4,259 results on '"LAPLACIAN operator"'
Search Results
2. Solutions for nonhomogeneous Kohn–Spencer Laplacian on Heisenberg group.
- Author
-
Razani, Abdolrahman
- Subjects
- *
LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
3. Bifurcation for indefinite‐weighted p$p$‐Laplacian problems with slightly subcritical nonlinearity.
- Author
-
Cuesta, Mabel and Pardo, Rosa
- Subjects
- *
ORLICZ spaces , *BOUNDARY value problems , *EIGENVALUES , *LAPLACIAN operator - Abstract
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]- Published
- 2024
- Full Text
- View/download PDF
4. Ground state solutions for a (p,q)-Choquard equation with a general nonlinearity.
- Author
-
Ambrosio, Vincenzo and Isernia, Teresa
- Subjects
- *
EQUATIONS , *LAPLACIAN operator , *SYMMETRY - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
5. Normalized solutions to a critical growth Choquard equation involving mixed operators.
- Author
-
Giacomoni, J., Nidhi, Nidhi, and Sreenadh, K.
- Subjects
- *
LAPLACIAN operator , *HEAT equation , *EQUATIONS , *LAGRANGE multiplier - Abstract
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 andg is a real valued function satisfying some L 2 -supercritical conditions. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
6. Improved Unsupervised Stitching Algorithm for Multiple Environments SuperUDIS.
- Author
-
Wu, Haoze, Bao, Chun, Hao, Qun, Cao, Jie, and Zhang, Li
- Subjects
- *
LAPLACIAN operator , *DIFFERENTIAL operators , *IMAGE fusion , *FEATURE extraction , *DEEP learning - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
7. Isotropic optimizations of finite difference discretization.
- Author
-
Huang, Yuhao, Liu, Qilin, Chai, Zhenhua, and Wen, Binghai
- Subjects
- *
FINITE differences , *LATTICE Boltzmann methods , *PARTIAL differential equations , *LAPLACIAN operator , *DIFFERENCE equations - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
8. Triple increasing positive solutions to fractional differential equations with p$$ p $$‐Laplacian operator.
- Author
-
Cai, Shan and Li, Xiaoping
- Subjects
- *
FRACTIONAL differential equations , *BOUNDARY value problems , *POSITIVE operators , *OPERATOR equations , *LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
9. Asymptotics for a parabolic problem of Kirchhoff type with singular critical exponential nonlinearity.
- Author
-
Boudjeriou, Tahir
- Subjects
- *
EQUATIONS , *PARABOLIC operators , *LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
10. Solution for nonvariational fractional elliptic system with concave and convex nonlinearities.
- Author
-
Santos, Gelson C. G. dos, Medeiros, Aldo H. S., and Figueiredo Sousa, Tarcyana S.
- Subjects
- *
GALERKIN methods , *LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
11. Generalised heat kernel invariants of a graph and application to object clustering.
- Author
-
Kalala Mutombo, Franck and Nanyanzi, Alice
- Subjects
- *
MELLIN transform , *LAPLACIAN operator , *PRINCIPAL components analysis , *FLOWGRAPHS , *HEAT equation , *LAPLACIAN matrices - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
12. Geometry and probability on the noncommutative 2-torus in a magnetic field.
- Author
-
Hounkonnou, M. N. and Melong, F.
- Subjects
- *
LAPLACIAN operator , *STOCHASTIC processes , *PARTICLE motion , *MAGNETIC fields , *CURVATURE , *NONCOMMUTATIVE algebras - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
13. Mathematical Models of Diffusion in Physiology.
- Author
-
JANÁČEK, Jiří
- Subjects
DIFFUSION coefficients ,PARTIAL differential equations ,LAPLACIAN operator ,MATHEMATICAL models ,CELL membranes ,CONFOCAL microscopy - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
14. Continuous data assimilation and feedback control of fractional reaction-diffusion equations.
- Author
-
Lv, Guangying and Shan, Yeqing
- Subjects
REACTION-diffusion equations ,LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
15. On homoclinic solutions of nonlinear Laplacian partial difference equations with a parameter.
- Author
-
Long, Yuhua
- Subjects
DIFFERENCE equations ,MOUNTAIN pass theorem ,LAPLACIAN operator ,SEMILINEAR elliptic equations - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
16. Multiplicity of solutions for variable-order fractional Kirchhoff problem with singular term.
- Author
-
Chammem, R., Sahbani, A., and Saidani, A.
