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We study existence and multiplicity of nontrivial solutions of the following problem { − Δ p u + (− Δ p) s u = λ | u | q − 2 u + | u | p ⁎ − 2 u in Ω , u = 0 on R N ∖ Ω , where Ω ⊂ R N is a bounded open set with smooth boundary, dimension N ≥ 2 , parameter λ > 0 , exponents 0 < s < 1 < p < N , while q ∈ (1 , p ⁎) with p ⁎ = N p N − p. The problem is driven by an operator of mixed order obtained by the sum of the classical p -Laplacian and of the fractional p -Laplacian. We analyze three different scenarios depending on exponent q. For this, we combine variational methods with some topological techniques, such as the Krasnoselskii genus and the Lusternik–Schnirelman category theories. [ABSTRACT FROM AUTHOR]

We prove that the curl operator on closed oriented 3-manifolds, i.e., the square root of the Hodge Laplacian on its coexact spectrum, generically has 1-dimensional eigenspaces, even along 1-parameter families of \mathcal {C}^k Riemannian metrics, where k\geq 2. We show further that the Hodge Laplacian in dimension 3 has two possible sources for nonsimple eigenspaces along generic 1-parameter families of Riemannian metrics: either eigenvalues coming from positive and from negative eigenvalues of the curl operator cross, or an exact and a coexact eigenvalue cross. We provide examples for both of these phenomena. In order to prove our results, we generalize a method of Teytel [Comm. Pure Appl. Math. 52 (1999), pp. 917–934], allowing us to compute the meagre codimension of the set of Riemannian metrics for which the curl operator and the Hodge Laplacian have certain eigenvalue multiplicities. A consequence of our results is that while the simplicity of the spectrum of the Hodge Laplacian in dimension 3 is a meagre codimension 1 property with respect to the \mathcal {C}^k topology as proven by Enciso and Peralta-Salas in [Trans. Amer. Math. Soc. 364 (2012), pp. 4207–4224], it is not a meagre codimension 2 property. [ABSTRACT FROM AUTHOR]

The structure stability from the Gromov-Hausdorff viewpoint can be used to describe the continuity of complete trajectories inside global attractors for nonlinear evolutionary partial differential equations defined on perturbed domains, such as the unsteady incompressible fluid flow models. This paper is concerned with the Gromov-Hausdorff stability for the three dimensional Navier-Stokes-Voigt equations by combining the properties of the inverse operator associated with the Laplacian to deal with transformations of different domains to avoid higher regularity estimates. The key technique lies in the construction of new equations on perturbed domains and the normalized method to obtain the compactness. [ABSTRACT FROM AUTHOR]

In the present study, we establish the existence and uniqueness of solutions for nonlinear initial value problems and nonlocal boundary value problems associated with implicit fractional differential equations involving k,Ψ$$ \left(k,\Psi \right) $$‐Hilfer derivative operator. Furthermore, we explore the stability properties of these solutions in the sense of the Ulam–Hyers, Ulam–Hyers–Rassias, and their generalized stability concepts by proving and applying generalized Gronwall inequality. Lastly, we present the fractional RC$$ \mathcal{RC} $$ electric circuit model as an example to show the practical applicability of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Schrödinger–Poisson system with discontinuous nonlinearity: ε 2 s (− Δ) s u + V (x) u + ϕ u = H (u − β) f (u) , in R 3 , ε 2 s (− Δ) s ϕ = u 2 , in R 3 , u > 0 , in R 3 , where ɛ > 0 is a small parameter, s ∈ ( 3 4 , 1) , β > 0, H is the Heaviside function, (−Δ)su is the fractional Laplacian operator, V : R 3 → R is a continuous potential and f : R → R is superlinear continuous nonlinearity with subcritical growth at infinity. By using nonsmooth analysis, we investigate the existence and concentration of solutions for the above problem. Moreover, we obtain some properties of these solutions, such as convergence and decay estimate. [ABSTRACT FROM AUTHOR]

