Beltrami Operator Research Articles

Functional Inequalities on Symmetric Spaces of Noncompact Type and Applications

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein–Weiss inequality, also known as a weighted Hardy–Littlewood–Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein–Weiss inequality, we deduce Hardy–Sobolev, Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace–Beltrami operator on symmetric spaces.

Quasiconformal deformation of the chordal Loewner driving function and first variation of the Loewner energy

We derive the variational formula of the Loewner driving function of a simple chord under infinitesimal quasiconformal deformations with Beltrami coefficients supported away from the chord. As an application, we obtain the first variation of the Loewner energy of a Jordan curve, defined as the Dirichlet energy of its driving function. This result gives another explanation of the identity between the Loewner energy and the universal Liouville action introduced by Takhtajan and Teo, which has the same variational formula. We also deduce the variation of the total mass of SLE8/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {SLE}_{8/3}$$\\end{document} loops touching the Jordan curve under quasiconformal deformations.

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} asymptotic convergence without the assumption of bi-K-invariance.

Нахождение погрешности усреднения уравнения Бельтрами на плоскости методом сдвига В.В. Жикова

Нахождение погрешности усреднения уравнения Бельтрами на плоскости методом сдвига В.В. Жикова

Perturbing isoradial triangulations

We consider an infinite planar Delaunay graph \mathbf{G}_{\epsilon} which is obtained by locally deforming the coordinate embedding of a general isoradial graph \mathbf{G}_{\mathrm{cr}} , with respect to a real deformation parameter \epsilon . Using Kenyon’s exact and asymptotic results for the critical Green’s function on an isoradial graph, we calculate the leading asymptotics of the first- and second-order terms in the perturbative expansion of the log-determinant of the Laplace–Beltrami operator \Delta(\epsilon) , the David–Eynard Kähler operator \mathcal{D}(\epsilon) , and the conformal Laplacian {\underline{\boldsymbol{\Delta}}}(\epsilon) on the deformed Delaunay graph \mathbf{G}_{\epsilon} . We show that the scaling limits of the second-order bi-local term for both the Laplace–Beltrami and David–Eynard Kähler operators exist and coincide, with a shared value independent of the choice of the initial isoradial graph \mathbf{G}_{\mathrm{cr}} . Our results allow us to define a discrete analog of the stress-energy tensor for each of the three operators. Furthermore, we can identify a central charge ( c=-2 ) in the case of both the Laplace–Beltrami and David–Eynard Kähler operators. While the scaling limit is consistent with the stress-energy tensor and the value of the central charge for the Gaussian free field (GFF), the discrete central charge value of c=-2 for the David–Eynard Kähler operator is, however, at odds with the value of c=-26 expected by Polyakov’s theory of 2D quantum gravity; moreover, there are problems with convergence of the scaling limit of the discrete stress-energy tensor for the David–Eynard Kähler operator. The second-order bi-local term for the conformal Laplacian involves anomalous terms corresponding to the creation of discrete curvature dipoles in the deformed Delaunay graph \mathbf{G}_{\epsilon} ; we examine the difficulties in defining a convergent scaling limit in this case. Connections with some discrete statistical models at criticality are explored.

Spectrum of the Laplace–Beltrami Operator on Antisymmetric Functions

Spectrum of the Laplace–Beltrami Operator on Antisymmetric Functions

On the Dirichlet problem for beltrami equations with sources in simply connected domains

In this paper, we present our recent results on the solvability of the Dirichlet problem Reω(z) → φ (ζ) as z → ζ, z∈ D, ζ∈ ∂D, with continuous boundary data φ: ∂D ???? R for degenerate Beltrami equations ωz =µ(z)ω2 + σ(z), |µ(z) ˂ 1 a.e., with sources σ: D → C that belong to the class Lp (D), p ˃ 2, and have compact supports in D. In the case of locally uniform ellipticity of the equations, we formulate, in arbitrary simply connected domains D of the complex plane C a series of eff ective integral criteria of the type of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz on singularities of the equations at the boundary for existence of locally Hölder continuous solutions in the class W1.2loc (D) of the Dirichlet problem with their representation through the so-called generalized analytic functions with sources.

