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Firstly, this paper establishes useful forms of the remainder term of Hardy-type inequalities on general domains where the weights are functions of the distance to the boundary. For weakly mean convex domains we use the resulting identities to establish nonexistence of extremizers for and improve known sharp Hardy inequalities. Secondly, we establish geometrically interesting remainders for the Davies-Hardy-Tidblom inequalities for the mean distance function, as well as generalize and improve several Hardy type inequalities in the spirit of Brezis and Marcus and spectral estimates of Davies. Lastly, we apply our results to obtain Sobolev inequalities for non-regular Riemannian metrics on geometric exterior domains. [ABSTRACT FROM AUTHOR]

We consider the eigenfrequency problem of the magnetic field of Maxwell's equation in a domain Ω of $ \mathbb {R}^3 $ R 3 with an inclusion O under perturbation $ O_\delta $ O δ , which is a thin layer at the wall boundary (skin depth) $ \partial O_\delta $ ∂ O δ . We write the asymptotic formula in terms of the function modelling: the oscillation and the electric permittivity. We establish the convergence of the solution and then we justify a rigorous asymptotic formula for the eigenfrequency. [ABSTRACT FROM AUTHOR]

We consider the Laplace operator in the exterior of a compact set in the plane, subject to Robin boundary conditions. If the boundary coupling is sufficiently negative, there are at least two discrete eigenvalues below the essential spectrum. We state a general conjecture that the second eigenvalue is maximised by the exterior of a disk under isochoric or isoperimetric constraints. We prove an isoelastic version of the conjecture for the exterior of convex domains. Finally, we establish a monotonicity result for the second eigenvalue under the condition that the compact set is strictly star-shaped and centrally symmetric. [ABSTRACT FROM AUTHOR]

For a homogeneous incompressible 2D fluid confined within a bounded Lipschitz simply connected domain, homogeneous Neumann pressure boundary conditions are equivalent to a constant boundary vorticity. We investigate the rigidity of such conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study the universal inequalities for eigenvalues of a clamped plate problem, and establish some new universal inequalities that are different from those already present in the literature, such as (Wang and Xia J. Funct. Anal. 245(1), 334-352 2007), (Wang and Xia Calc. Var. Partial Differential 653 Equations 40(1-2), 273-289 2011), (Chen, Zheng, and Lu Pacific J. Math. 255(1), 41-54 2012), and so on. In particular, our results can reveal the relationship between the (k + 1) -th eigenvalue and the first k eigenvalues relatively quickly. [ABSTRACT FROM AUTHOR]

This paper is concerned with spatiotemporal dynamics of a fractional diffusive susceptible-infected-susceptible (SIS) epidemic model with mass action infection mechanism. Concretely, we first focus on the existence and stability of the disease-free and endemic equilibria. Then, we give the asymptotic profiles of the endemic equilibrium on small and large diffusion rates, which can reveal the impact of dispersal rates and fractional powers simultaneously. It is worth noting that we have some counter-intuitive findings: controlling the flow of infected individuals will not eradicate the disease, but restricting the movement of susceptible individuals will make the disease disappear. [ABSTRACT FROM AUTHOR]

We consider the Schrödinger operator in a bounded domain $ \mathbf{R}^n (n=2,3) $ R n (n = 2 , 3) with Neumann boundary condition. We suppose that this domain contains small deformable inclusions, i.e. regions where the potentials do not have the same values as the exterior medium. Our goal is to construct an asymptotic formula for the case of multiple and simple eigenvalue problems. We find an expansion that highlights the relation between the deformation parameters and the eigenvalues. The problem is devoted to a domain containing a finite number of inhomogeneities. Also, we focus on the interface of a small inclusion. [ABSTRACT FROM AUTHOR]

In this paper we investigate the Riemannian extensibility of saturation phenomena treated first in the Euclidean framework by Brandolini et al. [Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem. Adv Math (N Y). 2011;228(4):2352–2365.]. The saturation problem is formulated in terms of the first eigenvalue of the perturbation of the Laplace-Beltrami operator by the integral of the unknown function: the first eigenvalue increases with the weight affecting the integral up to a finite critical value and then remains constant, i.e. it saturates. Given a Riemannian manifold with certain curvature constraints, by using symmetrization arguments and sharp isoperimetric inequalities, we reduce the general problem to a variational one, formulated on either positively or negatively curved Riemannian model spaces; in addition, the possible scenarios for the optimal domains turn to be either geodesic balls or the union of two disjoint geodesic balls. We then explicitly compute the eigenvalues and eigenfunctions in terms of the radii, curvature and weight. A sufficient condition (incompatibility of a system of nonlinear equations involving special functions) is given that implies similar saturation phenomena to the Euclidean case. Due to its highly nonlinear character of the reduced problem (arising from the presence of curvature and special functions), we provide only partial answers to the original problem. However, both analytical computations and numerical tests suggest that the required incompatibility always persists. In addition, in the limit cases when the curvature tends to zero (for both positive an negative curvature), our results reduce to the Euclidean version. [ABSTRACT FROM AUTHOR]