Effective exploration of the pattern dynamic behaviors of reaction-diffusion models is a popular but difficult topic. The Schnakenberg model is a famous reaction-diffusion system that has been widely used in many fields, such as physics, chemistry, and biology. Herein, we explore the stability, Turing instability, and weakly non-linear analysis of the Schnakenberg model; further, the pattern dynamics of the fractional-in-space Schnakenberg model was simulated numerically based on the Fourier spectral method. The patterns under different parameters, initial conditions, and perturbations are shown, including the target, bar, and dot patterns. It was found that the pattern not only splits and spreads from the bar to spot pattern but also forms a bar pattern from the broken connections of the dot pattern. The effects of the fractional Laplacian operator on the pattern are also shown. In most cases, the diffusion rate of the fractional model was higher than that of the integer model. By comparing with different methods in literature, it was found that the simulated patterns were consistent with the results obtained with other numerical methods in literature, indicating that the Fourier spectral method can be used to effectively explore the dynamic behaviors of the fractional Schnakenberg model. Some novel pattern dynamics behaviors of the fractional-in-space Schnakenberg model are also demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of solutions for a p–Laplacian boundary value problem defined by semilinear fractional system that involves Caputo–Katugampola fractional derivatives. Our main results rely on the implementation of the Banach and Schauder fixed point theorems. An example is introduced to expose the applicability of the theoretical findings. [ABSTRACT FROM AUTHOR]

We investigate the viscoelastic relaxation to equilibrium of an isolated disordered planar epithelium described using the cell vertex model. In its standard form, the model is formulated as coupled evolution equations for the locations of vertices of confluent polygonal cells. Exploiting the model's gradient-flow structure, we use singular-value decomposition to project modes of deformation of vertices onto modes of deformation of cells. Expressing the dynamics in terms of operators of a discrete calculus, we show how eigenmodes of discrete Laplacian operators (specified by constitutive assumptions related to dissipation and mechanical energy) provide a spatial basis for evolving fields, and demonstrate how the operators can incorporate approximations of conventional spatial derivatives. We relate the spectrum of relaxation times to the eigenvalues of the Laplacians, modified by corrections that account for the fact that the cell network (and therefore the Laplacians) evolve during relaxation to an equilibrium prestressed state, providing the monolayer with geometric stiffness. While dilational modes of the Laplacians capture rapid relaxation in some circumstances, showing diffusive dynamics, geometric stiffness is typically a dominant source of monolayer rigidity, as we illustrate for monolayers exposed to unsteady stretching deformations. [ABSTRACT FROM AUTHOR]

We consider a cooperative system of equations involving the fractional Laplacian operator with zero Dirichlet external condition on a ball. We prove that nonnegative solutions of such problems, with semipositone nonlinearities, are positive and hence radially symmetric. We use the method of moving planes to establish our result. [ABSTRACT FROM AUTHOR]

We construct the global existence for a wave equation involving the fractional Laplacian with a logarithmic nonlinear source by using the Galerkin approximations. The corresponding results for equations with classical Laplacian are considered as particular cases of our assertions. [ABSTRACT FROM AUTHOR]

We show maximal regularity estimates for the damped hyperbolic and strongly damped wave equations with periodic initial conditions in a cylindrical domain. We prove that this property strongly depends on a critical combination on the parameters of the equation. Noteworthy, our results are still valid for fractional powers of the negative Laplacian operator. We base our methods on the theory of operator-valued Fourier multipliers on vector-valued Lebesgue spaces of periodic functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of at least one bounded weak solution for Kohn–Spencer Laplacian with a weight depending on the solution and convection term of the form \[ -div_{\mathbb{H}^n}(\nu(\xi,u) |D_{\mathbb{H}^n}u|^{p-2}_{\mathbb{H}^n}D_{\mathbb{H}^n}u)=f(\xi,u,D_{\mathbb{H}^n} u) \] − di v H n (ν (ξ , u) | D H n u | H n p − 2 D H n u) = f (ξ , u , D H n u) in a bounded domain $ \Omega \subset \mathbb {H}^n $ Ω ⊂ H n . We show the set of solutions is uniformly bounded by a special Moser's iteration. [ABSTRACT FROM AUTHOR]