- Subjects
IMPLICIT functions ,LAPLACIAN operator ,SYMMETRIC functions ,CONTINUOUS functions ,MULTIPLICITY (Mathematics) - Abstract
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]- Published
- 2024
- Full Text
- View/download PDF
17. Ground state solutions for the fractional impulsive differential system with ψ‐Caputo fractional derivative and ψ–Riemann–Liouville fractional integral.
- Author
-
Li, Dongping, Li, Yankai, Feng, Xiaozhou, Li, Changtong, Wang, Yuzhen, and Gao, Jie
- Subjects
- *
FRACTIONAL calculus , *IMPULSIVE differential equations , *FRACTIONAL integrals , *CRITICAL point theory , *LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
18. Robustness and exploration between the interplay of the nonlinear co-dynamics HIV/AIDS and pneumonia model via fractional differential operators and a probabilistic approach.
- Author
-
Rashid, Saima, Hamidi, Sher Zaman, Raza, Muhammad Aon, Shafique, Rafia, Alsubaie, Assayel Sultan, and Elagan, Sayed K.
- Subjects
- *
DIFFERENTIAL operators , *PROBABILITY density function , *AIDS , *HIV , *PNEUMONIA , *LAPLACIAN operator , *LOTKA-Volterra equations - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
19. Existence of weak solutions of the fractional p(x,y)$$ p\left(x,y\right) $$‐Laplacian problem by topological degree.
- Author
-
Belhadi, Tahar, Lekhal, Hakim, and Slimani, Kamel
- Subjects
- *
TOPOLOGICAL degree , *SOBOLEV spaces , *LAPLACIAN operator , *INTEGRO-differential equations - Abstract
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)$$ {\mathcal{L}}_K^{p(x)} $$. 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]
- Published
- 2024
- Full Text
- View/download PDF
20. Energy behavior for Sobolev solutions to viscoelastic damped wave models with time‐dependent oscillating coefficient.
- Author
-
Lu, Xiaojun
- Subjects
- *
WAVE equation , *LAPLACIAN operator , *THRESHOLD energy , *WAVE energy - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
21. Computing compact finite difference formulas under radial basis functions with enhanced applicability.
- Author
-
Song, Yanlai, Barfeie, Mahdiar, and Soleymani, Fazlollah
- Subjects
- *
FINITE differences , *RADIAL basis functions , *LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
22. Numerical analysis of finite element method for a stochastic active fluids model.
- Author
-
Li, Haozheng, Wang, Bo, and Zou, Guang-an
- Subjects
- *
FINITE element method , *NUMERICAL analysis , *LAPLACIAN operator , *FLUIDS , *DISCRETIZATION methods , *EULER method - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
23. Optimal spectral Galerkin approximation for time and space fractional reaction-diffusion equations.
- Author
-
Hendy, A.S., Qiao, L., Aldraiweesh, A., and Zaky, M.A.
- Subjects
- *
RIESZ spaces , *REACTION-diffusion equations , *LAPLACIAN operator , *SPACETIME - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
24. Green's formulas and Poisson's equation for bosonic Laplacians.
- Author
-
Ding, Chao and Ryan, John
- Subjects
- *
POISSON'S equation , *POISSON integral formula , *DIFFERENTIAL operators , *EUCLIDEAN domains , *FUNCTION spaces , *LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
25. A Hopf type lemma for nonlocal pseudo-relativistic equations and its applications.
- Author
-
Wang, Pengyan
- Subjects
- *
SCHRODINGER operator , *NONLINEAR equations , *EQUATIONS , *LAPLACIAN operator , *SEMILINEAR elliptic equations - Abstract
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 0
0. 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]- Published
- 2024
- Full Text
- View/download PDF
26. Fourier spectral exponential time-differencing method for space-fractional generalized wave equations.
- Author
-
Mohammadi, S., Fardi, M., Ghasemi, M., Hendy, A. S., and Zaky, M. A.
- Subjects
- *
SEPARATION of variables , *LAPLACIAN operator , *RUNGE-Kutta formulas , *ENERGY function - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
27. Spectral constant rigidity of warped product metrics.
- Author
-
Chai, Xiaoxiang, Pyo, Juncheol, and Wan, Xueyuan
- Subjects
- *
ELLIPTIC operators , *HARMONIC functions , *CURVATURE , *LAPLACIAN operator , *EIGENVALUES , *SPACETIME - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
28. Periodic solution for Hamiltonian type systems with critical growth.
- Author
-
Guo, Yuxia, Wu, Shengyu, and Yan, Shusen
- Subjects
GREEN'S functions ,HAMILTONIAN systems ,OPERATOR functions ,LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
