Laplace operator

For scientists and engineers, the Laplacian operator is a fundamental tool that has made it possible to carry out important frontier studies involving wave propagation, potential theory, heat conduction, the distribution of stresses in a deformable solid and quantum mechanics. Knowing, understanding, and manipulating the Laplacian operator allows us to tackle complex and exciting physics, chemistry, and engineering problems. In this paper, contained in the Special Issue “Mathematics as the M in STEM Education”, we present an instructional derivation of the Laplacian operator in spherical coordinates. Our derivation is self-contained and employs well-known mathematical concepts used in all science, technology, engineering, and mathematics (STEM) disciplines. Our lengthy but straightforward procedure shows that this fundamental tool in mathematics is not intractable but accessible to anyone who studies any of the STEM disciplines. We consider that this work may be helpful for students...

In this paper, we will prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth: provided N > p 2 + p, where p is the p-Laplacian operator, 1 < p < N, p * = pN N -p , μ > 0 and Ω is an open bounded domain in R N .

We improve some recent results of Sagiv and Steinerberger that quantify the following uncertainty principle: for a function f with mean zero, either the size of the zero set of the function or the cost of transporting the mass of the positive part of f to its negative part must be big. We also provide a sharp upper estimate of the transport cost of the positive part of an eigenfunction of the Laplacian. This proves a conjecture of Steinerberger and provides a lower bound of the size of the nodal set of the eigenfunction.

In this paper we address sampling and approximation of functions on combinatorial graphs. We develop filtering on graphs by using Schrodinger's group of operators generated by combinatorial Laplace operator. Then we construct a sampling theory by proving Poincare and Plancherel-Polya-type inequalities for functions on graphs. These results lead to a theory of sparse approximations on graphs and have potential applications to filtering, denoising, data dimension reduction, image processing, image compression, computer graphics, visualization and learning theory.

We study the eigenvalues of the connection Laplacian on a graph with an orthogonal group or unitary group signature. We establish higher order Buser type inequalities, i.e., we provide upper bounds for eigenvalues in terms of Cheeger constants in the case of nonnegative Ricci curvature. In this process, we discuss the concepts of Cheeger type constants and a discrete Ricci curvature for connection Laplacians and study their properties systematically. The Cheeger constants are defined as mixtures of the expansion rate of the underlying graph and the frustration index of the signature. The discrete curvature, which can be computed efficiently via solving semidefinite programming problems, has a characterization by the heat semigroup for functions combined with a heat semigroup for vector fields on the graph.

Abstract: We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the degree is ...

We introduce two new spectral-geometric constructs on truncated one-sheet hyperboloids of revolution: (i) the S M Nazmuz Sakib Axial Band-Dominance Principle (Sakib Principle) asserting that beyond a geometry threshold the first positive Laplace-Beltrami band-edge is axially symmetric; and (ii) the Hyperboloidal ASCI Invariant (Sakib Invariant) quantifying anisotropy sensitivity via-∂ κ log λ (1) 0 (κ; U). We prove sharp asymptotics and provide largescale numerical evidence that: (a) for fixed truncation U , λ (1) 0 (κ; U) ∼ C 0 (U)/(a 2 κ 2) while each m ≥ 1 sector stabilizes at Θ(m 2 /a 2); hence the azimuthal-to-axial ratios grow as Θ(κ 2) (S M Nazmuz Sakib Quadratic Ratio Law). (b) A cylinder comparison yields a natural threshold κU ≈ π/2 for axial dominance. All figures are generated via pgfplots directly from representative data patterns consistent with our solver output. The setup is classical (Laplace-Beltrami in orthogonal coordinates, separation into azimuthal sectors), but the packaging-Sakib Principle, Sakib Invariant, and the threshold/quadratic laws-appears novel.

We optimize a selection of eigenvalues of the Laplace operator with Dirichlet or Neumann boundary conditions by adjusting the shape of the domain on which the eigenvalue problem is considered. Here, a phase-field function is used to represent the shapes over which we minimize. The idea behind this method is to modify the Laplace operator by introducing phase-field dependent coefficients in order to extend the eigenvalue problem on a fixed design domain containing all admissible shapes. The resulting shape and topology optimization problem can then be formulated as an optimal control problem with PDE constraints in which the phase-field function acts as the control. For this optimal control problem, we establish first-order necessary optimality conditions and we rigorously derive its sharp interface limit. Eventually, we present and discuss several numerical simulations for our optimization problem.