We study a superlinear elliptic boundary value problem involving the p$p$‐Laplacian operator, with changing sign weights. The problem has positive solutions bifurcating from the trivial solution set at the two principal eigenvalues of the corresponding linear weighted boundary value problem.Drabek's bifurcation result applies when the nonlinearity is of power growth. We extend Drabek's bifurcation result to slightly subcritical nonlinearities. Compactness in this setting is a delicate issue obtained via Orlicz spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of ground state solutions for the following (p , q) -Choquard equation: − Δ p u − Δ q u + | u | p − 2 u + | u | q − 2 u = (I α ⁎ F (u)) f (u) in R N , where 2 ≤ p < q < N , Δ s is the s -Laplacian operator, with s ∈ { p , q } , I α is the Riesz potential of order α ∈ ((N − 2 q) + , N) , F ∈ C 1 (R , R) is a general nonlinearity of Berestycki-Lions type and F ′ = f. Furthermore, we analyze the regularity, symmetry and decay properties of these solutions. In particular, we extend the results in [33] to the (p , q) -Laplacian setting. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and regularity results of normalized solutions to the following critical growth Choquard equation with mixed diffusion type operators: − Δ u + ( − Δ ) s u = λ u + g ( u ) + ( I α ∗ | u | 2 α ∗ ) | u | 2 α ∗ − 2 u in  R N , ∫ R N | u | 2 d x = τ 2 , where N ⩾ 3, τ > 0, I α is the Riesz potential of order α ∈ ( 0 , N ), ( − Δ ) s is the fractional laplacian operator, 2 α ∗ = N + α N − 2 is the critical exponent with respect to the Hardy Littlewood Sobolev inequality, λ appears as a Lagrange multiplier and g is a real valued function satisfying some L 2 -supercritical conditions. [ABSTRACT FROM AUTHOR]

Large field-of-view images are increasingly used in various environments today, and image stitching technology can make up for the limited field of view caused by hardware design. However, previous methods are constrained in various environments. In this paper, we propose a method that combines the powerful feature extraction capabilities of the Superpoint algorithm and the exact feature matching capabilities of the Lightglue algorithm with the image fusion algorithm of Unsupervised Deep Image Stitching (UDIS). Our proposed method effectively improves the situation where the linear structure is distorted and the resolution is low in the stitching results of the UDIS algorithm. On this basis, we make up for the shortcomings of the UDIS fusion algorithm. For stitching fractures of UDIS in some complex situations, we optimize the loss function of UDIS. We use a second-order differential Laplacian operator to replace the difference in the horizontal and vertical directions to emphasize the continuity of the structural edges during training. Combined with the above improvements, the Super Unsupervised Deep Image Stitching (SuperUDIS) algorithm is finally formed. SuperUDIS has better performance in both qualitative and quantitative evaluations compared to the UDIS algorithm, with the PSNR index increasing by 0.5 on average and the SSIM index increasing by 0.02 on average. Moreover, the proposed method is more robust in complex environments with large color differences or multi-linear structures. [ABSTRACT FROM AUTHOR]

The isotropy is a fundamental requirement for solving partial differential equations by the finite difference scheme. In this work, we first introduce the concept of the virtual nodes to construct the finite difference scheme, and propose several stencils of finite difference discretization to optimize the isotropy of the gradient and Laplacian operators. The isotropic error of the optimized gradient stencil is reduced to 4.1% and 7.5% of the conventional scheme and the typical 4th-order isotropic stencil that is widely used in the multiphase lattice Boltzmann method, while the optimized Laplacian stencil displays the best performance in the Fourier analysis. Then, the optimized stencils are applied to suppress the spurious currents in the multiphase lattice Boltzmann method. The spurious current is reduced to only 7.2% of that calculated by the typical 4th-order isotropic stencil at the reduced temperature 0.6, at which the liquid/gas density ratio is near to 1000. Furthermore, they are adopted to rectify the dendrite directions in the alloy solidification simulations by the phase field method. The angle deviation of dendrite growth is reduced to 15.7% of the conventional scheme. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solution to boundary value problem of fractional differential equations with p$$ p $$‐Laplacian operator. By using Avery–Peterson theorem, some new existence results of three increasing positive solutions are obtained. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to characterize stable sets based on the asymptotic behavior of solutions as t$t$ goes to infinity for the following class of parabolic Kirchhoff equations: ut+∥u∥(θ−1)Ns(−Δ)N/ssu=λ|u|q−2uexpα0|u|NN−s|x|γinΩ,t>0,u=0inRN∖Ω,t>0,u(x,0)=u0(x)inΩ,$$\begin{eqnarray*} \hspace*{13pc}{\left\lbrace \def\eqcellsep{&}\begin{array}{llc}u_{t}+\Vert u\Vert ^{\frac{(\theta -1)N}{s}}(-\Delta)^{s}_{N/s}u=\frac{\lambda |u|^{q-2}u\exp {\left(\alpha _{0}|u|^{\frac{N}{N-s}}\right)}}{|x|^{\gamma }} &\text{in}\ &\Omega,\;t>0, \\ u =0 &\text{in} & \mathbb {R}^{N}\backslash \Omega,\;t > 0, \\ u(x,0)=u_{0}(x)& \text{in} &\Omega, \end{array} \right.} \end{eqnarray*}$$where ∥u∥Ns=∫R2N|u(x,t)−u(y,t)|N/s|x−y|2Ndxdy,$$\begin{equation*} \hspace*{7pc}\Vert u\Vert ^{\frac{N}{s}}=\int _{\mathbb {R}^{2N}}\frac{|u(x,t)-u(y,t)|^{N/s}}{|x-y|^{2N}}\,dxdy, \end{equation*}$$Ω⊂RN(N≥2)$\Omega \subset \mathbb {R}^N \, (N\ge 2)$ is a bounded domain with a Lipschitz boundary, 0∈Ω$0\in \Omega$, α0,λ>0$\alpha _{0},\lambda >0$, θ≥1$\theta \ge 1$, γ∈[0,N)$\gamma \in [0, N)$, q>Nθ/s$q>N\theta /s$, and (−Δ)N/ss$(-\Delta)_{N/s}^{s}$ is the fractional N/s$N/s$‐Laplacian operator, s∈(0,1)$s\in (0,1)$. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence of a positive solution for a class of nonvariational fractional elliptic system with concave and convex nonlinearities in two cases. The paper is divided in two parts: In the first one, for general nonlinearity with subcritical or critical growth, we use Galerkin's method and an approximation argument to show the existence of a solution for the system considered. In the second part, for special cases (which include the power case), we remove the restriction on the growth of the nonlinearity and use sub-supersolution, monotone iteration and a comparison argument to obtain a solution for the system considered. [ABSTRACT FROM AUTHOR]