We provide a detailed study of the spectral properties of the linear operator H (ε) = - (ε 2 χ Ω ε + χ Ω ε c) Δ modeling, through the wave equation (∂ tt + H (ε)) u = 0 , the dynamics of acoustic waves in the presence of a small inhomogeneity of size ε having high contrast ε - 2 . In particular, we give precise results on the localization of the resonances of H (ε) and their first-order ε -expansions; the latter are explicitly expressed in terms of the eigenvalues and eigenvectors of the Newton potential operator of the set Ω whose rescaling of size ε defines Ω ε . [ABSTRACT FROM AUTHOR]

We consider the problem of minimising the k-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are well-posed for Dirichlet eigenvalues in any dimension d ≥ 2 and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as k → + ∞ . In this paper, we show that the same is true in the case of Neumann eigenvalues under diameter constraint in any dimension and under perimeter constraint in dimension d = 2 . We also consider these problems for Robin eigenvalues and mixed Dirichlet–Neumann eigenvalues, under an additional geometric constraint. [ABSTRACT FROM AUTHOR]

We study the magnetic Laplacian in a two-dimensional exterior domain with Neumann boundary condition and uniform magnetic field. For the exterior of the disk we establish accurate asymptotics of the low-lying eigenvalues in the weak magnetic field limit. For the exterior of a star-shaped domain, we obtain an asymptotic upper bound on the lowest eigenvalue in the weak field limit, involving the 4 -moment, and optimal for the case of the disk. Moreover, we prove that, for moderate magnetic fields, the exterior of the disk is a local maximizer for the lowest eigenvalue under a p -moment constraint. [ABSTRACT FROM AUTHOR]

We provide bounds for the sequence of eigenvalues {λi(Ω)}i{\left\{{\lambda }_{i}\left(\Omega )\right\}}_{i} of the Dirichlet problem (I−Δ)lnu=λuinΩ,u=0inRN\Ω,{\left(I-\Delta )}^{\mathrm{ln}}u=\lambda u\hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\hspace{1.0em}u=0\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N}\setminus \Omega , where (I−Δ)ln{\left(I-\Delta )}^{\mathrm{ln}} is the Klein-Gordon operator with Fourier transform symbol ln(1+∣ξ∣2)\mathrm{ln}\left(1+{| \xi | }^{2}). The purpose of this study is to obtain the upper and lower bounds for the sum of the first k-eigenvalues by extending the Li-Yau’s method and Kröger’s method, respectively.

In this paper, we study the first eigenvalue of the Laplacian on doubly connected domains when Robin and Dirichlet conditions are imposed on the outer and the inner part of the boundary, respectively. We provide that the spherical shell reaches the maximum of the first eigenvalue of this problem among a suitable class of domains when the measure, the outer perimeter and inner (n - 1) th quermassintegral are fixed. [ABSTRACT FROM AUTHOR]

The phrase solotone effect refers to a persistent irregular pattern within a set of eigenvalues. This effect arises when the underlying differential equations contain discontinuities. Models of the Earth contain such discontinuities and research into solotones was originally motivated by vibration problems in geophysics. More recently, solotone effects have been discovered within heat conduction, and within quantum wells. However, at present, only one-dimensional arrangements have been shown to give rise to a solotone effect. In this paper, we compute the eigenvalues of an inhomogeneous vibrating rectangular membrane. The resulting eigenspectra, when taken as a whole, fail to display a solotone effect. Nevertheless, we show that irregular patterns do exist within selected subsets of eigenvalues. Our analysis therefore provides the first example of a solotone effect in a spatial dimension greater than one. Furthermore, this discovery could be used to extend the solotone inverse method already implemented in one dimension. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the eigenvalue problem of a time-periodic parabolic operator on a finite network. The network under consideration can support various types of flows, such as water, wind, or traffic. Our focus is to determine the asymptotic behavior of the principal eigenvalue as the diffusion rate approaches zero, or the advection rate approaches infinity, under reasonable boundary conditions that can be derived from ecosystems. Our results demonstrate that such asymptotics is primarily influenced by the boundary conditions at the upstream and downstream vertices of the network, rather than the geometric structure of the finite network itself provided that it is simple and connected. We then apply our results to a single-species population model and two SIS epidemic systems on networks and reveal the substantial impact of the diffusion and advection rates as well as the boundary conditions on the long-time dynamics of the population and the transmission of infectious diseases. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We improve the work [R. L. Frank and J. Sabin, Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math. 317 2017, 157–192] concerning the spectral cluster bounds for orthonormal systems at p = ∞ , on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from [ λ 2 , (λ + 1) 2) to [ λ 2 , (λ + ϵ ⁢ (λ) ) 2) , where ϵ ⁢ (λ) is a function of λ that goes to 0 as λ goes to ∞ . In achieving this, we invoke the method developed in [J. Bourgain, P. Shao, C. D. Sogge and X. Yao, On L p -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys. 333 2015, 3, 1483–1527]. [ABSTRACT FROM AUTHOR]