29. Tunneling effect in two dimensions with vanishing magnetic fields.
- Author
-
Alfa, Khaled Abou
- Subjects
SCHRODINGER operator ,PSEUDODIFFERENTIAL operators ,EIKONAL equation ,SCHRODINGER equation ,LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
30. Neural Dynamics in Parkinson’s Disease: Integrating Machine Learning and Stochastic Modelling with Connectomic Data
- Author
-
Shaheen, Hina, Melnik, Roderick, Hartmanis, Juris, Founding Editor, van Leeuwen, Jan, Series Editor, Hutchison, David, Editorial Board Member, Kanade, Takeo, Editorial Board Member, Kittler, Josef, Editorial Board Member, Kleinberg, Jon M., Editorial Board Member, Kobsa, Alfred, Series Editor, Mattern, Friedemann, Editorial Board Member, Mitchell, John C., Editorial Board Member, Naor, Moni, Editorial Board Member, Nierstrasz, Oscar, Series Editor, Pandu Rangan, C., Editorial Board Member, Sudan, Madhu, Series Editor, Terzopoulos, Demetri, Editorial Board Member, Tygar, Doug, Editorial Board Member, Weikum, Gerhard, Series Editor, Vardi, Moshe Y, Series Editor, Goos, Gerhard, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Franco, Leonardo, editor, de Mulatier, Clélia, editor, Paszynski, Maciej, editor, Krzhizhanovskaya, Valeria V., editor, Dongarra, Jack J., editor, and Sloot, Peter M. A., editor
- Published
- 2024
- Full Text
- View/download PDF
31. Digital Calculus Frameworks and Comparative Evaluation of Their Laplace-Beltrami Operators
- Author
-
Weill–Duflos, Colin, Coeurjolly, David, Lachaud, Jacques-Olivier, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Brunetti, Sara, editor, Frosini, Andrea, editor, and Rinaldi, Simone, editor
- Published
- 2024
- Full Text
- View/download PDF
32. Single‐peak solution for a fractional slightly subcritical problem with non‐power nonlinearity.
- Author
-
Deng, Shengbing and Yu, Fang
- Subjects
- *
LYAPUNOV-Schmidt equation , *LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
33. A Visual Measurement Method for Deep Holes in Composite Material Aerospace Components.
- Author
-
Meng, Fantong, Yang, Jiankun, Yang, Guolin, Lu, Haibo, Dong, Zhigang, Kang, Renke, Guo, Dongming, and Qin, Yan
- Subjects
- *
AEROSPACE materials , *ROBOTIC assembly , *LAPLACIAN operator , *MATERIALS texture , *SURFACES (Technology) , *COMPOSITE materials - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
34. Localization for general Helmholtz.
- Author
-
Cheng, Xinyu, Li, Dong, and Yang, Wen
- Subjects
- *
LAPLACIAN operator , *HELMHOLTZ equation - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
35. The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces II.
- Author
-
Neto, Paulo Mendes Carvalho and Júnior, Renato Fehlberg
- Subjects
- *
FRACTIONAL integrals , *BANACH spaces , *COMPACT operators , *LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
36. Fractional graph Laplacian for image reconstruction.
- Author
-
Aleotti, Stefano, Buccini, Alessandro, and Donatelli, Marco
- Subjects
- *
IMAGE reconstruction , *LAPLACIAN operator , *KRYLOV subspace , *IMAGE reconstruction algorithms , *TOMOGRAPHY - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
37. Optimal control problem with nonlinear fractional system constraint applied to image restoration.
- Author
-
Atlas, Abdelghafour, Attmani, Jamal, Karami, Fahd, and Meskine, Driss
- Subjects
- *
IMAGE reconstruction , *NONLINEAR equations , *NONLINEAR systems , *IMAGE denoising , *PARTIAL differential equations , *OPTIMAL control theory , *LAPLACIAN operator - Abstract
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$$ {H}&#x0005E;{-s} $$‐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]
- Published
- 2024
- Full Text
- View/download PDF
38. Stability and Numerical Simulation of a Nonlinear Hadamard Fractional Coupling Laplacian System with Symmetric Periodic Boundary Conditions.
- Author
-
Lv, Xiaojun, Zhao, Kaihong, and Xie, Haiping
- Subjects
- *
FRACTIONAL calculus , *BOUNDARY value problems , *COMPUTER simulation , *NONLINEAR boundary value problems , *NONLINEAR analysis , *LAPLACIAN operator - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
39. Statistical Warped Product Immersions into Statistical Manifolds of (Quasi-)Constant Curvature.
- Author
-
Siddiqui, Aliya Naaz, Khan, Meraj Ali, and Chaubey, Sudhakar Kumar
- Subjects
- *
CURVATURE , *LAPLACIAN operator , *SUBMANIFOLDS , *RIEMANNIAN manifolds - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
40. Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls.