In recent years, the heat kernel has proven to be a valuable tool for graph characterization and graph-based object clustering. It serves as the fundamental solution to the heat diffusion equation associated with the discrete graph Laplacian, describing how information flows across graph edges over time by exponentiating the Laplacian eigensystem. This paper focuses on the novel concept of the generalized heat kernel of a network, recently introduced by Kalala Mutombo et al. These authors build upon the k-path Laplacian operator for graphs, pioneered by Estrada et al., to incorporate long-range interactions (LRI) into information transmission across nodes and edges of a network/graph. LRI are handled through the use of Mellin and Laplace transforms. This paper makes a notable contribution by highlighting the practical application of the generalized heat kernel. We achieve this by demonstrating the effectiveness of the generalised heat kernel invariants in object clustering with real data using principal component analysis (PCA). Moreover, we investigate how incorporating long-range interactions (LRI) impacts object characterization and clustering, revealing superior results compared to conventional diffusion methods. Through experimentation, we show that object clustering remains achievable even with small values of the Mellin and Laplace parameters, contrasting with the requirement of an infinite value in the absence of LRI. • Application of graph generalised heat kernel invariants in object clustering via PCA. • Long-range interactions enhance graph-based object clustering. • Object clustering achieved with small Mellin/Laplace parameters. [ABSTRACT FROM AUTHOR]

We describe the geometric and probabilistic properties of a noncommutative -torus in a magnetic field. We study the volume invariance, integrated scalar curvature, and the volume form by using the operator method of perturbation by an inner derivation of the magnetic Laplacian operator on the noncommutative -torus. We then analyze the magnetic stochastic process describing the motion of a particle subject to a uniform magnetic field on the noncommutative -torus, and discuss the related main properties. [ABSTRACT FROM AUTHOR]

Diffusion is a mass transport phenomenon caused by chaotic thermal movements of molecules. Studying the transport in specific domain is simplified by using evolutionary differential equations for local concentration of the molecules instead of complete information on molecular paths [1]. Compounds in a fluid mixture tend to smooth out its spatial concentration inhomogeneities by diffusion. Rate of the transport is proportional to the concentration gradient and coefficient of diffusion of the compound in ordinary diffusion. The evolving concentration profile c(x,t) is then solution of evolutionary partial differential equation ∂c/∂t = DΔC where D is diffusion coefficient and Δ is Laplacian operator. Domain of the equation may be a region in space, plane or line, a manifold, such as surface embedded in space, or a graph. The Laplacian operates on smooth functions defined on given domain. We can use models of diffusion for such diverse tasks as: a) design of method for precise measurement of receptors mobility in plasmatic membrane by confocal microscopy [2], b) evaluation of complex geometry of trabeculae in developing heart [3] to show that the conduction pathway within the embryonic ventricle is determined by geometry of the trabeculae. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the tempered fractional parabolic problem $$ \begin{align*}\frac{\partial u}{\partial t}(x, t)-\left(\Delta+\lambda\right)^{\frac{\alpha}{2}} u(x, t)=f(u(x, t)), \end{align*} $$ where $-\left (\Delta +\lambda \right)^{\frac {\alpha }{2}}$ is a tempered fractional operator with $\alpha \in (0,2)$ and $\lambda $ is a sufficiently small positive constant. We first establish maximum principle principles for problems involving tempered fractional parabolic operators. And then, we develop the direct sliding methods for the tempered fractional parabolic problem, and discuss how they can be used to establish monotonicity results of solutions to the tempered fractional parabolic problem in various domains. We believe that our theory and methods can be conveniently applied to study parabolic problems involving other nonlocal operators. [ABSTRACT FROM AUTHOR]