L ν operator is an important extrinsic differential operator of divergence type and has profound geometric settings. In this paper, we consider the clamped plate problem of L ν 2 operator on a bounded domain of the complete Riemannian manifolds. A general formula of eigenvalues of L ν 2 operator is established. Applying this general formula, we obtain some estimates for the eigenvalues with higher order on the complete Riemannian manifolds. As several fascinating applications, we discuss this eigenvalue problem on the complete translating solitons, minimal submanifolds on the Euclidean space, submanifolds on the unit sphere and projective spaces. In particular, we get a universal inequality with respect to the L I I operator on the translating solitons. Usually, it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds. Therefore, this work can be viewed as a new contribution to universal estimate. [ABSTRACT FROM AUTHOR]

We obtain upper bounds on the number of nodal domains of Laplace eigenfunctions on chain domains with Neumann boundary conditions. The chain domains consist of a family of planar domains, with piecewise smooth boundary, that are joined by thin necks. Our work does not assume a lower bound on the width of the necks in the chain domain. As a consequence, we prove an upper bound on the eigenvalue of Courant sharp eigenfunctions that is independent of the widths of the necks. [ABSTRACT FROM AUTHOR]

In the article we study the Neumann (p, q)-eigenvalue problems in bounded Hölder γ -singular domains Ω γ ⊂ R n . In the case 1 < p < ∞ and 1 < q < p γ ∗ we prove solvability of this eigenvalue problem and existence of the minimizer of the associated variational problem. In addition, we establish some regularity results of the eigenfunctions and some estimates of (p, q)-eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper we develop a comprehensive study on principal eigenvalues and both the (weak and strong) maximum and comparison principles related to an important class of nonlinear systems involving fractional m-Laplacian operators. Explicit lower bounds for principal eigenvalues of this system in terms of the diameter of bounded domain Ω ⊂ R N are also proved. As application, we measure explicitly how small has to be diam (Ω) so that weak and strong maximum principles associated to this problem hold in Ω . [ABSTRACT FROM AUTHOR]

We consider a Dirac operator with constant magnetic field defined on a half-plane with boundary conditions that interpolate between infinite mass and zigzag. By a detailed study of the energy dispersion curves, we show that the infinite mass case generically captures the profile of these curves, which undergoes a continuous pointwise deformation into the topologically different zigzag profile. Moreover, these results are applied to the bulk-edge correspondence. In particular, by means of a counterexample, we show that this correspondence does not always hold true in the zigzag case. [ABSTRACT FROM AUTHOR]

We establish a framework for fiberwise symmetrization to find a lower bound of a Dirichlet-type energy functional in a variational problem on a fibred Riemannian manifold, and use it to prove a comparison theorem of the first eigenvalue of the Laplacian on a warped product manifold. [ABSTRACT FROM AUTHOR]

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and provide a number of geometric applications. In particular, we derive an inequality which relates the eigenvalues of the Jacobi operator for f-minimal hypersurfaces and the spectrum of the Hodge Laplacian. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p-Laplace operator, namely: λ F (β , Ω) = min ψ ∈ W 1 , p (Ω) \ { 0 } ∫ Ω F (∇ ψ) p d x + β ∫ ∂ Ω | ψ | p F (ν Ω) d H N - 1 ∫ Ω | ψ | p d x , where p ∈ ] 1 , + ∞ [ , Ω is a bounded, convex domain in R N , ν Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on R N. We show an upper bound for λ F (β , Ω) in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on the volume and the anisotropic perimeter of Ω , in the spirit of the classical estimates of Pólya (J Indian Math Soc (NS) 24:413–419, 1961) for the Euclidean Dirichlet Laplacian. We will also provide a lower bound for the torsional rigidity τ p (β , Ω) p - 1 = max ψ ∈ W 1 , p (Ω) \ { 0 } ∫ Ω | ψ | d x p ∫ Ω F (∇ ψ) p d x + β ∫ ∂ Ω | ψ | p F (ν Ω) d H N - 1 when β > 0. The obtained results are new also in the case of the classical Euclidean Laplacian. [ABSTRACT FROM AUTHOR]