Let G be a simple connected graph with n vertices and m edges, sequence of vertex degrees , where µ i are the Laplacian eigenvalues of graph G, the Kirchhoff index of G, and by NK = n i=1 d i the Narumi-Katayama index. In this paper we prove some inequalities that exhibit relationship between the Kirchhoff and Narumi-Katayama indices. One can easily observe that Π 1 (G) = (NK(G)) 2 . Current name, i.e. the Narumi-Katayama index, was proposed in . Details on this topological index can be found in [8-10, 18, 34, 35].

Let G be a simple connected graph with degree sequence (d 1 , d 2 , . . . , dn) i=1 µ i denote the Kirchhoff index and the number of spanning trees of G, respectively. In this paper we establish several lower bounds for Kf (G) in terms of τ (G), the order, the size and maximum degree of G.

Let G = (V, E), V = {1, 2, . . . , n}, be a simple connected graph of order n and size m, with sequence of vertex degrees the Kirchhoff index and the number of spanning trees of G, respectively. In this paper we determine several lower bounds for K f (G) depending on t(G) and some of the graph parameters n, m or ∆.

For a simple connected graph G of order n and size m, the Laplacian energy of G is defined as In this note, some new lower bounds on the graph invariant LE(G) are derived. The obtained results are compared with some already known lower bounds of LE(G).

Let G be an undirected connected graph with n, n ≥ 3, vertices and m edges. If 0 are the Laplacian and the normalized Laplacian eigenvalues of G, then the Kirchhoff and the degree Kirchhoff indices obey the relations K f i , respectively. The inequalities that determine lower bounds for some invariants of G, that contain K f (G) and DK f (G) , are obtained in this paper. Lower bounds for K f (G) and DK f (G), known in the literature, are obtained as a special case.

Let G be a simple connected graph with n vertices and m edges, sequence of vertex degrees , where µ i are the Laplacian eigenvalues of graph G, the Kirchhoff index of G, and by NK = n i=1 d i the Narumi-Katayama index. In this paper we prove some inequalities that exhibit relationship between the Kirchhoff and Narumi-Katayama indices. One can easily observe that Π 1 (G) = (NK(G)) 2 . Current name, i.e. the Narumi-Katayama index, was proposed in . Details on this topological index can be found in [8-10, 18, 34, 35].

Let G = (V, E), V = {1, 2, . . . , n}, be a simple graph without isolated vertices, with vertex degree sequence is measure of irregularity of graph G with the property I(G) = 0 if and only if G is regular, and I(G) > 0 otherwise. In this paper we introduce some new irregularity measures.

Let ∆M be the Laplace operator on a compact n-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions u : ∆M u + λu = 0. In dimension n = 2 we refine the Donnelly-Fefferman estimate by showing that H 1 ({u = 0}) ≤ Cλ 3/4-β for some β ∈ (0, 1/4). The proof employs the Donnelly-Fefferman estimate and a combinatorial argument, which also gives a lower (nonsharp) bound in dimension n = 3: H 2 ({u = 0}) ≥ cλ α for some α ∈ (0, 1/2). The positive constants c, C depend on the manifold, α and β are universal.

Let g  be the Laplace Beltrami operator on a manifold M with Dirichlet (resp., Neumann) boundary conditions. We compare the spectrum of on a Riemannian manifold for Neumann boundary condition and Dirichlet boundary condition. Then we construct an effective method of obtaining small eigenvalues for Neumann's problem.

Differential equations with deviating arguments
serve as a fundamental cornerstone in mathematical modeling,
offering a robust framework for characterizing the dynamics of
various systems across multiple disciplines. Meanwhile, oscillatory
theorems are essential in analyzing the intrinsic vibrational
patterns within dynamic systems, providing critical insights
into their stability and periodicity. This study focuses on
examining the behavior of half-linear second-order difference
equations of an unstable type when their arguments are altered
from the form