Self‐organization of earth's inner magnetospheric multi‐ion plasma

AbstractThe self‐organization of a magnetized multi‐ion plasma, composed of inertialess electrons and inertial H+, He+, and O+ ions, leads to the formation of quadruple Beltrami (QB) field structures. The QB self‐organized state is a linear combination of four single Beltrami fields, and it is a non‐force‐free state that shows strong magnetofluid coupling. Moreover, the QB state is characterized by four relaxed state structures of different length scales. The investigation reveals that the generalized helicities of plasma species and the densities of ion species have a significant impact on the characteristics of the self‐organized vortices in the QB state. The study also highlights the potential consequences of QB field structures on earth's inner magnetosphere, including diamagnetic and paramagnetic trends as well as heating effects resulting from disparate length scales.

Turing pattern formation on the sphere is robust to the removal of a hole

The formation of buds on the cell membrane of budding yeast cells is thought to be driven by reactions and diffusion involving the protein Cdc42. These processes can be described by a coupled system of partial differential equations known as the Schnakenberg system. The Schnakenberg system is known to exhibit diffusion-driven pattern formation, thus providing a mechanism for bud formation. However, it is not known how the accumulation of bud scars on the cell membrane affect the ability of the Schnakenberg system to form patterns. We have approached this problem by modelling a bud scar on the cell membrane with a hole on the sphere. We have studied how the spectrum of the Laplace–Beltrami operator, which determines the resulting pattern, is affected by the size of the hole, and by numerically solving the Schnakenberg system on a sphere with a hole using the finite element method. Both theoretical predictions and numerical solutions show that pattern formation is robust to the introduction of a bud scar of considerable size, which lends credence to the hypothesis that bud formation is driven by diffusion-driven instability.

Quadruple Beltrami field structures in electron–positron multi-ion plasma

Abstract A quadruple Beltrami (QB) equilibrium state for a four-component plasma that consists of inertial electrons, positrons, lighter positive (H +) ions and heavier negative ions O 2 − $\left({\mathrm{O}}_{2}^{-}\right)$ is derived and investigated. The QB relaxed state is a linear superposition of four distinct single Beltrami fields and provides the possibility of the formation of four self-organized vortices of different length scales. In addition, robust magnetofluid coupling characterizes this non-force-free state. The analysis of the QB state also shows that by adjusting the generalized helicities and densities of plasma species, the formation of multiscale structures as well as the paramagnetic and diamagnetic behavior of the relaxed state can be controlled.

Unconditional energy-stable method for the Swift–Hohenberg equation over arbitrarily curved surfaces with second-order accuracy

This paper presents an unconditional energy-stable method for the Swift-Hohenberg equation over arbitrarily curved surfaces. We directly define the Laplace–Beltrami operator on the triangular mesh and its dual mesh, which are the discretizations of regular surface. The direct discretization method has several advantages including intrinsic geometry, convergence property, and mass conservation. A second-order temporal accuracy method has been proposed with the Crank-Nicolson type formulation. We have applied the stabilized splitting method to solve the governing system. Our proposed scheme is easy to implement since the two elliptic equations are both linear with low computational burden. The proposed scheme is provable to be Unconditional energy-stable. Several computational tests are conducted to demonstrate that the numerical method is efficient.

