Laplace-type Operators Research Articles

Crank-Nicolson finite element methods for nonlocal problems with p -Laplace-type operator

A theoretical analysis of a Crank-Nicolson Galerkin finite element method for a class of nonlinear nonlocal diffusion problems associated with p -Laplace-type operator is presented here. It is shown, by a rigorous analysis that the unconditionally optimal error estimates for the fully discrete scheme are established. The presence of the nonlocal term in the models destroys the sparsity of the Jacobian matrices when solving the problem numerically using finite element method and Newton-Raphson method. As a consequence, computations consume more time and space in contrast to local problems. To overcome this difficulty, a new algorithm is proposed to avoid the full Jacobian matrix. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Dirac operators with torsion, spectral Einstein functionals and the noncommutative residue

Recently Dabrowski et al. [Adv. Math. 427, 109128 (2023)] obtained the metric and Einstein functionals by two vector fields and Laplace-type operators over vector bundles, giving an interesting example of the spinor connection and square of the Dirac operator. Pfäffle and Stephan [Commun. Math. Phys. 321, 283–310 (2013)] considered orthogonal connections with arbitrary torsion on compact Riemannian manifolds and computed the spectral action. Motivated by the spectral functionals and Dirac operators with torsion, we give some new spectral functionals which is the extension of spectral functionals to the noncommutative realm with torsion, and we relate them to the noncommutative residue for manifolds with boundary. Our method of producing these spectral functionals is the noncommutative residue and Dirac operators with torsion.

On variational competing [formula omitted]Laplacian Dirichlet problem with gradient depending weight

We investigate the existence of variational generalized solutions to coercive competitive equations driven by a competing p,q−Laplacian type operator with weigth depending on the gradient. An abstract existence result based on a Galerkin scheme is therefore obtained and subsequently applied.

Fractional differentiability for elliptic double obstacle problems with measure data

We consider elliptic double obstacle problems involving p -Laplacian type operators and measure data. We prove maximal fractional differentiability results in a completely linearized form under suitable differentiability assumptions on the obstacles. We also investigate fractional Sobolev–Morrey regularity for such problems under a density condition on the measures.

On spectral properties of the Schreier graphs of the Thompson group 𝐹

In this article we study spectral properties of the family of Schreier graphs associated to the action of the Thompson group F F on the interval [ 0 , 1 ] [0,1] . In particular, we describe spectra of Laplace type operators associated to these Schreier graphs and calculate certain spectral measures associated to the Schreier graph Υ \Upsilon of the orbit of 1 / 2 1/2 . As a byproduct we calculate the asymptotics of the return probabilities of the simple random walk on Υ \Upsilon starting at 1 / 2 1/2 . In addition, given a Laplace type operator L L on a tree-like graph we study relations between the spectral measures of L L associated to delta functions of different vertices and the spectrum of L L .

Multiple solutions to Kirchhoff-Schrödinger equations involving the $ p(\cdot) $-Laplace-type operator

<abstract><p>This paper is devoted to deriving several multiplicity results of nontrivial weak solutions to Kirchhoff-Schrödinger equations involving the $ p(\cdot) $-Laplace-type operator. The aims of this paper are stated as follows. First, under some conditions on a nonlinear term, we show that our problem has a sequence of infinitely many large energy solutions. Second, we obtain the existence of a sequence of infinitely many small energy solutions to the problem on a new class of nonlinear term. The primary tools to obtain such multiplicity results are the fountain theorem and the dual fountain theorem, respectively.</p></abstract>

Solutions for a class of problems driven by an anisotropic $ (p, q) $-Laplacian type operator

<abstract><p>In this manuscript, existence and multiplicity results are obtained for a problem involving an anisotropic $ (p, q) $-Laplacian-type operator by means of sub-supersolutions and variational techniques. This problem arises in various applications such as in the study of the enhancement of images, the spread of epidemic disease and in the dynamic of fluids. Under a general condition, the existence of a solution is proved, and the multiplicity of solutions is obtained by considering an additional natural hypothesis.</p></abstract>

