Extremal domains for the first eigenvalue of the Laplace-Beltrami operator

The Laplace-Beltrami-Operator on Riemannian Manifolds

This report mainly illustrates a way to compute the Laplace-Beltrami-Operator on a Riemannian Manifold and gives information to why and where it is used in the Analysis of 3D Shapes. After a brief introduction, an overview over the necessary properties of manifolds for calculating the Laplacian is given. Furthermore the two operators needed for defining the Laplace-Beltrami-Operator the gradien...

Laplace-Beltrami operator for Jack polynomials

We introduce a Laplace-Beltrami type operator on the Fock space of symmetric functions and show that the Jack symmetric functions are the only family of eigenvectors of the differential operator, thus giving a new characterization of Jack polynomials. This was achieved by explicit computation of its action on generalized homogeneous symmetric functions. Using this new method we give a combinato...

Data driven estimation of Laplace-Beltrami operator

Approximations of Laplace-Beltrami operators on manifolds through graph Laplacians have become popular tools in data analysis and machine learning. These discretized operators usually depend on bandwidth parameters whose tuning remains a theoretical and practical problem. In this paper, we address this problem for the unnormalized graph Laplacian by establishing an oracle inequality that opens ...

Boundary Integral Equations for the Laplace–Beltrami Operator

High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates

We present a new AFEM for the Laplace-Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally W 1 ∞ and piecewise in a suitable Besov class embedded in C1,α with α ∈ (0, 1]. The idea is to have the surface sufficiently well resolved in W 1 ∞ relative to the current resolution of the PDE in H1. This gives rise to a conditional contraction property of the PDE...