The operator square root of the Laplacian (−△) can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some proper...

This paper concerned with the existence of solutions of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator. Firstly, the definition a pair of coupled lower and upper solutions of the problem is introduced. Then, under the approach of coupled upper and lower solutions together with Nagumo condition, we prove that there exists at least one solutio...

The Laplace-Beltrami operator on a Riemannian manifold, its spectral theory and the relations between its first eigenvalue and the geometrical data of the manifold, such as curvatures, diameter, injectivity radius, volume, has been extensively studied in the recent mathematical literature. In the last few years, another operator, called p-Laplacian, arising from problems on Non-Newtonian Fluids...

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 ...

We study 22 systems of variational inequalities which are only weakly elliptic; in particular, these systems are not necessarily monotone. The prototype diierential operator is the (vector-valued) p-Laplacian. We prove, under certain conditions, the existence of solutions to the unilateral obstacle problem. In addition, we address the question of determining function spaces on which the p-Lapla...

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...

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.