p-Laplacian Operator Research Articles

A multiway p-spectral clustering algorithm

The original p-spectral clustering algorithm can obtain more balanced clustering results by introducing p-Laplacian operator. However, we may not get an ideal clustering result when clustering the multi-model or multi-scale data sets. Moreover, the original p-spectral clustering algorithm is suitable to deal with bipartition situation. when solving the multi-class partition problem, we have to recursively implement bipartition process, which will bring about ineffectiveness of the graph partition, and clustering result is not stable. Therefore, we propose a multiway p-spectral clustering algorithm, which employs local scaling parameter to optimize the calculation of similarity of the data objects. Furthermore, we use multi-eigenvectors to solve the multi-class partition problem by introducing idea of classical spectral clustering NJW algorithm, which can avoid the instability of the clustering result due to the information losing. Accordingly, we can attain the approximate optimal solution of the multi-class partition problem. Experiments show that multiway p-spectral clustering algorithm has much stronger adaptability and robustness, and can produce more balanced clusters.

Blow-up and global existence of solutions to a parabolic equation associated with the fraction <i>p</i>-Laplacian

We consider a nonlocal parabolic equation associated with the fractional p-laplace operator, which was studied by Gal and Warm in [On some degenerate non-local parabolic equation associated with the fractional p-Laplacian. Dyn. Partial Differ. Equ., 14(1): 47-77, 2017]. By exploiting the boundary condition and the variational structure of the equation, according to the size of the initial dada, we prove the finite time blow-up, global existence, vacuum isolating phenomenon of the solutions. Furthermore, the upper and lower bounds of the blow-up time for blow-up solutions are also studied. The results generalize the results got by Gal and Warm.

Positive solutions for a system of fractional differential equations with p-Laplacian operator and multi-point boundary conditions

We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann–Liouville fractional differential equations with parameters and p-Laplacian operator subject to multi-point boundary conditions, which contain fractional derivatives. The proof of our main existence results is based on the Guo–Krasnosel'skii fixed-point theorem.

Ground State Solutions for Fractional p-Kirchhoff Equation with Subcritical and Critical Exponential Growth

In this paper, we show the existence of nontrivial ground state solutions of fractional p-Kirchhoff problem $$\begin{aligned} \left\{ \begin{array}{ll} m\left( \int _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\mathrm{d}x\mathrm{d}y\right) (-\Delta )_p^s u=f(x,u) ~&{}\text {in}~\Omega , \\ u=0 ~&{}\text {in}~{\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$where $$(-\Delta )_p^s$$ is the fractional p-Laplacian operator with $$0<s<1<p<\infty $$, $$\Omega $$ is a bounded domain in $${\mathbb {R}}^N$$ with smooth boundary, m is continuous function and the nonlinearity f(x, u) has subcritical or critical exponential growth at $$\infty $$. For the purpose of obtaining our existence results, we used minimax techniques combined with the fractional Moser–Trudinger inequality.

English

The paper is devoted to the study of the existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces involving anti-periodic boundary conditions. Our approach in this study relies on the theory of monotone and maximal monotone operators combined with the Schaefer fixed-point theorem and the monotonicity method. We apply our abstract results in order to solve a diffusion equation of Kirchhoff type involving the Dirichlet p-Laplace operator.

On the First Robin Eigenvalue of a Class of Anisotropic Operators

The paper is devoted to the study of some properties of the first eigenvalue of the anisotropic p-Laplace operator with Robin boundary condition involving a function $${\beta}$$ which in general is not constant. In particular we obtain sharp lower bounds in terms of the measure of the domain and we prove a monotonicity property of the eigenvalue with respect the set inclusion.

Homoclinic solutions for Hamiltonian system with impulsive effects

In this article, we investigate a class of impulsive Hamiltonian systems with a p-Laplacian operator. By establishing a series of new sufficient conditions, the existence of homoclinic solutions to such type of systems is revealed. We show the existence of homoclinic orbit induced by impulses by introducing some conditions. To illustrate the applications of the main results in this article, we create an example.

