We propose a newmethod to analyze and represent data recorded on a domain of general shape in R by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problemon such a domain via theHelmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the ...

In this article, we discuss the existence of solutions to boundaryvalue problems for a coupled system of fractional differential equations with p-Laplacian operator at resonance. We prove the existence of solutions when dim ker L ≥ 2, using the coincidence degree theory by Mawhin.

Every generalized laplacian L defined on a manifold M determines a sheaf of "L-harmonic" sections namely the sheaf of local solutions of Lu = 0. We study the converse problem: to what extent this sheaf determines the operator. Our main result states that the sheaf of L-harmonic sections determines the operator up to a conformal factor. Moreover, when the operator is a covariant laplacian and th...

Abstract. We discuss a general framework for the numerical solution of a family of semilinear elliptic problems whose leading differential operator is the Laplacian. A problem is first transformed to one on a standard domain via a conformal mapping. The boundary value problem on the standard domain is then reduced to an equivalent integral operator equation. We employ the Galerkin method to sol...

The laplacian operator applied to the coordinates of a manifold provides the mean curvature vector. Manipulating the metric of the manifold or interpreting its coordinates in various ways provide useful tools for shape and image processing and representation. We will review some of these tools focusing on scale invariant geometry, curvature flow with respect to an embedding of the image manifol...

The Seiberg-Witten equations are studied from the viewpoint of gauge potential decomposition. We find a determinant equation ∆Aμ = −λAμ for the twisting U(1) potential Aμ of the Seiberg-Witten theory, which is in itself an eigenvalue problem of the Laplacian operator, with the eigenvalue being the vacuum expectation value of the field function, λ = ‖Φ‖ /2. This establishes a direct relationship...

This paper mainly studies the existence of multiple positive solutions a class Riemann-Liouville fractional q-difference equations under four-point boundary value condition with p-Laplacian operator. The two equation is verified by monotonic iterative method. Finally, an example used to prove validity main results obtained.

We show global and interior higher-order log-Hölder regularity estimates for solutions of Dirichlet integral equations where the operator has a nonintegrable kernel with singularity at origin that is weaker than any fractional Laplacian. As consequence, under mild assumptions on right hand side, we existence classical problems involving logarithmic Laplacian Schrödinger operator.

The paper develops a sub-supersolution approach for quasilinear elliptic equations driven by degenerated p-Laplacian and containing convection term. presence of the operator forces substantial change to functional setting previous works. existence location solutions through is established. abstract result applied find nontrivial, nonnegative bounded solutions.