Keywords px-Laplace operator, p x-Kirchhoﬀ-type equations, variable exponent Sobolev spaces, variational method, mountain pass theorem, Ekeland variational principle MSC: 35D05; 35J60; 3[r]

Riesz Means of Dirichlet Eigenvalues for the Sub-Laplace Operator on the Engel Group Jingjing Xue Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, China E[r]

In this paper, we study an n-dimensional fractional diﬀerential system with p-**Laplace** **operator**, which involves multi-strip integral boundary conditions. By using the Leggett–Williams ﬁxed point theorem, the existence results of at least three positive solutions are established. Besides, we also get the nonexistence results of positive solutions. Finally, two examples are presented to validate the main results.

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The existence of at least three weak solutions is established for a class of quasilinear elliptic systems involving the p ( x )-**Laplace** **operator** with Neumann boundary condition. The technical approach is mainly based on a three critical points theorem due to Ricceri. MSC: 35D05; 35J60; 58E05.

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[12] P. Li and S. T. Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the **Laplace** **operator** (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), 205–239, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980. [13] J. Ling, The first eigenvalue of a closed manifold with positive Ricci curvature, Proc. Amer.

this reason some of the authors, in particular those who work in ODEs, call equations involving the p-Laplacian “half-linear” equations. The word “half-linear” reﬂects the fact that “one half” of the properties of linearity (i.e. additivity) is lost, while “one half” of the properties is preserved (i.e. homogeneity). The p-**Laplace** **operator** is very popular in nonlinear analysis and appears particularly in describing the behavior of compressible ﬂuid in a homogeneous isotropic rigid porous medium (see e.g. [8, 12, 18]).

Mourou, Transmutation Operators Associated with a Bessel Type Op-erator on The Half Line and Certain of Their Applications, Tamsui Oxford Journal of Information and Mathematical Scienc[r]

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which is the one phase obstacle problem for the p-Laplacian **operator**. Through a private com- munication with H. Shahgholian, we found that many questions related to the free boundary of (3.19) are yet to be settled, see [11] and [13]. However, when p = 2, the free boundary of the problem (3.19) is extensively studied, see for example [15].

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In section 2, we establish the shape optimization results. And we prove the nec- essary conditions of optimality that is the existence of Lagrange multiplier.Section 3 is devoted to auxiliary lemmas based on maximum principal theory [24]. These lemmas play a fundamental part in the following sections. In section 4, we give, under geometrical assumptions, an uniqueness results which ensure the statement of an algorithm and its convergence. In the last section, we discuss on a general elliptical **operator** and we end by some classical numerical illustrations. But this numerical analysis gives interesting information and allows us to talk on futures.

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application related to the above model problem are described, for example, in [1, 11] (see also the references therein). Let us mention the articles [2, 5, 10, 15, 17] on the inverse boundary value problems for the **Laplace** equation in a bounded domain. Our problem presents additional diﬃculties related to the unbounded nature of the domain and, to our knowledge, it was not considered in the literature. In this work we generalize the potential method used in the precursor works of Kress and Rundel [9, 10].

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Abstract ⎯ Many imaging systems produce images with deficient sharpness due to different real limitations. Hence, various image sharpening techniques have been used to improve the acutance of digital images. One of such is the well-known Laplacian sharpening technique. When implementing the basic Laplacian technique for image sharpening, two main drawbacks were detected. First, the amount of introduced sharpness cannot be increased or decreased. Second, in many situations, the resulted image suffers from a noticeable increase in brightness around the sharpened edges. In this article, an improved version of the basic Laplacian technique is proposed, wherein it contains two key modifications of weighting the **Laplace** **operator** to control the introduced sharpness and tweaking the second order derivatives to provide adequate brightness for recovered edges. To perform reliable experiments, only real-degraded images were used, and their accuracies were measured using a specialized no-reference image quality assessment metric. From the obtained experimental results, it is evident that the proposed technique outperformed the comparable techniques in terms of recorded accuracy and visual appearance.