In the present paper, by the variational method coupled with mountain pass type theorems, we study nontrivial homoclinic solutions of nonlinear $ (p, q) $-Laplacian partial difference equations with a parameter $ \lambda>0 $:$ \begin{equation*} \begin{split} &\Delta^2_1(\phi_{p_2}(\Delta^2_1u(k-2, l)))+\Delta^2_2(\phi_{p_2}(\Delta^2_2u(k, l-2)))-a[\Delta_1(\phi_{p_1}(\Delta_1u(k-1, l)))\\ &+\Delta_2(\phi_{p_1}(\Delta_2u(k, l-1)))]+V(k, l)\phi_q(u(k, l))\\& = \lambda f((k, l), u(k, l)). \end{split} \end{equation*} $We prove that the above equation admits two nontrivial homoclinic solutions if $ \lambda $ is sufficiently large and one nontrivial homoclinic solutions if $ \lambda>0 $. Our obtained results generalize and improve some existing ones. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of singular variable-order fractional Kirchhoff problem of the form: where is a bounded domain, is the variable-order fractional Laplacian operator, [u]s(·) is the Gagliardo seminorm and is a continuous and symmetric function. We assume that λ is a non-negative parameter, with and. We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem. [ABSTRACT FROM AUTHOR]

We introduce a new inequality similar to the fractional Poincar$ \acute{e} $ inequality and obtain the continuous data assimilation and feedback control of fractional reaction-diffusion equations. The feedback control scheme has finite number of determining parameters. The continuous data assimilation is obtained based on finite-dimensional feedback controls. [ABSTRACT FROM AUTHOR]

This article examines a new family of (p,q)‐Laplacian type nonlinear fractional impulsive differential coupled equations involving both the ψ$$ \psi $$‐Caputo fractional derivative and ψ$$ \psi $$–Riemann–Liouville fractional integral. With the help of Nehari manifold in critical point theory and fractional calculus properties, we obtain the existence of at least one nontrivial ground state solution for the coupled system with some natural and easily verifiable superlinear conditions on the nonlinearity. [ABSTRACT FROM AUTHOR]

In this article, we considered a nonlinear compartmental mathematical model that assesses the effect of treatment on the dynamics of HIV/AIDS and pneumonia (H/A-P) co-infection in a human population at different infection stages. Understanding the complexities of co-dynamics is now critically necessary as a consequence. The aim of this research is to construct a co-infection model of H/A-P in the context of fractional calculus operators, white noise and probability density functions, employing a rigorous biological investigation. By exhibiting that the system possesses non-negative and bounded global outcomes, it is shown that the approach is both mathematically and biologically practicable. The required conditions are derived, guaranteeing the eradication of the infection. Furthermore, adequate prerequisites are established, and the configuration is tested for the existence of an ergodic stationary distribution. For discovering the system's long-term behavior, a deterministic-probabilistic technique for modeling is designed and operated in MATLAB. By employing an extensive review, we hope that the previously mentioned approach improves and leads to mitigating the two diseases and their co-infections by examining a variety of behavioral trends, such as transitions to unpredictable procedures. In addition, the piecewise differential strategies are being outlined as having promising potential for scholars in a range of contexts because they empower them to include particular characteristics across multiple time frame phases. Such formulas can be strengthened via classical techniques, power law, exponential decay, generalized Mittag-Leffler kernels, probability density functions and random procedures. Furthermore, we get an accurate description of the probability density function encircling a quasi-equilibrium point if the effect of H/A-P minimizes the propagation of the co-dynamics. Consequently, scholars can obtain better outcomes when analyzing facts using random perturbations by implementing these strategies for challenging issues. Random perturbations in H/A-P co-infection are crucial in controlling the spread of an epidemic whenever the suggested circulation is steady and the amount of infection eliminated is closely correlated with the random perturbation level. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence result of weak solutions to a class fractional p(x,y)$$ p\left(x,y\right) $$‐Laplacian problems involving the nonlocal integro‐differential operator of elliptique type LKp(x)$$ {\mathcal{L}}_K&#x0005E;{p(x)} $$. The main tool used here is based on the topological degree theory combined with the theory fractional Sobolev spaces with variable exponent. Our results generalize and improve the existing results. [ABSTRACT FROM AUTHOR]