This approach finds new upper bounds for the first positive eigenvalue of the p-Laplacian operator using the mean and constant sectional curvatures on Riemannian manifolds. In particular, we provide several estimates for the first nonzero eigenvalue of the p-Laplacian operator on closed orientated totally real submanifolds of dimension m in a generalized complex space form M n (κ , ε) . Moreover, we generalize the Reilly-inequality of Laplacian (Reilly in Comment Math Helv 52(4):525–533, 1977) to the p-Laplacian for totally real submanifold in complex projective space and complex Euclidean space for κ = 1 and κ = 0 , as applications, respectively. [ABSTRACT FROM AUTHOR]

We use a Gaussian Process Regression (GPR) strategy to analyze different types of curves that are commonly encountered in parametric eigenvalue problems. We employ an offline-online decomposition method. In the offline phase, we generate the basis of the reduced space by applying the proper orthogonal decomposition (POD) method on a collection of pre-computed, full-order snapshots at a chosen set of parameters. Then we generate our GPR model using four different Matérn covariance functions. In the online phase, we use this model to predict both eigenvalues and eigenvectors at new parameters. We then illustrate how the choice of each covariance function influences the performance of GPR. Furthermore, we discuss the connection between Gaussian Process Regression and spline methods and compare the performance of the GPR method against linear and cubic spline methods. We show that GPR outperforms other methods for functions with a certain regularity. [ABSTRACT FROM AUTHOR]

In this paper, we prove sharp isoperimetric inequalities for lower order eigenvalues of Neumann Laplacian on bounded domains in both compact and noncompact rank-1 symmetric spaces. Our results generalize the work of Wang and Xia for bounded domains in the hyperbolic space (Xia and Wang in Math Ann 385(1–2):863–879, 2023), and Szegö–Weinberger inequalities in rank-1 symmetric spaces obtained by Aithal and Santhanam (Trans Am Math Soc 348(10):3955–3965, 1996). [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R d be a bounded open connected set with Lipschitz boundary. Let A N and A D be the Stokes Neumann operator and Stokes Dirichlet operator on Ω , respectively. We study the associated Stokes version of the Dirichlet-to-Neumann operator and show a Krein formula which relates these three Stokes version operators. We also prove a Stokes version of the Friedlander inequalities, which relates the Dirichlet eigenvalues and the Neumann eigenvalues. [ABSTRACT FROM AUTHOR]

We obtain estimates of Neumann eigenvalues of the divergence form elliptic operators in Sobolev extension domains. The suggested approach is based on connections between divergence form elliptic operators and quasiconformal mappings. The connection between Neumann eigenvalues of elliptic operators and the smallest-circle problem (initially suggested by J. J. Sylvester in 1857) is given. [ABSTRACT FROM AUTHOR]

We study the first nonzero eigenvalues for the p-Laplacian on quaternionic Kähler manifolds. Our first result is a lower bound for the first nonzero closed (Neumann) eigenvalue of the p-Laplacian on compact quaternionic Kähler manifolds. Our second result is a lower bound for the first Dirichlet eigenvalue of the p-Laplacian on compact quaternionic Kähler manifolds with smooth boundary. Our results generalize corresponding results for the Laplace eigenvalues on quaternionic Kähler manifolds proved in Li and Wang (J Geom Anal 33(3):20, 2023). [ABSTRACT FROM AUTHOR]

The space-time behavior for the Cauchy problem of the 3D compressible bipolar Navier-Stokes-Poisson (BNSP) system with unequal viscosities is given. The space-time estimate of the electric field ∇ϕ = ∇(−Δ)−1(n − Zρ) is the most important in deducing generalized the Huygens' principle for the BNSP system and it requires proving that the space-time estimate of n − Zρ only contains the diffusion wave due to the singularity of the operator ∇(−Δ)−1. A suitable linear combination of unknowns reformulating the original system into two small subsystems for the special case (with equal viscosities) in Wu and Wang (2017) is so crucial for both linear analysis and nonlinear estimates, especially for the space-time estimate of ∇ϕ. However, the benefits from this reformulation will not exist any longer for general cases. Here, we study an 8×8 Green's matrix directly. More importantly, each entry in Green's matrix contains wave operators in the low frequency part, which will generally produce the Huygens' wave; as a result, one cannot achieve that the space-time estimate of n − Zρ only contains the diffusion wave as before. We overcome this difficulty by taking more detailed spectral analysis and developing new estimates arising from subtle cancellations in Green's function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the first Steklov–Dirichlet eigenvalue of the Laplace operator in annular domains with a spherical hole. We prove a monotonicity result with respect to the hole, when the outer region is centrally symmetric. [ABSTRACT FROM AUTHOR]

We consider the magnetic Schrödinger operator H = (i ∇ + A) 2 - V with a non-negative potential V supported over a strip which is a local deformation of a straight one, and the magnetic field B : = rot (A) is assumed to be non-zero and local. We show that the magnetic field does not change the essential spectrum of this system, and investigate a sufficient condition for the discrete spectrum of H to be empty. [ABSTRACT FROM AUTHOR]