Beltrami Operator Research Articles

Elliptic differential inclusions on non-compact Riemannian manifolds

We investigate a large class of elliptic differential inclusions on non-compact complete Riemannian manifolds which involves the Laplace–Beltrami operator and a Hardy-type singular term. Depending on the behavior of the nonlinear term and on the curvature of the Riemannian manifold, we guarantee non-existence and existence/multiplicity of solutions for the studied differential inclusion. The proofs are based on nonsmooth variational analysis as well as isometric actions and fine eigenvalue properties on Riemannian manifolds. The results are also new in the smooth setting.

Data-Driven Fuzzy Transform

The Fuzzy transform is applied mainly to 1-D signals and 2-D data organized as a regular grid (e.g., 2-D images), thus, limiting its potential application to arbitrary data in terms of dimensionality and structure. This article defines and analyzes the properties of the data-driven F-transform, with a focus on the construction of the class of data-driven membership functions, which are multiscale, local, linearly independent, intrinsic, and robust to data discretization. Data-driven membership functions are defined by applying a 1-D filter to the Laplace–Beltrami operator, which encodes the geometric and topological properties of the input data. Then, we address the efficient computation of the data-driven F-transform through a polynomial or a rational polynomial approximation of the input filter. In this way, the computation of the data-driven F-transform is independent of the evaluation of the membership functions at any point of the input domain and reduces to the solution of a small set of sparse and symmetric linear systems. Finally, the data-driven F-transform is efficiently evaluated on large and arbitrary data, in terms of dimensionality, structure, and size.

Covariant derivatives of eigenfunctions along parallel tensors over space forms and a conjecture motivated by the vertex algebraic structure

We study the covariant derivatives of an eigenfunction for the Laplace–Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. We show that along every parallel tensor, the covariant derivative is a scalar multiple of the eigenfunction. We also show that the scalar is a polynomial depending on the eigenvalue and prove some properties. A conjecture motivated by the study of vertex algebraic structure on space forms is also announced, suggesting the existence of interesting structures in these polynomials that awaits further exploration.

Fractional Klein–Gordon equation on AdS2+1

We propose a covariant definition of the fractional Klein–Gordon equation (KGE) with long-range interactions independent of the metric of the underlying manifold. As an example we consider the fractional KGE on AdS2+1, computing the explicit kernel representation of the fractional Laplace–Beltrami operator as well as the two-point propagator of the fractional KGE. Our results suggest that the propagator only exists if the mass is small compared to the inverse AdS radius, presumably because the AdS space expands faster with distance as a flat space of the same dimension. Our results are expected to be useful in particular for new applications of the AdS/CFT correspondence within statistical mechanics and quantum information.

Quantum statistical mechanics of vortices

The asymptotic partition function for quantized abelian Higgs vortices at high temperature T is found to leading and subleading order, and from this the equation of state of the vortex gas is derived, including the first quantum correction. It is assumed that the Hamiltonian is proportional to the Laplace–Beltrami operator on the moduli space of static N-vortex solutions. The partition function is calculated using the total volume and total scalar curvature of the moduli space.

Almost positive kernels on compact Riemannian manifolds

We show how to build a kernel K_X(x,y)=sum _{m=0}^Xh(lambda _m/{lambda _X})varphi _m(x)overline{varphi _m(y)} on a compact Riemannian manifold {{,mathrm{mathcal {M}},}}, which is positive up to a negligible error and such that K_X(x,x)approx X. Here 0=lambda _0^2le lambda _1^2le cdots are the eigenvalues of the Laplace–Beltrami operator on {{,mathrm{mathcal {M}},}}, listed with repetitions, and varphi _0,,varphi _1,ldots an associated system of eigenfunctions, forming an orthonormal basis of L^2({{,mathrm{mathcal {M}},}}). The function h is smooth up to a certain minimal degree, even, compactly supported in [-1,1] with h(0)=1, and K_X(x,y) turns out to be an approximation to the identity.

Generalized anisotropic elliptic Wentzell problems with nonstandard growth conditions

Let Ω⊆RN be a bounded Lipschitz domain with the W1,p→(⋅)-extension property, for N≥3. We investigate the solvability and global regularity of a class of quasi-linear elliptic equations involving the anisotropic p→(⋅)-Laplace operator Δp→(⋅), with nonhomogeneous anisotropic Wentzell boundary conditions ∑i=1N|∇u|pi(⋅)−2∂u∂xiνi−Δq→(⋅),Γu+β|u|qM(⋅)−2u∋0onΓ≔∂Ω,for β∈L∞(Γ)+ with a positive lower bound, where p→∈C0,1(Ω¯)N and q→∈C0,1(Γ)N−1 fulfill 1<pm−≤pM+<∞, and 1<qm−≤qM+<∞. Here Δq→(⋅),Γ denotes the anisotropic q→(⋅)-Laplace–Beltrami operator. We first show existence and uniqueness of weak solutions for the elliptic problem, and moreover, we prove that such solutions are globally bounded over Ω¯. Key L∞-type a priori estimates for the difference of weak solutions are provided, as well as Maximum principles and inverse positivity results. At the end, we establish a sort of “nonlinear anisotropic Fredholm alternative” for the corresponding anisotropic Wentzell problem of nonstandard structure.