An approximation approach for solving degenerated quasilinear problems

<p style='text-indent:20px;'>The present paper develops an approximation approach for solving a quasilinear Dirichlet boundary value problem that exhibits a degenerated <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian and full dependence on the solution and its gradient (convection term). The results establish that the solution set is nonempty and bounded. The principal part of the equation is driven by a <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian type operator containing a weight depending on the solution without any monotonicity assumption. The existence of a weak solution results through a limit process by means of approximate solutions arising from finite-dimensional problems. The a priori estimates are obtained in adequate Sobolev spaces with weights. An example provides an explicit illustration of the involved technique.</p>

Critical Kirchhoff equations involving the p-sub-Laplacians operators on the Heisenberg group

In this paper, we deal with a class of Kirchhoff-type critical elliptic equations involving the [Formula: see text]-sub-Laplacians operators on the Heisenberg group of the form [Formula: see text] where [Formula: see text] is the [Formula: see text]-sub-Laplacian, [Formula: see text], [Formula: see text] is the horizontal Sobolev space on [Formula: see text]. And [Formula: see text] is the homogeneous dimension of [Formula: see text], [Formula: see text] is a real parameter, [Formula: see text] is the critical Sobolev exponent on the Heisenberg group. Under some proper assumptions on the Kirchhoff function [Formula: see text], the potential function [Formula: see text] and [Formula: see text], together with the mountain pass theorem and the concentration-compactness principles for classical Sobolev spaces on the Heisenberg group, we prove the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases on the Heisenberg group. The results of this paper extend or complete recent papers and are new in several directions for the non-degenerate and degenerate critical Kirchhoff equations involving the [Formula: see text]-Laplacian type operators on the Heisenberg group.

Multiplicity of solutions to class of nonlocal elliptic problems with critical exponents

In this paper, we establish existence of infinitely many weak solutions for a class of quasilinear stationary Kirchhoff‐type equations, which involves a general variable exponent elliptic operator with critical growth. Precisely, we study the following nonlocal problem: where Ω is a bounded smooth domain of , with homogeneous Dirichlet boundary conditions on ∂Ω, the nonlinearity is a continuous function, is a function of the class is a continuous function, whose properties will be introduced later, λ is a positive parameter and . We assume that , where is the critical Sobolev exponent. We will prove that the problem has infinitely many solutions and also we obtain the asymptotic behavior of the solution as λ → 0+. Furthermore, we emphasize that a difference with previous researches is that the conditions on a(·) are general overall enough to incorporate some interesting differential operators. Our work covers a feature of the Kirchhoff's problems, that is, the fact that the Kirchhoff's function M in zero is different from zero, it also covers a wide class of nonlocal problems for p(x) > 1, for all . The main tool to find critical points of the Euler–Lagrange functional associated with this problem is through a suitable truncation argument, concentration‐compactness principle for variable exponent found in Bonder and Silva (2010), and the genus theory introduced by Krasnoselskii. The result of this paper extends or complements, or else completes recent papers and is new in several directions for the stationary Kirchhoff equations involving the p(x)‐Laplacian type operators.

Shape Optimization of a Weighted Two-Phase Dirichlet Eigenvalue

Let $m$ be a bounded function and $\alpha$ a nonnegative parameter. This article is concerned with the first eigenvalue $\lambda\_\alpha(m)$ of the drifted Laplacian type operator $\mathcal L\_m$ given by $\mathcal L\_m(u)= -\operatorname{div} \left((1+\alpha m)\nabla u\right)-mu$ on a smooth bounded domain, with Dirichlet boundary conditions. Assuming uniform pointwise and integral bounds on $m$, we investigate the issue of minimizing $\lambda\_\alpha(m)$ with respect to $m$. Such a problem is related to the so-called "two phase extremal eigenvalue problem" and arises naturally, for instance in population dynamics where it is related to the survival ability of a species in a domain. We prove that unless the domain is a ball, this problem has no "regular" solution. We then provide a careful analysis in the case of a ball by: (1) characterizing the solution among all radially symmetric resources distributions, with the help of a new method involving a homogenized version of the problem; (2) proving in a more general setting, a stability result for the centered distribution of resources with the help of a monotonicity principle for second order shape derivatives which significantly simplifies the analysis.