In this work, we study the asymptotic behavior of the structurally damped wave equations arising from the viscoelastic mechanics. We are particularly interested in the complicated interaction of the time‐dependent oscillating coefficients on the Dirichlet Laplacian operator and the structurally damped terms. On the one hand, by the application of WKB analysis, we explore the asymptotic energy estimates of the wave equations influenced by four types of oscillating mechanisms. On the other hand, in order to prove the optimality of the energy estimates for the critical cases, typical coefficients and initial Cauchy data will be constructed to show the lower bound of the energy growth rate by the application of instability arguments. [ABSTRACT FROM AUTHOR]

In this work, we use the combination of radial basis functions (RBFs) and polynomials to derive compact finite difference (FD) formulas. We use RBFs and integrated RBFs (IRBFs), to obtain two sets of formulas: RBF-Hermite FD (RBF-HFD) and IRBF weight formulas. The analytical coefficients and truncation errors for these formulations are computed for the first derivative, second derivative, and 2D Laplacian operator. Our presented approach expands the usability of the weights by introducing a general framework for deriving weighting formulas. Finally, the computational results underscore the effectiveness of the introduced approach. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate the well-posedness and regularity of mild solution to a stochastic active fluids model driven by the additive noise. A fully-discrete scheme is proposed for solving the given model, which is based on the finite element method for spatial discretization and the backward Euler method for temporal discretization. By overcoming the difficulty of error analysis caused by the discrete Laplacian operator, we obtain the convergence results of the developed scheme. Finally, some numerical examples are provided to validate the theoretical results and we also simulate the motion states of the active fluids. [ABSTRACT FROM AUTHOR]

A one-dimensional space-time fractional reaction-diffusion problem is considered. We present a complete theory for the solution of the time-space fractional reaction-diffusion model, including existence and uniqueness in the case of using the spectral representation of the fractional Laplacian operator. An optimal error estimate is presented for the Galerkin spectral approximation of the problem under consideration. [ABSTRACT FROM AUTHOR]

A bosonic Laplacian is a conformally invariant second‐order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher‐order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita–Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson's equation in Euclidean space is also provided. In the end, we provide Green's formulas for bosonic Laplacians in scalar‐valued and Clifford‐valued cases, respectively. These formulas reveal that bosonic Laplacians are self‐adjoint with respect to a given L2 inner product on certain compact supported function spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear equation involving the nonlocal pseudo-relativistic operators \[ (-\Delta +m^2)^s u(x) = f(x,u(x)) , \] (− Δ + m 2) s u (x) = f (x , u (x)) , where 00. The nonlocal pseudo-relativistic operator includes the pseudo-relativistic Schrödinger operator $ \sqrt {-\Delta +m^2} $ − Δ + m 2 . When $ m\rightarrow 0^+ $ m → 0 + , the nonlocal pseudo-relativistic operator $ (-\Delta +m^2)^s $ (− Δ + m 2) s is also closely related to the fractional Laplacian operator $ (-\Delta)^s $ (− Δ) s . But these two operators are quite different. We first establish a Hopf type lemma for anti-symmetric functions to nonlocal pseudo-relativistic operators, which play a key role in the method of moving planes. The main difficulty is to construct a suitable sub-solution to nonlocal pseudo-relativistic operators. Then we prove a pointwise estimate to nonlocal pseudo-relativistic operators. As an application, combined with the Hopf type lemma and the pointwise estimate, we obtain the radial symmetry and monotonicity of positive solutions to the above nonlinear nonlocal pseudo-relativistic equation in the whole space. We believe that the Hopf type lemma will become a powerful tool in applying the method of moving planes on nonlocal pseudo-relativistic equations to obtain qualitative properties of solutions. [ABSTRACT FROM AUTHOR]