Exact decay estimate on solutions of critical p-Laplacian equations in [formula omitted

Consider the critical p-Laplacian equation in R+N{−Δpu=0,inR+N,|∇u|p−2∂u∂n=|u|p‾−2u,onRN−1, where R+N=RN−1×(0,∞), 1<p<N,p‾=(N−1)pN−p is the critical exponent of the Sobolev imbedding from W1,p(R+N) to Lq(RN−1), p≤q≤p‾ and Δp is the p-Laplacian operator, Δpu=∇(|∇u|p−2∇u). We prove polynomial decay of the solutions|u(x)|≤c(1+|x|)−N−pp−1,forx∈R+N. The decay exponent is the best possible.

Existence and Nonexistence of Positive Solutions for Mixed Fractional Boundary Value Problem with Parameter and p-Laplacian Operator

This paper mainly studies a class of mixed fractional boundary value problem with parameter and p-Laplacian operator. Based on the Guo-Krasnosel’skii fixed point theorem, results on the existence and nonexistence of positive solutions for the fractional boundary value problem are established. An example is then presented to illustrate the effectiveness of the results.

Bifurcation results for the critical Choquard problem involving fractional p-Laplacian operator

By using an abstract critical point theorem based on a pseudo-index related to the cohomological index, we prove the bifurcation results for the critical Choquard problems involving fractional p-Laplacian operator: \t\t\t(−Δ)psu=λ|u|p−2u+(∫Ω|u|pμ,s∗|x−y|μdy)|u|pμ,s∗−2uin Ω,u=0in RN∖Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$(- \\Delta)_{p}^{s} u = \\lambda \\vert u \\vert ^{p-2} u + \\biggl( \\int_{\\Omega}\\frac{ \\vert u \\vert ^{p_{\\mu,s}^{*}}}{ \\vert x-y \\vert ^{\\mu}}\\,dy \\biggr) \\vert u \\vert ^{p_{\\mu,s}^{*}-2}u \\quad \\text{in } \\Omega,\\qquad u = 0\\quad \\text{in } {\\mathbb {R}}^{N} \\setminus \\Omega, $$\\end{document} where Ω is a bounded domain in {mathbb {R}}^{N} with Lipschitz boundary, λ is a real parameter, pin(1,infty), sin (0,1), N>sp, and p_{mu,s}^{*}=frac{(N-frac{mu}{2})p}{N-sp} is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. These extend results in the literature for the fractional Choquard problems, and they are still new for a p-Laplacian case.

Optimal Shape Design for the p-Laplacian Eigenvalue Problem

In this paper, a shape optimization problem corresponding to the p-Laplacian operator is studied. Given a density function in a rearrangement class generated by a step function, find the density such that the principal eigenvalue is as small as possible. Considering a membrane of known fixed mass and with fixed boundary of prescribed shape consisting of two different materials, our results determine the way to distribute these materials such that the basic frequency of the membrane is minimal. We obtain some qualitative aspects of the optimizer and then we determine nearly optimal sets which are approximations of the minimizer for specific ranges of parameters values. A numerical algorithm is proposed to derive the optimal shape and it is proved that the numerical procedure converges to a local minimizer. Numerical illustrations are provided for different domains to show the efficiency and practical suitability of our approach.

Multiplicity for fractional differential equations with p-Laplacian

This paper investigates the existence of positive solution for a boundary value problem of fractional differential equations with p-Laplacian operator. Our analysis relies on the research of properties of the corresponding Green’s function. By the use of Krasnosel’skii’s fixed-point theorem, the multiplicity results of some positive solutions are obtained.