Towards a Unified theory of Fractional and Nonlocal Vector Calculus

Abstract Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or anomalous behavior. This has driven a desire for a vector calculus that includes nonlocal and fractional gradient, divergence and Laplacian type operators, as well as tools such as Green’s identities, to model subsurface transport, turbulence, and conservation laws. In the literature, several independent definitions and theories of nonlocal and fractional vector calculus have been put forward. Some have been studied rigorously and in depth, while others have been introduced ad-hoc for specific applications. The goal of this work is to provide foundations for a unified vector calculus by (1) consolidating fractional vector calculus as a special case of nonlocal vector calculus, (2) relating unweighted and weighted Laplacian operators by introducing an equivalence kernel, and (3) proving a form of Green’s identity to unify the corresponding variational frameworks for the resulting nonlocal volume-constrained problems. The proposed framework goes beyond the analysis of nonlocal equations by supporting new model discovery, establishing theory and interpretation for a broad class of operators, and providing useful analogues of standard tools from the classical vector calculus.

Quantum dynamics of the classical harmonic oscillator

A correspondence is established between measure-preserving, ergodic dynamics of a classical harmonic oscillator and a quantum mechanical gauge theory on two-dimensional Minkowski space. This correspondence is realized through an isometric embedding of the L2(μ) space on the circle associated with the oscillator’s invariant measure, μ, into a Hilbert space H of sections of a C-line bundle over Minkowski space. This bundle is equipped with a covariant derivative induced from an SO+(1, 1) gauge field on the corresponding inertial frame bundle, satisfying the Yang–Mills equations. Under this embedding, the Hamiltonian operator of a Lorentz-invariant quantum system, constructed as a geometrical Laplace-type operator on bundle sections, pulls back to the generator of the unitary group of Koopman operators governing the evolution of classical observables of the harmonic oscillator, with Koopman eigenfunctions of zero, positive, and negative eigenfrequencies corresponding to quantum eigenstates of zero (“vacuum”), positive (“matter”), and negative (“antimatter”) energies. The embedding also induces a pair of operators acting on classical observables of the harmonic oscillator, exhibiting canonical position–momentum commutation relationships. These operators have the structure of order-1/2 fractional derivatives and therefore display a form of non-locality. In a second part of this work, we study a quantum mechanical representation of the classical harmonic oscillator using a one-parameter family of reproducing kernel Hilbert spaces, K^τ, associated with the time-τ transition kernel of a fractional diffusion on the circle. As shown in recent work, these spaces are unital Banach *-algebras of functions. It is found that the evolution of classical observables in these spaces takes place via a strongly continuous, unitary Koopman evolution group, which exhibits a stronger form of classical–quantum consistency than the L2(μ) case. Specifically, for every real-valued classical observable in K^τ, there exists a quantum mechanical observable, whose expectation value is consistent with classical function evaluation. This allows for a description of classical state space dynamics, classical statistics, and quantum statistics of the harmonic oscillator within a unified framework.

Fractional elliptic problems with nonlinear gradient sources and measures

In this manuscript we deal with existence/uniqueness and regularity issues of suitable weak solutions to nonlocal problems driven by fractional Laplace type operators. Different from previous researches, in our approach we consider gradient non-linearity sources with subcritical growth, as well as appropriated measures as sources and boundary datum. We provide an in-depth discussion on the notions of solutions involved together with existence/uniqueness results in different regimes and for different boundary value problems. Finally, this work extends previous ones by dealing with more general nonlocal operators, source terms and boundary data.

The tunneling effect for Schr\xf6dinger operators on a vector bundle

In the semiclassical limit hbar rightarrow 0, we analyze a class of self-adjoint Schrödinger operators H_hbar = hbar ^2 L + hbar W + Vcdot {mathrm {id}}_{mathscr {E}} acting on sections of a vector bundle {mathscr {E}} over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m^1,ldots m^r in M, called potential wells. Using quasimodes of WKB-type near m^j for eigenfunctions associated with the low lying eigenvalues of H_hbar , we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension ell + 1. This dimension ell determines the polynomial prefactor for exponentially small eigenvalue splitting.