Efficient and practical phase-field method for the incompressible multi-component fluids on 3D surfaces with arbitrary shapes

The fluid dynamics on a curved surface can be used to approximately describe many physical phenomena in the natural world, such as the atmospherical circulation on a planet. The present work develops a simple, practical, and efficient numerical method for the multi-phase fluid simulations on arbitrarily curved surfaces in 3D space. The multi-component Cahn–Hilliard type phase-field fluid model is adopted to capture the motion of multiple interfaces. The velocity field and pressure are updated by solving the incompressible Navier–Stokes equations. To achieve the decoupled computations of pressure and velocity, the Chorin's projection method with pressure correction is considered. A linear semi-implicit time-marching scheme is designed for the phase-field equations. By adopting the embedding method and appropriate pseudo-Neumann boundary conditions, the direct calculations on the surface can be extended into a 3D narrow band domain including the surface. The standard Laplacian operator can be used to replace the surface Laplace–Beltrami operators and the finite difference method is used to perform the discretization in space. In each time step, all variables can be updated in a step-by-step manner because the proposed method is totally decoupled. Therefore, the whole algorithm is highly efficient and easy to implement. Extensive numerical experiments indicate that the proposed method has good performance for the multi-component simulations on curved surfaces with complex shapes.

Nonlinear Beltrami equation and asymptotics of its solution

We continue to study regular homeomorphic solutions to the nonlinear Beltrami equation introduced in [24]. Estimates of the Schwarz Lemma type have been obtained using the length-area method.

Restriction of Laplace operator on one-forms: From Rn+2 and Rn+1 ambient spaces to embedded (A)dSn submanifolds

The Laplace–de Rham operator acting on a one-form a: □a in Rn+2 or Rn+1 spaces is restricted to n-dimensional pseudo-spheres. This includes, in particular, the n-dimensional de Sitter and anti-de Sitter space-times. The restriction is designed to extract the corresponding n-dimensional Laplace–de Rham operator acting on the corresponding n-dimensional one-form on pseudo-spheres. Explicit formulas relating these two operators are given in each situation. The converse problem of extending an n-dimensional operator composed of the sum of the Laplace–de Rham operator and additional terms to the Laplace–de Rham operator on ambient spaces is also studied. We show that for any additional term, this operator on the embedded space is the restriction of the Laplace–de Rham operator on the embedding space. These results are translated to the Laplace–Beltrami operator thanks to the Weitzenböck formula, for which a proof is also given.

Brownian loops and the central charge of a Liouville random surface

We explore the geometric meaning of the so-called zeta-regularized determinant of the Laplace–Beltrami operator on a compact surface, with or without boundary. We relate the (−c/2)th power of the determinant of the Laplacian to the appropriately regularized partition function of a Brownian loop soup of intensity c on the surface. This means that, in a certain sense, decorating a random surface by a Brownian loop soup of intensity c corresponds to weighting the law of the surface by the (−c/2)th power of the determinant of the Laplacian. Next, we introduce a method of regularizing a Liouville quantum gravity (LQG) surface (with some matter central charge parameter c) to produce a smooth surface. And we show that weighting the law of this random surface by the (−c′/2)th power of the Laplacian determinant has precisely the effect of changing the matter central charge from c to c+c′. Taken together with the earlier results, this provides a way of interpreting an LQG surface of matter central charge c as a pure LQG surface decorated by a Brownian loop soup of intensity c. Building on this idea, we present several open problems about random planar maps and their continuum analogs. Although the original construction of LQG is well defined only for c≤1, some of the constructions and questions also make sense when c>1.

Strengthened Belinskii theorem and its applications

The remarkable theorem of P. Belinskii is a deep underlying result in the variational calculus for quasiconformal maps. It is valid only for maps with small sufficiently regular Beltrami coefficients. We provide a global version of this theorem connected with the classical results on quasiconformal extensions of conformal maps and its new applications.