A [formula omitted] mortar spectral element method for the [formula omitted]-Laplacian equation

We present a constrained-vertex variant of the mortar spectral element method to solve the p-Laplacian equation. To show reliability of the method, first we investigate convergence rate of h-version and k-version for a sufficiently smooth solution. Then for solutions with limited regularity, we use a hk-strategy in which the size of elements is geometrically reduced towards the singularity while the polynomial degree is linearly reduced. In fact, we numerically study convergence rate of the hk mortar spectral element method in the L2-norm and H1-norm using geometrical meshes with elements of non-uniform polynomial degree. In this regard, we consider several benchmarks with various choices of the parameter p as well as geometric grading factors. To deal with possible degeneracy in the p-Laplacian operator, we add artificial diffusion to the original equation and then study the effects of this extra diffusion on convergence rate. To find the solution of highly nonlinear system arising from discretization we use an optimization technique based on the trust region method in which an analytical Jacobian matrix is utilized.

Boundary higher integrability for very weak solutions of quasilinear parabolic equations

We prove boundary higher integrability for the (spatial) gradient of very weak solutions of quasilinear parabolic equations of the formut−divA(x,t,∇u)=0onΩ×R, where the non-linear structure divA(x,t,∇u) is modelled after the p-Laplace operator. To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary. In order to do this, we construct a suitable test function which is Lipschitz continuous and preserves the boundary values. These results are new even for linear parabolic equations on domains with smooth boundary and make no assumptions on the smoothness ofA(x,t,∇u). These results are also applicable for systems as well as higher order parabolic equations.

Singular p-Laplacian equations with superlinear perturbation

We consider a nonlinear Dirichlet problem driven by the p-Laplace operator and with a right-hand side which has a singular term and a parametric superlinear perturbation. We are interested in positive solutions and prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ>0 varies. In addition, we show that for every admissible parameter λ>0 the problem has a smallest positive solution u‾λ and we establish the monotonicity and continuity properties of the map λ→u‾λ.

Existence of solutions for quasilinear Dirichlet problems with gradient terms

In this paper we prove an existence theorem for positive solutions of a nonlinear Dirichlet problem involving the p-Laplacian operator on a smooth bounded domain when a nonlinearity depending on the gradient is considered. Our main theorem extends a previous result by Ruiz in [ 19 ], in which a slight modification of the celebrated blowup technique due to Gidas and Spruck, [ 11 ] and [ 12 ] is introduced.

Composition Operators on Sobolev Spaces and Neumann Eigenvalues

In this paper we discuss applications of the geometric theory of composition operators on Sobolev spaces to the spectral theory of non-linear elliptic operators. Lower estimates of the first non-trivial Neumann eigenvalues of the p-Laplace operator in cusp domains $$\Omega \subset \mathbb R^n$$ , $$n\ge 2$$ , are given.

Asymptotics for quasilinear obstacle problems in bad domains

We study two obstacle problems involving the p-Laplace operator in domains with n-th pre-fractal and fractal boundary. We perform asymptotic analysis for \begin{document} $p \to \infty $ \end{document} and \begin{document} $n \to \infty $ \end{document} .

Energy dependent potential problems for the one dimensional [formula omitted]-Laplacian operator

In this work we analyze a nonlinear eigenvalue problem for the p-Laplacian operator with zero Dirichlet boundary conditions. We assume that the problem has a potential which depends on the eigenvalue parameter, and we show that, for n big enough, there exists a real eigenvalue λn, and their corresponding eigenfunctions have exactly n nodal domains.We characterize the asymptotic behavior of these eigenvalues, obtaining two terms in the asymptotic expansion of λn in powers of n.Finally, we study the inverse nodal problem in the case of energy dependent potentials, showing that some subset of the zeros of the corresponding eigenfunctions is enough to determine the main term of the potential.

Symmetry properties for solutions of nonlocal equations involving nonlinear operators

We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional p-Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincaré inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions. Most of the methods and ideas applied in the current article are applicable to nonlocal operators with general kernels where the famous extension problem, given by Caffarelli and Silvestre, is not necessarily known.