- Author
-
Birindelli, Isabeau, Demengel, Françoise, and Leoni, Fabiana
- Subjects
- *
NONLINEAR equations , *EIGENFUNCTIONS , *EIGENVALUES , *LAPLACIAN operator , *ELLIPTIC equations - Abstract
This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions ( λ ¯ γ , u γ) of the equation F (D 2 u γ) + λ ¯ γ u γ r γ = 0 in B (0 , 1) ∖ { 0 } , u γ = 0 on ∂ B (0 , 1) where u γ > 0 in B (0 , 1) ∖ { 0 } and γ > 0. We prove existence of radial solutions which are continuous on B (0 , 1) ‾ in the case γ < 2 , existence of unbounded solutions in the case γ = 2 and a non existence result for γ > 2. We also give, in the case of Pucci's operators, the explicit value of λ ¯ 2 , which generalizes the Hardy–Sobolev constant for the Laplacian. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. SOME QUENCHING PROBLEMS FOR ω-DIFFUSION EQUATIONS ON GRAPHS WITH A POTENTIAL AND A SINGULAR SOURCE.
- Author
-
B., EDJA Kouamé, A. T., DIABATÉ Paterne, C., N'DRI Kouakou, and A., TOURÉ Kidjegbo
- Subjects
- *
LAPLACIAN operator , *EQUATIONS , *HYPOTHESIS - Abstract
In this paper, we study the quenching phenomenon related to the ω-diffusion equation on graphs with a potential and a singular source ut(x, t) = Δωu(x, t) + b(x)(1 - u(x, t))-p, where Δω is called the discrete weighted Laplacian operator. Under some appropriate hypotheses, we prove the existence and uniqueness of the local solution via Banach fixed point theorem. We also show that the solution of the problem quenches in a finite time and that the time-derivative blows up at the quenching time. Moreover, we estimate the quenching time and the quenching rate. Finally, we verify our results through some numerical examples. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. A numerical scheme based on the Taylor expansion and Lie product formula for the second‐order acoustic wave equation and its application in seismic migration.
- Author
-
Araujo, Edvaldo S. and Pestana, Reynam C.
- Subjects
- *
SEISMIC migration , *WAVE equation , *SOUND waves , *LAPLACIAN operator , *MATRIX exponential , *TAYLOR'S series , *COSINE function - Abstract
We have developed a numerical scheme for the second‐order acoustic wave equation based on the Lie product formula and Taylor‐series expansion. The scheme has been derived from the analytical solution of the wave equation and in the approximation of the time derivative for a wavefield. Through these two equations, we obtained the first‐order differential equation in time, where the time evolution operator of the analytic solution of this differential equation is written as a product of exponential matrices. The new numerical solution using a Lie product formula may be combined with Taylor‐series, Chebyshev, Hermite and Legendre polynomial expansion or any other expansion for the cosine function. We use the proposed scheme combined with the second‐ or fourth‐order Taylor approximations to propagate the wavefields in a recursive procedure, in a stable manner, accurately and efficiently with even larger time steps than the conventional finite‐difference method. Moreover, our numerical scheme has provided results with the same quality as the rapid expansion method but requiring fewer computations of the Laplacian operator per time step. The numerical results have shown that the proposed scheme is efficient and accurate in seismic modelling, reverse time migration and least‐squares reverse time migration. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. STABLE CONES IN THE THIN ONE-PHASE PROBLEM.
- Author
-
FERNÁNDEZ-REAL, XAVIER and ROS-OTON, XAVIER
- Subjects
- *
MINIMAL surfaces , *LAPLACIAN operator - Abstract
The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions ≥ 3 is completely open. In this context, axially symmetric solutions are expected to play the same role as Simons' cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open. The goal of this paper is twofold. On the one hand, our first main contribution is to find, for the first time, the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of "large solutions" for the fractional Laplacian, which blow up on the free boundary. On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions ≤ 5 is one-dimensional, independently of the parameter s ∈ (0,1). [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. Studying behavior of the asymptotic solutions to P-Laplacian type diffusion-convection model.
- Author
-
Qasim, Ruba H. and Aal-Rkhais, Habeeb A.
- Subjects
LAPLACIAN operator ,DIFFUSION ,CAUCHY problem ,ADVECTION ,CONSERVATION laws (Mathematics) - Abstract
Copyright of Journal of University of Anbar for Pure Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
- Published
- 2024
- Full Text
- View/download PDF
45. Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side.
