The Laplace-Beltrami operator is an extension of the Laplacian from flat domains to curved manifolds. It was proven to be useful for color image processing as it models a meaningful coupling between the color channels. This coupling is naturally expressed in the Beltrami framework in which a color image is regarded as a two dimensional manifold embedded in a hybrid, five dimensional, spatial-ch...

Abstract We study the removable singularities for solutions to the Beltrami equation ∂f = μ∂f , where μ is a bounded function, ‖μ‖∞ ≤ K−1 K+1 < 1, and such that μ ∈ W 1,p for some p ≤ 2. Our results are based on an extended version of the well known Weyl’s lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly s...

We study the removable singularities for solutions to the Beltrami equation ∂f = μ∂f , where μ is a bounded function, ‖μ‖∞ ≤ K−1 K+1 < 1, and such that μ ∈ W 1,p for some p ≤ 2. Our results are based on an extended version of the well known Weyl’s lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly supported ...

We consider the Klein–Gordon equation associated with the Laplace– Beltrami operator ∆ on real hyperbolic spaces of dimension n≥2; as ∆ has a spectral gap, the wave equation is a particular case of our study. After a careful kernel analysis, we obtain dispersive and Strichartz estimates for a large family of admissible couples. As an application, we prove global well–posedness results for the c...

We present a novel approach for computing and solving the Poisson equation over the surface of a mesh. As in previous approaches, we define the Laplace-Beltrami operator by considering the derivatives of functions defined on the mesh. However, in this work, we explore a choice of functions that is decoupled from the tessellation. Specifically, we use basis functions (second-order tensor-product...

In this paper, we investigate optimal control problems for Allen–Cahn equations with singular nonlinearities and a dynamic boundary condition involving singular nonlinearities and the Laplace– Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribe...

We study the problem of Michell trusses when the system of applied equilibrated forces is a vector measure with compact support. We introduce a class of stress tensors which can be written as a superposition of rank-one tensors carried by curves (lines of principal strains). Optimality conditions are given for such families showing in particular that optimal stress tensors are carried by mutual...

Abstract. In this paper, we consider the harmonic extension problem, which is widely used in many applications of machine learning. We formulate the harmonic extension as solving a LaplaceBeltrami equation with Dirichlet boundary condition. We use the point integral method (PIM) proposed in [14, 19, 13] to solve the Laplace-Beltrami equation. The basic idea of the PIM method is to approximate t...