In this paper, we study the spectrum of the operator which results when the Perfectly Matched Layer (PML) is applied in Cartesian geometry to the Laplacian on an unbounded domain. This is often thought of as a complex change of variables or "complex stretching." The reason that such an operator is of interest is that it can be used to provide a very effective domain truncation approach for approximating acoustic scattering problems posed on unbounded domains. Stretching associated with polar or spherical geometry lead to constant coefficient operators outside of a bounded transition layer and so even though they are on unbounded domains, they (and their numerical approximations) can be analyzed by more standard compact perturbation arguments. In contrast, operators associated with Cartesian stretching are non-constant in unbounded regions and hence cannot be analyzed via a compact perturbation approach. Alternatively, to show that the scattering problem PML operator associated with Cartesian geometry is stable for real nonzero wave numbers, we show that the essential spectrum of the higher order part only intersects the real axis at the origin. This enables us to conclude stability of the PML scattering problem from a uniqueness result given in a subsequent publication.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

We study a recently proposed modification of the Skyrme model that possesses an exact self-dual sector leading to an infinity of exact Skyrmion solutions with arbitrary topological (baryon) charge. The self-dual sector is made possible by the introduction, in addition to the usual three SU (2) Skyrme fields, of six scalar fields assembled in a symmetric and invertible three dimensional matrix h. The action presents quadratic and quartic terms in derivatives of the Skyrme fields, but instead of the group indices being contracted by the SU (2) Killing form, they are contracted with the h-matrix in the quadratic term, and by its inverse on the quartic term. Due to these extra fields the static version of the model, as well as its self-duality equations, are conformally invariant on the three dimensional space IR 3 . We show that the static and self-dual sectors of such a theory are equivalent, and so the only non-self-dual solution must be time dependent. We also show that for any configuration of the Skyrme SU (2) fields, the h-fields adjust themselves to satisfy the self-duality equations, and so the theory has plenty of non-trivial topological solutions. We present explicit exact solutions using a holomorphic rational ansatz, as well as a toroidal ansatz based on the conformal symmetry. We point to possible extensions of the model that break the conformal symmetry as well as the self-dual sector, and that can perhaps lead to interesting physical applications.

The power graph $\mathcal{G}(G)$ of a group $G$ is a simple graph whose vertices are the elements of $G$ and two distinct vertices are adjacent if one is a power of other. In this paper, we investigate the Laplacian spectrum of the power graph $\mathcal{G}(\mathbb{Z}_{p^m}^n)$ of finite abelian $p$-group $\mathbb{Z}_{p^m}^n$. In particular, we prove that the spectrum of group $\mathbb{Z}_{p^m}^n$ is contained in the Laplacian spectrum of graph $\mathcal{G}(\mathbb{Z}_{p^m}^n)$. For a finite abelian group $G$ whose power graph $\mathcal{G}(G)$ is planar, we also prove that the spectrum of group $G$ is contained in the Laplacian spectrum of graph $\mathcal{G}(G)$.

We study the time-independent Schrödinger equation with radially symmetric potential k|x| α , k ≥ 0, k ∈ R, α ≥ 2 on a bounded domain Ω in R n , (n ≥ 2) with Dirichlet boundary conditions. In particular, we compare the eigenvalue λ2(Ω) of the operator -∆ + k|x| α on Ω with the eigenvalue λ2(S1) of the same operator -∆ + kr α on a ball S1, where S1 has radius such that the first eigenvalues are the same (λ1(Ω) = λ1(S1)). The main result is to show λ2(Ω) ≤ λ2(S1). We also give an extension of the main result to the case of a more general elliptic eigenvalue problem on a bounded domain Ω with Dirichlet boundary conditions.

In this paper we give sufficient conditions for the nonexistence of nonnegative nontrivial entire weak solutions of p-Laplacian elliptic inequalities, with possibly singular weights and gradient terms, of the form div{g(|x|)|Du| p-2 Du} ≥ h(|x|)f (u) ± h(|x|) (|Du|), under the main request that h and h are continuous on R + . We achieve our conclusions introducing a generalized version of the well known Keller-Ossermann condition.

In this paper we obtain generalized Keller-Osserman conditions for wide classes of differential inequalities on weighted Riemannian manifolds of the form Lu ≥ b(x)f (u)ℓ(|∇u|) and Lu ≥ b(x)f (u)ℓ(|∇u|)g(u)h(|∇u|), where L is a non-linear diffusiontype operator. Prototypical examples of these operators are the p-Laplacian and the mean curvature operator. While we concentrate on non-existence results, in many instances the conditions we describe are in fact necessary for non-existence. The geometry of the underlying manifold does not affect the form of the Keller-Osserman conditions, but is reflected, via bounds for the modified Bakry-Emery Ricci curvature, by growth conditions for the functions b and ℓ. We also describe a weak maximum principle related to inequalities of the above form which extends and improves previous results valid for the ϕ-Laplacian.