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positive minimal solution for all γ ∈ (0, γ ∗ ] and admits no positive solution for γ > γ ∗ . We prove Theorem 1.1 by the method of monotonic iteration, also known as the super and subsolution method, which is a basic tool in nonlinear partial diﬀerential equations. In this paper, we discuss a fractional **Laplace** **operator** version of this method compared with second order linear or quasilinear elliptic **operator**. With respect to the classical case of the Laplacian, here some estimates are more delicate, due to the non-local nature of the **operator** (–) s .

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This paper is devoted to asymptotic formulas for functions related with the spectrum of the standard Laplace operator in two and three dimensional bounded doubly connected domains with i[r]

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and Lamberti have developed in [13] some preliminary abstract results for the dependence of the eigenvalues upon perturbation. Their applications to the Dirichlet eigenvalue problem for the **Laplace** **operator** appear clearly in Section 3 of their paper and in Theorem 3.21 when they justify the analyticity result for some symmetric functions of eigenvalues. Our analysis and uniform asymptotic formulas of the eigenfunctions, which are represented by the single-layer potential involving the Green function, are considerably different from those in [12, 13, 10]. Next, Our method differ, essentially, from the classical methods used to study the analytic dependence of the eigenfunctions of a real or complex parameter and used to give the asymp- totic formulae for the eigenvalues.

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In this paper we describe the set of all Schrödinger operators on graph and branched manifold, defined as a self-adjoint extension of the **operator**, originally defined on smooth functions with supports, not contained in the branch points manifold. Thus, given a description of the various options, we determine the **Laplace** **operator** on the space of the functions defined on a branched manifold. Description of the definition of each of the self-ad- joint extensions is given in terms of linear relations satisfied by the limit at the branch points and the boundary points of the graph function value in the domain of **operator** and the its derivative. Each of the **Laplace** operators corresponds to the Markov process, whose behavior in a neighborhood of branch points, we determined by the choice of the domain of the **Laplace** **operator**, obtained in this paper results, which is an extension of the study work [8] describes the self-adjoint extensions of a graph with a single vertex and two edges, to the case of a graph with an arbitrary number of edges. In addition, this paper summarizes the results of [6] in the case of Lap- lace operators, for which the linear relation in the space of boundary values that define the domain of the opera- tor, do not admit the possibility of expressing the limit function values at the boundary points and branch points of the graph of the limiting values of its derivative.

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n example is the **Laplace** transform, which renders a useful class of differential equations trivially solvable, converting them into algebraic ones instead. Another is the very well-known Fourier transform, which maps unctions of frequencies (Berian J. James, 2008). These transforms play an important role in the analysis of all kinds of physical phenomenon. As a link between the various applications of these transforms the authors

Wave equation, radial symmetry, boundary value problem, eigenvalue problem, Hilbert space, weighted Sobolev space, Fredholm operator, Laplace transform, Bessel functions.. 1980 AMS SUBJE[r]

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The NI Vision Builder AI processing module has multiple edge detection functions such as **Laplace** edge detection **operator**, Prewitt edge detection **operator**, Sobel edge detection **operator**, and Robert edge detection **operator**. Among them, The characteristic of **Laplace** edge detection **operator** is that the edge detection is more continuous, accurate and clear, and the false edge is effectively avoided [11], therefore, this experiment uses this **operator** to process the arc image.

The classical **Laplace** transform theory violates the requirements of modern engineering implementations. (a) We show that the ILT is valid only for infinite duration signals. On the other hand the modern engineering uses finite duration signals. (b) ILT models have poles; the FLT models do not generate poles. Thus ILT models may introduce instability in engineering implementations. (c) ILT requires convolution which is not valid for finite duration signals. (d) ILT methods are based on steady state concepts. Microprocessors do not see steady state in engineering implementations. (e) Infinite time theory for

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