This manuscript deals with a space-fractional generalized wave problem involving the fractional Laplacian operator of order α for 1 < α ≤ 2 . We propose an accurate numerical method to solve the mentioned fractional wave problem. The problem is discretized in spatial direction by the Fourier spectral method, and in temporal direction by using the fourth-order exponential time-differencing Runge–Kutta method. One of the main features of this method is reducing the mentioned fractional wave model to an ODE by using the Fourier transform. Then the fourth-order exponential time-differencing Runge–Kutta method is used to solve this ODE. We define the discrete energy function and check the energy-conserving properties. The convergence of this method is proved. Various numerical experiments are conducted to confirm the accuracy and dependability of the suggested approach. [ABSTRACT FROM AUTHOR]

A theorem of Llarull says that if a smooth metric g$g$ on the n$n$‐sphere Sn$\mathbb {S}^n$ is bounded below by the standard round metric and the scalar curvature Rg$R_g$ of g$g$ is bounded below by n(n−1)$n (n - 1)$, then the metric g$g$ must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound Rg⩾n(n−1)$R_g \geqslant n (n - 1)$ by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature Rg$R_g$. We utilize two methods: spinor and spacetime harmonic function. [ABSTRACT FROM AUTHOR]

Based on the variational methods and critical-point theory, we are concerned with the existence results for a second-order impulsive boundary value problem involving an ordinary differential equation with p(x)-Laplacian operator, and Neumann conditions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the semiclassical 2D magnetic Schrödinger operator in the case where the magnetic field vanishes along a smooth closed curve. Assuming that this curve has an axis of symmetry, we prove that semiclassical tunneling occurs. The main result is an expression of the splitting of the first two eigenvalues and an explicit tunneling formula. [ABSTRACT FROM AUTHOR]

We consider an elliptic system of Hamiltonian type in a strip in R N , satisfying the periodic boundary condition for the first k variables. In the superlinear case with critical growth, we prove the existence of a single bubbling solution for the system under an optimal condition on k. The novelty of the paper is that all the estimates needed in the proof of the existence result can be obtained once the Green's function of the Laplacian operator in a strip with periodic boundary conditions is found. [ABSTRACT FROM AUTHOR]

We consider the following fractional problem involving slightly subcritical non‐power nonlinearity, (−Δ)su=|u|2s∗−2u[ln(e+|u|)]εinΩ,[2mm]u=0on∂Ω,$$\begin{equation*} {\hspace*{60pt}\left\lbrace \def\eqcellsep{&}\begin{array}{lll}(-\Delta)^s u =\frac{|u|^{2_s^*-2}u}{[\ln (e+|u|)]^\epsilon }\ \ &{\rm in}\ \Omega, [2mm] u= 0 \ \ & {\rm on}\ \partial \Omega, \end{array} \right.} \end{equation*}$$where Ω$\Omega$ is a smooth bounded domain in Rn$\mathbb {R}^n$, n≥2s+1$n\ge 2s+1$, s∈(12,1)$s\in (\frac{1}{2},1)$, 2s∗=2nn−2s$2_s^*=\frac{2n}{n-2s}$ is the fractional critical Sobolev exponent and ε>0$\epsilon >0$ is a small parameter, (−Δ)s$(-\Delta)^s$ is the spectral fractional Laplacian operator. We construct a positive bubbling solution, which concentrates at a nondegenerate critical point of the Robin function by Lyapunov–Schmidt reduction procedure. [ABSTRACT FROM AUTHOR]

In this paper, first we prove a nonexistence theorem for α‐harmonic mappings between Riemannian manifolds. Second, the instability of nonconstant α‐harmonic maps is studied with regard to the Ricci curvature criterion of their codomain. Then, we estimate the Morse index for measuring the degree of instability of some particular α‐harmonic maps. Furthermore, the notion of α‐stable manifolds and its applications are considered. Finally, we investigate the α‐stability of any compact Riemannian manifolds admitting a nonisometric conformal vector field and any Einstein Riemannian manifold under certain assumptions on the smallest positive eigenvalue of its Laplacian operator on functions. [ABSTRACT FROM AUTHOR]

The visual measurement of deep holes in composite material workpieces constitutes a critical step in the robotic assembly of aerospace components. The positioning accuracy of assembly holes significantly impacts the assembly quality of components. However, the complex texture of the composite material surface and mutual interference between the imaging of the inlet and outlet edges of deep holes significantly challenge hole detection. A visual measurement method for deep holes in composite materials based on the radial penalty Laplacian operator is proposed to address the issues by suppressing visual noise and enhancing the features of hole edges. Coupled with a novel inflection-point-removal algorithm, this approach enables the accurate detection of holes with a diameter of 10 mm and a depth of 50 mm in composite material components, achieving a measurement precision of 0.03 mm. [ABSTRACT FROM AUTHOR]