A Maximization Problem Involving a Fractional Laplace Type Operator

Fractional Laplacian is an important nonlocal operator which has many applications in different kinds of differential equations. Recently, optimization problems involving the fractional Laplacian have been studied a lot by many authors. However, most of these papers are focusing on the optimization problems related to the first eigenvalue of the equation. Optimization problems related to the energy functional of the equation have not been investigated well enough. In this paper, we are going to study a maximization problem related to the energy functional of an equation involving a fractional Laplace type operator. Firstly, by using suitable variational framework in a fractional Sobolev space, we can show that a fractional equation has a solution which is in fact the global minimum of the corresponding energy functional. Moreover, by using reduction to absurdity we can obtain the uniqueness of the solution of the fractional equation. Then, we focus on a maximization problem related to the equation which takes the energy functional as the objective functional. Finally, by carefully analysing the properties of an arbitrarily choosen minimizing sequence and the tools of the rearrangement theory, we can prove that the maximization problem is solvable.

Generalized Picone inequalities and their applications to (p,q)-Laplace equations

Abstract We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the ( p , q ) (p,q) -Laplace-type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation − Δ p u − Δ q u = f μ ( x , u , ∇ u ) -\hspace{-0.25em}{\text{Δ}}_{p}u-{\text{Δ}}_{q}u={f}_{\mu }(x,u,\nabla u) in a bounded domain Ω ⊂ ℝ N \text{Ω}\hspace{0.25em}\subset {{\mathbb{R}}}^{N} under certain assumptions on the nonlinearity and with a special attention to the resonance case f μ ( x , u , ∇ u ) = λ 1 ( p ) | u | p − 2 u + μ | u | q − 2 u {f}_{\mu }(x,u,\nabla u)={\lambda }_{1}(p)|u{|}^{p-2}u+\mu |u{|}^{q-2}u , where λ 1 ( p ) {\lambda }_{1}(p) is the first eigenvalue of the p-Laplacian.

Tensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints

We introduce the tensor numerical method for solution of the d-dimensional optimal control problems (d=2,3) with spectral fractional Laplacian type operators in constraints discretized on large n⊗d tensor-product Cartesian grids. The approach is based on the rank-structured approximation of the matrix valued functions of the corresponding fractional finite difference Laplacian. We solve the equation for the control function, where the system matrix includes the sum of the spectral fractional d-dimensional Laplacian and its inverse. The matrix valued functions of discrete Laplace operator on a tensor grid are diagonalized by using the fast Fourier transform (FFT). Then the low rank approximation of the d-dimensional tensors obtained by folding of the corresponding large diagonal matrices of eigenvalues are computed, which allows to solve the governing equation for the control function in a tensor-structured format. The existence of low rank canonical approximation to the class of matrix valued functions involved is justified by using the sinc quadrature approximation method applied to the Laplace transform of the generating function. The linear system of equations for the control function is solved by the PCG iterative method with the rank truncation at each iteration step, where the low Kronecker rank preconditioner is pre-computed. The right-hand side, the solution vector, and the governing system matrix are maintained in the rank-structured tensor format which beneficially reduces the numerical cost to O(nlog⁡n), outperforming the standard FFT based methods of complexity O(n3log⁡n) for 3D case. Numerical tests for the 2D and 3D control problems confirm the linear complexity scaling of the method in the univariate grid size n.

Existence of Solutions for Implicit Obstacle Problems of Fractional Laplacian Type Involving Set-Valued Operators

The paper is devoted to a new kind of implicit obstacle problem given by a fractional Laplacian-type operator and a set-valued term, which is described by a generalized gradient. An existence theorem for the considered implicit obstacle problem is established, using a surjectivity theorem for set-valued mappings, Kluge’s fixed point principle and nonsmooth analysis.

Generalized Evans–Krylov and Schauder type estimates for nonlocal fully nonlinear equations with rough kernels of variable orders

We establish the generalized Evans–Krylov and Schauder type estimates for nonlocal fully nonlinear elliptic equations with rough kernels of variable orders. In contrast to the fractional Laplacian type operators having a fixed order of differentiability σ∈(0,2), the operators under consideration have variable orders of differentiability. Since the order is not characterized by a single number, we consider a function φ describing the variable orders of differentiability, which is allowed to oscillate between two functions rσ1 and rσ2 for some 0<σ1≤σ2<2. By introducing the generalized Hölder spaces, we provide Cφψ estimates that generalizes the standard Evans–Krylov and Schauder type Cσ+α estimates.