet eigenfunctions on the boundary and in the interior of compact Riemannian manifolds. Most efforts are devoted to obtaining the sharp doubling inequalities for the gradients. New idea is developed to overcome the difficulties on the unavailability of the double manifold in obtaining doubling inequalities in smooth manifolds. The sharp upper bounds of critical sets in analytic Riemannian manifolds are consequences of the doubling inequalities.

Estimates of eigenvalues and eigenfunctions in elliptic homogenization with rapidly oscillating potentials

In this paper, for a family of second-order elliptic equations with rapidly oscillating periodic coefficients and rapidly oscillating periodic potentials, we are interested in the $H^1$ convergence rates and the Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The $H^1$ convergence rates rely on the Dirichlet correctors and the first-order corrector for the oscillating potentials. And the bound results rely on an $O(\varepsilon)$ estimate in $H^1$ for solutions with Dirichlet condition.

Crystallographic Groups, Strictly Tessellating Polytopes, and Analytic Eigenfunctions

The mathematics of crystalline structures connects analysis, geometry, algebra, and number theory. The planar crystallographic groups were classified in the late 19th century. One hundred years later, Bérard proved that the fundamental domains of all such groups satisfy a very special analytic property: the Dirichlet eigenfunctions for the Laplace eigenvalue equation are all trigonometric functions. In 2008, McCartin proved that in two dimensions, this special analytic property has both an equivalent algebraic formulation, as well as an equivalent geometric formulation. Here we generalize the results of Bérard and McCartin to all dimensions. We prove that the following are equivalent: the first Dirichlet eigenfunction for the Laplace eigenvalue equation on a polytope is real analytic, the polytope strictly tessellates space, and the polytope is the fundamental domain of a crystallographic Coxeter group. Moreover, we prove that under any of these equivalent conditions, all of the eigenfunctions are trigonometric functions. To conclude, we connect these topics to the Fuglede and Goldbach conjectures and give a purely geometric formulation of Goldbach’s conjecture.

Nodal line estimates for the second Dirichlet eigenfunction

We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser–Jerison in a series of works on convex planar domains and rectangles with one curved edge and a large aspect ratio. Here, we study the structure of the nodal curve in greater detail, in that we find precise bounds on its curvature, with uniform estimates up to the two points where it meets the domain at right angles, and show that many of our results hold for relatively small aspect ratios of the side lengths. We also discuss applications of our results to Courant-sharp eigenfunctions and spectral partitioning.

A parametric analysis of discrete Hamiltonian functional maps

AbstractIn this paper we develop an in‐depth theoretical investigation of the discrete Hamiltonian eigenbasis, which remains quite unexplored in the geometry processing community. This choice is supported by the fact that Dirichlet eigenfunctions can be equivalently computed by defining a Hamiltonian operator, whose potential energy and localization region can be controlled with ease. We vary with continuity the potential energy and study the relationship between the Dirichlet Laplacian and the Hamiltonian eigenbases with the functional map formalism. We develop a global analysis to capture the asymptotic behavior of the eigenpairs. We then focus on their local interactions, namely the veering patterns that arise between proximal eigenvalues. Armed with this knowledge, we are able to track the eigenfunctions in all possible configurations, shedding light on the nature of the functional maps. We exploit the Hamiltonian‐Dirichlet connection in a partial shape matching problem, obtaining state of the art results, and provide directions where our theoretical findings could be applied in future research.

Approximation of the Zero-Index Transmission Eigenvalues with a Conductive Boundary and Parameter Estimation

In this paper, we present a spectral-Galerkin method to approximate the zero-index transmission eigenvalues with a conductive boundary condition. This is a new eigenvalue problem derived from the scalar inverse scattering problem for an isotropic media with a conductive boundary condition. In our analysis, we will consider the equivalent fourth-order eigenvalue problem where we establish the convergence when the approximation space is the span of finitely many Dirichlet eigenfunctions for the Laplacian. We establish the convergence rate of the spectral approximation by appealing to Weyl’s law. Numerical examples for computing the eigenvalues and eigenfunctions for the unit disk and unit square are presented. Lastly, we provide a method for estimating the refractive index assuming the conductivity parameter is either sufficiently large or small but otherwise unknown.

GRADIENT AND HESSIAN ESTIMATES FOR DIRICHLET AND NEUMANN EIGENFUNCTIONS

Abstract We establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, sharp gradient and Hessian estimates are derived for the Dirichlet and Neumann eigenfunctions.

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.

Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions

We consider a spectral optimal design problem involving the Neumann traces of the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset $\Omega$ of $\R^n$. The cost functional measures the amount of energy that Dirichlet eigenfunctions concentrate on the boundary and that can be recovered with a bounded density function. We first prove that, assuming a $L^1$ constraint on densities, the so-called {\it Rellich functions} maximize this functional. Motivated by several issues in shape optimization or observation theory where it is relevant to deal with bounded densities, and noticing that the $L^\infty$-norm of {\it Rellich functions} may be large, depending on the shape of $\Omega$, we analyze the effect of adding pointwise constraints when maximizing the same functional. We investigate the optimality of {\it bang-bang} functions and {\it Rellich densities} for this problem. We also deal with similar issues for a close problem, where the cost functional is replaced by a spectral approximation. Finally, this study is completed by the investigation of particular geometries and is illustrated by several numerical simulations.

Restrictions of Laplacian Eigenfunctions to Edges in the Sierpinski Gasket

In this paper, we study the restrictions of both the harmonic functions and the eigenfunctions of the symmetric Laplacian to edges of pre-gaskets contained in the Sierpinski gasket. For a harmonic function, its restriction to any edge is either monotone or having a single extremum. For an eigenfunction, it may have several local extrema along edges. We prove general criteria, involving the values of any given function at the endpoints and midpoint of any edge, to determine which case it should be, as well as the asymptotic behavior of the restriction near the endpoints. Moreover, for eigenfunctions, we use spectral decimation to calculate the exact numbers of the local extrema along edges. This confirms, in a more general situation, a conjecture of Dalrymple et al. (J Fourier Anal Appl 5:203–284, 1999) on the behavior of the restrictions to edges of the basis Dirichlet eigenfunctions, suggested by the numerical data.