We investigate the existence of positive solutions for the system of fourth-orderp-Laplacian boundary value problems(|u′′|p-1u′′)′′=f1(t,u,v),  (|v′′|q-1v′′)′′=f2(t,u,v),  u(2i)(0)=u(2i)(1)=0,  i=0,1,  v(2i)(0)=v(2i)(1)=0,  i=0,1,wherep,q&gt;0andf1,f2∈C([0,1]×ℝ+2,ℝ+)  (ℝ+:=[0,∞)). Based on a priori estimates achieved by utilizing Jensen’s integral inequalities and nonnegative matrices, we use fixed point index theory to establish our main results.

The motive behind this work is to obtain some sufficient conditions for the existence of solution to a nonlinear problem of implicit fractional differential equations (IFDEs) involving integral boundary conditions with p-Laplacian operator, using prior estimate method. The method applied here does not require compactness of the operator, which makes it distinguished from other methods. Besides developing the respective conditions, we also investigate Hyers-Ulam type

This work presents a novel theoretical framework proposing a deep connection between two major relational approaches to quantum gravity: Shape Dynamics (SD) and Emergent Gravity (EG). Despite both eliminating absolute spacetime structures, SD (a global Hamiltonian theory of conformal geometry) and EG (a local thermodynamic theory deriving spacetime from entanglement) have fundamentally different mathematical structures. This paper argues that this disparity is only apparent and that their core principles—SD's "best-matching" and EG's "entanglement equilibrium"—may share a common origin in the quantum thermodynamics of shape degrees of freedom.

The proposed synthesis is built on several key pillars:

1. Relational Thermodynamic Variables: The definition of fundamental thermodynamic quantities (entropy, energy flux, temperature) within the configuration space of Shape Dynamics, making them invariant under diffeomorphisms and conformal transformations.
2. A Connecting Principle: A detailed argument, based on information geometry, linking the minimization of kinematic energy in SD (best-matching) to the maximization of entanglement entropy in EG.
3. A Novel Lagrangian: The introduction of a new action principle that incorporates a kinetic term from SD, a potential term from General Relativity, and a new entropic term based on a conformally invariant definition of relational entropy.
4. An Entropic Clock: A proposed solution to the "problem of time" in quantum gravity by defining a relational time parameter based on the rate of change of total entanglement entropy.

This work is primarily a research proposal and a theoretical framework. It includes detailed mathematical derivations of foundational results from both SD and EG to ensure clarity and provide a common starting point. A comprehensive research program is outlined to develop this framework into a testable theory, including steps for formal derivation, quantization, and the investigation of phenomenological implications in cosmology and black hole physics.

In this paper we prove the existence of a nontrivial solution in D 1 , p ( R N ) ∩ D 1 , q ( R N ) for the following ( p , q ) -Laplacian problem: { − Δ p u − Δ q u = λ g ( x ) | u | r − 1 u + | u | p ⋆ − 2 u , u ( x ) ≥ 0 , x ∈ R N , where 1 &lt; q ≤ p &lt; r + 1 &lt; p ⋆ : = N p N − p , p &lt; N , λ &gt; 0 is a parameter, Δ m u : = div ( | ∇ u | m − 2 ∇ u ) is the m-Laplacian operator and g ∈ L p ⋆ p ⋆ − r − 1 ( R N ) is positive in an open set. MSC: 35J92, 47J30.

This paper is concerned with the existence of nontrivial solutions for a class of degenerate quasilinear elliptic systems involving critical Hardy-Sobolev type exponents. The lack of compactness is overcame by using the Brezis-Nirenberg approach, and the multiplicity result is obtained by combining a version of the Ekeland's variational principle due to Mizoguchi with the Ambrosetti-Rabinowitz mountain pass theorem.

This paper studies degenerate quasilinear elliptic systems involving p, q-superlinear and critical nonlinearities with singularities. Existence results are obtained by using properties of the best Hardy-Sobolev constant together with an approach developed by Brezis and Nirenberg.

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on $\cn$. The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with Hörmander's $L^2$ estimates for the $\bar{\partial}$ operator are key ingredients in the proof.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Using the machinery of unitary spherical harmonics due to Koornwinder, Folland and other authors, we obtain expansions for the Szegö and the weighted Bergman kernels of M -harmonic functions, i.e. functions annihilated by the invariant Laplacian on the unit ball of the complex n-space. This yields, among others, an explicit formula for the M -harmonic Szegö kernel in terms of multivariable as well as single-variable hypergeometric functions, and also shows that most likely there is no explicit ("closed") formula for the corresponding weighted Bergman kernels.

Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. In this article, first we define and study statistical solitons on Ricci-symmetric statistical warped products R × f N 2 and N 1 × f R . Second, we study statistical warped products as submanifolds of statistical manifolds. For statistical warped products statistically immersed in a statistical manifold of constant curvature, we prove Chen’s inequality involving scalar curvature, the squared mean curvature, and the Laplacian of warping function (with respect to the Levi–Civita connection). At the end, we establish a relationship between the scalar curvature and the Casorati curvatures in terms of the Laplacian of the warping function for statistical warped product submanifolds in the same ambient space.

We resolve two problems in Mathematical Physics. First, we prove that any L^∞ connection Γ on the tangent bundle of an arbitrary differentiable manifold with L^∞ Riemann curvature can be smoothed by coordinate transformation to optimal regularity, Γ∈ W^1,p, any p&lt;∞, (i.e., one derivative smoother than the curvature). For Lorentzian metrics in GR this implies that shock wave solutions of the Einstein-Euler equations are non-singular—geodesic curves, locally inertial coordinates and the Newtonian limit exist. The proof is based on extending authors&#39; existence theory for the RT-equations, a system of nonlinear elliptic partial differential equations, to the level of L^∞ connections, and for this we introduce the reduced RT-equations. Secondly, we prove that this existence theory suffices to extend Uhlenbeckcompactness from the case of connections on vector bundles over Riemannian manifolds (2019 Abel Prize and 2007 Steele Prize), to the case of connections on the tangent bundle ...

In this paper we prove Schauder estimates at the boundary for sub-Laplacian type operators in Carnot groups. While internal Schauder estimates have been deeply studied, up to now subriemannian estimates at the boundary are known only in the Heisenberg groups. The proof of these estimates in the Heisenberg setting, due to Jerison ([32]), is based on the Fourier transform technique and can not be repeated in general Lie groups. After the result of Jerison no new contribution to the boundary problem has been provided. In this paper we introduce a new approach, which allows to build a Poisson kernel starting from the fundamental solution, from which we deduce the Schauder estimates at non characteristic boundary points.

In this paper we construct a fundamental solution for the Laplace operator on the contact complex in Heisenberg groups H n (Rumin's complex) relying on the notion of currents in H n given recently by Franchi, Serapioni and Serra Cassano. This operator is of order 2 on k intrinsic forms for k = n, but is of order 4 on n intrinsic forms. As an application, we prove sharp L p a priori estimates for horizontal derivatives.

We prove multidimensional analogs of the trace formula obtained previously for one-dimensional Schrödinger operators. For example, let V be a continuous function on [0, 1] ν ⊂ R ν . For A ⊂ {1, . . . , ν}, let -∆ A be the Laplace operator on [0, 1] ν with mixed Dirichlet-Neumann boundary conditions ϕ(x) = 0, x j = 0 or x j = 1 for j ∈ A, ∂ϕ ∂x j (x) = 0, x j = 0 or x j = 1 for j / ∈ A.

We consider the Laplace-Beltrami operator ∆ g on a smooth, compact Riemannian manifold (M, g) and the determinantal point process X λ on M associated with the spectral projection of -∆ g onto the subspace corresponding to the eigenvalues up to λ 2 . We show that the pull-back of X λ by the exponential map exp p : T * p M → M under a suitable scaling converges weakly to the universal determinantal point process on T * p M as λ → ∞.

Extending a result of D. V. Vassilevich , we obtain the asymptotic expansion for the trace of a spatially regularized heat operator L Θ (f )e -t△ Θ , where △ Θ is a generalized Laplacian defined with Moyal products and L Θ (f ) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples , the spectral action introduced in noncommutative geometry by A. Chamseddine and A. Connes [6] is computed. This result generalizes the Connes-Lott action previously computed by Gayral [23] for symplectic Θ.

Extending a result of Vassilevich, we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ(f)e−tΔΘ, where ΔΘ is a generalized Laplacian defined with Moyal products and LΘ(f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples, the spectral action introduced in noncommutative geometry by Chamseddine and Connes is computed. This result generalizes the Connes–Lott action previously computed by Gayral for symplectic Θ.