An efficient numerical method for reaction–diffusion equation on the general curved surfaces

In this paper, we propose an efficient numerical algorithm for reaction–diffusion equation on the general curved surface. The surface is discretized by a mesh consisting of triangular grids. The partial differential operators are defined based on the surface mesh and its dual surface polygonal tessellation. The proposed method has three advantages including intrinsic geometry, conservation law, and convergence property. The proposed method only needs the information of 1-ring of neighboring vertices for the divergence of a vector field and the Laplace–Beltrami operators, while the numerical conservation laws still hold. The proposed method avoids the global surface triangulation and its implementation is simple since we can explicitly define the Laplace–Beltrami operator by using the information of the neighborhood of each triangular grid. In order to obtain second-order temporal accuracy, we utilize the Crank–Nicolson formula to the reaction–diffusion system. The discrete system is solved by the biconjugate gradient stabilized method. The proposed algorithm is simple to implement and is second-order accurate both in space and time. Various numerical experiments are presented to demonstrate the efficiency of our algorithm.

Trudinger–Moser type inequalities with a symmetric conical metric and a symmetric potential

Let (M,g) be a closed Riemann surface, where the metric g has certain conical singularities at finite points. Suppose Γ is a group with elements of isometries acting on (M,g). In this paper, Trudinger–Moser inequalities involving Γ and the operador Δg+V are established, where Δg denotes the Laplace–Beltrami operator associated to g and the potential V:M→(0,∞) belongs to a class of symmetric and continuous functions. Moreover, via the method of blow-up analysis, the corresponding extremals are also obtained.

Estimates for the Locally Periodic Homogenization of the Riemann–Hilbert Problem for a Generalized Beltrami Equation

Estimates for the Locally Periodic Homogenization of the Riemann–Hilbert Problem for a Generalized Beltrami Equation

Multiscale structures in three species magnetoplasmas with two positive ions

The self-organization in a multi-ion plasma composed of electrons and two species of positively charged ions is investigated. It is shown that when canonical vorticities and velocities of all the plasma fluids are aligned, the magnetic field self-organizes to Quadruple Beltrami state (superposition of four Beltrami fields). The self-organized magnetic and velocity fields strongly depend on the relative strengths of the generalized vorticities, flows, inertia and densities of the plasma species. Thus, it is possible to generate a wide variety of multiscale magnetic field and flow structures. It is also shown that relaxed magnetic fields and velocities can vary on vastly different length scales simultaneously and are coupled together through singular perturbation generated by Hall effect. In this multi Beltrami self-organized states, then, the dynamo mechanism emerges naturally. The scale separation also suggests the heating of the plasma through a dissipative process. The work could be useful to study the dynamics and morphology of the multiscale magnetic field configurations in laboratory and astrophysical plasmas.

The p-Weil–Petersson Teichmüller Space and the Quasiconformal Extension of Curves

We consider the correspondence between the space of p-Weil–Petersson curves gamma on the plane and the p-Besov space of u=log gamma ' on the real line for p > 1. We prove that the variant of the Beurling–Ahlfors extension defined by using the heat kernel yields a holomorphic map for u on a domain of the p-Besov space to the space of p-integrable Beltrami coefficients. This in particular gives a global real-analytic section for the Teichmüller projection from the space of p-integrable Beltrami coefficients to the p-Weil–Petersson Teichmüller space.

Simple and practical method for the simulations of two-component PFC models for binary colloidal crystals on curved surfaces

Herein, we propose a practical numerical method for solving two-component phase-field crystal model with vacancy effect on curved surfaces and perform various numerical simulations. In the present work, the curved surfaces are defined by the zero level-set of signed distance functions. By adopting the pseudo-Neumann boundary condition and closest-point type method, we can extend the computation into a three-dimensional narrow band domain which contains the curved surfaces. The spatial discretization is performed by using the standard finite difference method and the seven-point Laplacian operator can be used to replace the surface Laplace–Beltrami operator. Therefore, the numerical implementation is simple and efficient. The semi-implicit discretization with stabilization term is considered to update the solutions in time. By simulating the pattern formations on curved surfaces with various shapes, we validate the good performance of the proposed method.

On Properties of Riemannian Metrics Associated with B-Elliptic Operators

In this paper, we consider a Riemannian metric in which the Laplace–Beltrami operator coincides with a B-elliptic operator up to a factor.

Quantum geometric confinement and dynamical transmission in Grushin cylinder

We classify the self-adjoint realizations of the Laplace–Beltrami operator minimally defined on an infinite cylinder equipped with an incomplete Riemannian metric of Grushin type, in the class of metrics yielding an infinite deficiency index. Such realizations are naturally interpreted as Hamiltonians governing the geometric confinement of a Schrödinger quantum particle away from the singularity, or the dynamical transmission across the singularity. In particular, we characterize all physically meaningful extensions qualified by explicit local boundary conditions at the singularity. Within our general classification we retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting one.