A general procedure is presented for computing axisymmetric swirling vortices which are steady with respect to an inviscid flow that is either uniform at infinity or includes shear. We consider cases both with and without a spherical obstacle. Choices of numerical parameters are given which yield vortex rings with swirl, attached vortices with swirl analogous to spherical vortices found by Moff...

Using Fefferman’s classical result on the boundary singularity of the Bergman kernel, we give an analogous description of the boundary behaviour of various related quantities like the Bergman invariant, the coefficients of the Bergman metric, of the associated Laplace-Beltrami operator, of its curvature tensor, Ricci curvature and scalar curvature. The main point is that even though one would e...

Abstract. We consider a linear-quadratic optimization problem with pointwise bounds on the state for which the constraint is given by the Laplace-Beltrami equation (to have uniqueness we add an lower order term) on a two-dimensional surface . By using finite elements we approximate the optimization problem by a family of discrete problems and prove convergence rates for the discrete controls an...

Sequences of nodal counts store information on the geometry (metric) of the domain where the wave equation is considered. To demonstrate this statement, we consider the eigenfunctions of the Laplace-Beltrami operator on surfaces of revolution. Arranging the wave functions by increasing values of the eigenvalues, and counting the number of their nodal domains, we obtain the nodal sequence whose ...

We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u+ ω2u = f on Sn ⊂ Rn+1. Here ∆ is the Laplace-Beltrami operator on Sn, ω is a non-zero constant and f belongs to C2k−2(Sn), where k ≥ n/4 + 1, k is an integer. The shifts of a spherical basis function φ with φ ∈ H τ (Sn) and τ > 2k ≥ n/2 + 2 are used to construct an approximate solution. An H1(Sn)error ...

One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a lowdimensional manifold embedded in a high-dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections ...

Let (M, g) be a Riemannian surface (i.e. a Riemannian manifold of dimension 2), orientable or not. We assume that M is either compact or a compact perturbation of the euclidian space, so that the Sobolev embeddings are true. Consider ∆ = ∆ g the Laplace-Beltrami operator. In this paper we are interested in constructing WKB approximations for the non linear cubic Schrödinger equation i∂ t u(t, x...

In this article, we consider the existence and asymptotic behavior of solutions for the Hénon equation −∆BN u = (d(x)) α|u|p−2u, x ∈ Ω u = 0 x ∈ ∂Ω, where ∆BN denotes the Laplace Beltrami operator on the disc model of the Hyperbolic space BN , d(x) = dBN (0, x), Ω ⊂ BN is geodesic ball with radius 1, α > 0, N ≥ 3. We study the existence of hyperbolic symmetric solutions when 2 < p < 2N+2α N−2 ....

A smooth map f: M ! N between semi-riemannian manifolds is called a harmonic morphism if f pulls back harmonic functions (i.e., local solutions of the Laplace{Beltrami equation) on N into harmonic functions on M. It is shown that a harmonic morphism is the same as a harmonic map which is moreover horizontally weakly conformal, these two notions being likewise carried over from the riemannian ca...