Related papers: Discrete Laplacians -- spherical and hyperbolic
We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a…
The Hodge decomposition provides a very powerful mathematical method for the analysis of 2D and 3D vector fields. It states roughly that any vector field can be $L^2$-orthogonally decomposed into a curl-free, divergence-free, and a harmonic…
We extend the discontinuous Galerkin (DG) framework to the analysis of first-order hyperbolic and advection-dominated problems posed on implicitly defined surfaces. The focus will be on the hyperbolic part, which is discretised using a…
The structural properties of packed soft-core particles provide a platform to understand the cross-pollinated physical concepts in solid-state- and soft-matter physics. Confined on spherical surface, the traditional differential geometry…
Spheres can be written as homogeneous spaces $G/H$ for compact Lie groups in a small number of ways. In each case, the decomposition of $L^2(G/H)$ into irreducible representations of $G$ contains interesting information. We recall these…
For $\Pi \subset \mathbb{R}^2$ a connected, open, bounded set whose boundary is a finite union of disjoint polygons whose vertices have integer coordinates, the logarithm of the discrete Laplacian on $L\Pi \cap \mathbb{Z}^2$ with Dirichlet…
Differential calculus on discrete spaces is studied in the manner of non-commutative geometry by representing the differential calculus by an operator algebra on a suitable Krein space. The discrete analogue of a (pseudo-)Riemannian metric…
We investigate in this paper the existence of a metric which maximizes the first eigenvalue of the Laplacian on Riemannian surfaces. We first prove that, in a given conformal class, there always exists such a maximizing metric which is…
We discuss bases of the space of holomorphic quadratic differentials that are dual to the differentials of Fenchel-Nielsen coordinates and hence appear naturally when considering functions on the set of hyperbolic metrics which are…
We consider Laplacian eigenfunctions in circular, spherical and elliptical domains in order to discuss three kinds of high-frequency localization: whispering gallery modes, bouncing ball modes, and focusing modes. Although the existence of…
We survey the distributional properties of progressively dilating sets under projection by covering maps, focusing on manifolds of constant sectional curvature. In the Euclidean case, we review previously known results and formulate some…
We discuss the local differential geometry of convex affine spheres in $\re^3$ and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In each case, there is a natural metric and cubic differential holomorphic with respect to the…
We classify weakly complete constant Gaussian curvature $-1<K<0$ surfaces in the hyperbolic three-space in terms of holomorphic quadratic differentials. For this purpose, we first establish a loop group method for constant Gaussian…
The Dedekind eta functions plays important role in different branches of Mathematics and Theoretical Physics. One way to construct Dedekind Eta function to use the explicit formula (Kroncker limit formula) for the regularized determinants…
In this paper, under suitable geometric constraints, we have successfully obtained characterizations for the extremum values of the functional of mixed eigenvalues of the Laplacian on triangles (or trapezoids) in the Euclidean plane…
In this paper the conformal Dirac operator on the sphere is defined to be operating on the space of square-integrable Clifford algebra-valued functions. The spinorial Laplacian of order d>0 is defined and used to establish Sobolev embedding…
We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite…
The immersed boundary method is a mathematical framework for modeling fluid-structure interaction. This formulation describes the momentum, viscosity, and incompressibility of the fluid-structure system in Eulerian form, and it uses…
We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part,…
It is shown that a superconformal surface with arbitrary codimension in flat Euclidean space has a (necessarily unique) dual superconformal surface if and only if the surface is S-Willmore, the latter a well-known necessary condition to…