- Author
-
Ferrari, Fausto and Lederman, Claudia
- Subjects
LIPSCHITZ continuity ,VISCOSITY solutions ,LAPLACIAN operator - Abstract
We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are C 1 , γ . No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when p (x) ≡ p , i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. On the Pohozaev identity for the fractional p$p$‐Laplacian operator in RN$\mathbb {R}^N$.
- Author
-
Ambrosio, Vincenzo
- Subjects
LAPLACIAN operator ,NONLINEAR equations - Abstract
In this paper, we show the existence of a nontrivial weak solution for a nonlinear problem involving the fractional p$p$‐Laplacian operator and a Berestycki–Lions type nonlinearity. This solution satisfies a Pohozaev identity. Moreover, we prove that any sufficiently smooth solution fulfills the Pohozaev identity. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. Investigation on integro-differential equations with fractional boundary conditions by Atangana-Baleanu-Caputo derivative.
- Author
-
Harisa, Samy A., Faried, Nashat, Vijayaraj, V., Ravichandran, C., and Morsy, Ahmed
- Subjects
- *
INTEGRO-differential equations , *FRACTIONAL differential equations , *HILBERT space , *EXISTENCE theorems , *BANACH spaces , *LAPLACIAN operator - Abstract
We establish, the existence and uniqueness of solutions to a class of Atangana-Baleanu (AB) derivative-based nonlinear fractional integro-differential equations with fractional boundary conditions by using special type of operators over general Banach and Hilbert spaces with bounded approximation numbers. The Leray-Schauder alternative theorem guarantees the existence solution and the Banach contraction principle is used to derive uniqueness solutions. Furthermore, we present an implicit numerical scheme based on the trapezoidal method for obtaining the numerical approximation to the solution. To illustrate our analytical and numerical findings, an example is provided and concluded in the final section. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. Normalized solutions for (p,q)-Laplacian equations with mass supercritical growth.
- Author
-
Cai, Li and Rădulescu, Vicenţiu D.
- Subjects
- *
GROUND state energy , *LAGRANGE multiplier , *EQUATIONS , *LAPLACIAN operator - Abstract
In this paper, we study the following (p , q) -Laplacian equation with L p -constraint: { − Δ p u − Δ q u + λ | u | p − 2 u = f (u) , in R N , ∫ R N | u | p d x = c p , u ∈ W 1 , p (R N) ∩ W 1 , q (R N) , where 1 < p < q < N , Δ i = div (| ∇ u | i − 2 ∇ u) , with i ∈ { p , q } , is the i -Laplacian operator, λ is a Lagrange multiplier and c > 0 is a constant. The nonlinearity f is assumed to be continuous and satisfying weak mass supercritical conditions. The purpose of this paper is twofold: to establish the existence of ground states, and to reveal the basic behavior of the ground state energy E c as c > 0 varies. Moreover, we introduce a new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints intersected with the closed ball of radius c p in L p (R N). The analysis developed in this paper allows to provide the general growth assumptions imposed to the reaction f. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. Spectral analysis in broken sheared waveguides.
- Author
-
Suarez Bello, Diana Carolina and Verri, Alessandra A.
- Subjects
- *
LAPLACIAN operator , *MULTIPLICITY (Mathematics) - Abstract
Let Ω ⊂ R³ be a broken sheared waveguide; that is, Ω is built by translating a cross-section (an arbitrary bounded connected open set of R²) in a constant direction along a broken line in R³. We prove that the discrete spectrum of the Dirichlet Laplacian operator in Ω is non-empty and finite. Furthermore, we show a particular geometry for Ω, which implies that the total multiplicity of the discrete spectrum is equal to 1. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
50. Existence of periodic solutions for a class of (ϕ1,ϕ2)-Laplacian difference system with asymptotically (p,q)-linear conditions.
- Author
-
Deng, Hai-yun, Lin, Xiao-yan, and He, Yu-bo
- Subjects
- *
LAPLACIAN operator , *MOUNTAIN pass theorem - Abstract
In this paper, we consider a (ϕ 1 , ϕ 2) -Laplacian system as follows: { Δ ϕ 1 (Δ u (t − 1)) + ∇ u F (t , u (t) , v (t)) = 0 , Δ ϕ 2 (Δ v (t − 1)) + ∇ v F (t , u (t) , v (t)) = 0 , where F (t , u (t) , v (t)) = − K (t , u (t) , v (t)) + W (t , u (t) , v (t)) is T-periodic in t. By using the mountain pass theorem, we obtain that the (ϕ 1 , ϕ 2) -Laplacian system has at least one periodic solution if W is asymptotically (p , q) -linear at infinity. Our results improve and extend some known works. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.