In [4] , Guan, Murugan and Wei established the equivalence of the classical Helmholtz equation with a "fractional Helmholtz" equation in which the Laplacian operator is replaced by the nonlocal fractional Laplacian operator. More general equivalence results are obtained for symbols which are complete Bernstein and satisfy additional regularity conditions. In this work we introduce a novel and general set-up for this Helmholtz equivalence problem. We show that under very mild and easy-to-check conditions on the Fourier multiplier, the general Helmholtz equation can be effectively reduced to a localization statement on the support of the symbol. [ABSTRACT FROM AUTHOR]

In this work we study the Riemann-Liouville fractional integral of order α ∈ (0 , 1 / p) as an operator from L p (I ; X) into L q (I ; X) , with 1 ≤ q ≤ p / (1 - p α) , whether I = [ t 0 , t 1 ] or I = [ t 0 , ∞) and X is a Banach space. Our main result provides necessary and sufficient conditions to ensure the compactness of the Riemann-Liouville fractional integral from L p (t 0 , t 1 ; X) into L q (t 0 , t 1 ; X) , when 1 ≤ q < p / (1 - p α) . [ABSTRACT FROM AUTHOR]

Image reconstruction problems, like image deblurring and computer tomography, are usually ill-posed and require regularization. A popular approach to regularization is to substitute the original problem with an optimization problem that minimizes the sum of two terms, an ℓ 2 term and an ℓ q term with 0 < q ≤ 1. The first penalizes the distance between the measured data and the reconstructed one, the latter imposes sparsity on some features of the computed solution. In this work, we propose to use the fractional Laplacian of a properly constructed graph in the ℓ q term to compute extremely accurate reconstructions of the desired images. A simple model with a fully automatic method, i.e., that does not require the tuning of any parameter, is used to construct the graph and enhanced diffusion on the graph is achieved with the use of a fractional exponent in the Laplacian operator. Since the fractional Laplacian is a global operator, i.e., its matrix representation is completely full, it cannot be formed and stored. We propose to replace it with an approximation in an appropriate Krylov subspace. We show that the algorithm is a regularization method under some reasonable assumptions. Some selected numerical examples in image deblurring and computer tomography show the performance of our proposal. [ABSTRACT FROM AUTHOR]

This study aims to investigate a novel nonlinear optimization problem that incorporate a partial differential equation (PDE) constraint for image denoising in the context of mixed noise removal. Based on H−s$$ {H}&amp;#x0005E;{-s} $$‐norm and decomposition approach, we develop a nonlinear system that involves the fractional Laplacian operator. Based on Schauder's fixed point theorem, we establish the existence and uniqueness of weak solution for the direct problem. Furthermore, we study the well‐posedness of the optimal control, and we also prove the existence of the weak solutions for the adjoint problem by using Galerkin's method. In order to numerically compute the solution of the proposed model, we introduce the numerical discretization scheme and the primal–dual algorithm used to solve our problem. Finally, we provide comparative numerical experiments to evaluate the efficiency and effectiveness of our proposed model. [ABSTRACT FROM AUTHOR]

The Hadamard fractional derivative and integral are important parts of fractional calculus which have been widely used in engineering, biology, neural networks, control theory, and so on. In addition, the periodic boundary conditions are an important class of symmetric two-point boundary conditions for differential equations and have wide applications. Therefore, this article considers a class of nonlinear Hadamard fractional coupling (p 1 , p 2) -Laplacian systems with periodic boundary value conditions. Based on nonlinear analysis methods and the contraction mapping principle, we obtain some new and easily verifiable sufficient criteria for the existence and uniqueness of solutions to this system. Moreover, we further discuss the generalized Ulam–Hyers (GUH) stability of this problem by using some inequality techniques. Finally, three examples and simulations explain the correctness and availability of our main results. [ABSTRACT FROM AUTHOR]

Warped products provide an elegant and versatile framework for exploring and understanding a wide range of geometric structures. Their ability to combine two distinct manifolds through a warping function introduces a rich and diverse set of geometries, thus making them a powerful tool in various mathematical, physical, and computational applications. This article addresses the central query related to warped product submanifolds in the context of statistics. It focuses on deriving two new and distinct inequalities for a statistical warped product submanifold in a statistical manifold of a constant (quasi-constant) curvature. This article then finishes with some concluding remarks. [ABSTRACT FROM AUTHOR]