APPLICATION OF PINN AND THE METHOD OF DIFFERENTIAL CONSTRUCTION OF ADAPTIVE ONE-DIMENSIONAL COMPUTATIONAL GRIDS

This paper presents an approach to constructing adaptive one-dimensional computational grids using the Beltrami equation and Physics-Informed Neural Networks (PINNs). The main focus is on exploring the potential for precise control of grid node density through the control function ω(s), which allows the grid to adapt to the local features of the problem. The Beltrami equation used, being a key component of the method, regulates the distribution of nodes by modifying the function’s derivatives depending on the values of the control function. The effectiveness of this approach is demonstrated through examples involving one and two regions of node clustering. The results showed that the PINN method combined with the Beltrami equation allows for the creation of computational grids with a high degree of adaptation to given conditions, providing detailed modeling in critical regions. This approach has advantages over traditional numerical methods, as integrating physical laws in the grid construction process minimizes numerical errors and improves modeling accuracy. The use of neural networks offers flexibility in model tuning and the ability to account for complex nonlinear dependencies. The discussion of the results highlights the potential of using PINNs for adaptive grid construction in various fields requiring precise and efficient modeling. In conclusion, this study confirms that the combination of the Beltrami equation and PINNs is a powerful tool for adaptive grid construction, opening new possibilities for numerical modeling of complex physical processes.

FINDING THE SOLUTION OF THE ELLIPTIC SYSTEM OF THE FIRSTORDER IN THE FORM OF A VECTOR-FUNCTION

The relationship between Douglis analytic functions and the analytic functions of the Laplace equation for elliptic equations with constant coefficients serves the same purpose. In this sense, some boundary value problems solved by Douglis using analytic functions remain relevant today. Douglis’s Bitsadze-Samarsky problem on analytic functions was reconsidered by A.P. Soldatov. Meanwhile, for certain types of matrices and regions, V.G. Nikolaev proved the existence and uniqueness of the Schwarz problem solution. In any region G of the complex plane C, we considered a first order system y J x =F, where the eigenvalues of the constant matrix J Cl l lie in the upper half-plane, Imv > 0. In the case where l = 1 J = J(z) is continuous and ImJ > 0, we obtain the Beltrami equation. In this paper, to find the solution of elliptic equation with constant coefficient, the corresponding J-analytic system was constructed by choosing the matrix J of size 2 x 2andJ has no real eigenvalues. In the postcomponential notation, the system consists of two differential equations depending on the variable z. Thus, the solution of this equation was found using the vector- function.

Investigating Helical Hypersurfaces Within 7‐Dimensional Euclidean Space

Differential geometry of a kind of helical hypersurface family that depends on six parameters within the seven‐dimensional Euclidean space is explored. The curvatures of these hypersurfaces are determined, and their minimality is examined, along with illustrative examples being provided. Lastly, the Laplace–Beltrami operator for that kind of hypersurfaces is calculated.

RIGHT CONOID HYPERSURFACES IN FOUR-SPACE

The right conoid hypersurfaces in the four-dimensional Euclidean space $\mathbb{E}^{4}$ are introduced. The matrices corresponding to the fundamental form, Gauss map, and shape operator of these hypersurfaces are calculated. By utilizing the Cayley--Hamilton theorem, the curvatures of these specific hypersurfaces are determined. Furthermore, the conditions for minimality are presented. Additionally, the Laplace--Beltrami operator of this family is computed, and some examples are provided.

Fractional anisotropic Calderón problem on complete Riemannian manifolds

We prove that the metric tensor [Formula: see text] of a complete Riemannian manifold is uniquely determined, up to isometry, from the knowledge of a local source-to-solution operator associated with a fractional power of the Laplace–Beltrami operator [Formula: see text]. Our result holds under the condition that the metric tensor [Formula: see text] is known in an arbitrary small subdomain. We also consider the case of closed manifolds and provide an improvement of the main result in [A. Feizmohammadi, T. Ghosh, K. Krupchyk and G. Uhlmann, Fractional anisotropic Calderón problem on closed Riemannian manifolds, preprint (2021); arXiv:2112.03480].

Uniform Resolvent Estimates for Laplace–Beltrami Operator on the Flat Euclidean Cone

Uniform Resolvent Estimates for Laplace–Beltrami Operator on the Flat Euclidean Cone

Nonlinear Beltrami equation: lower estimates of Schwarz lemma’s type

Abstract We study a nonlinear Beltrami equation $f_\theta =\sigma \,|f_r|^m f_r$ in polar coordinates $(r,\theta ),$ which becomes the classical Cauchy–Riemann system under $m=0$ and $\sigma =ir.$ Using the isoperimetric technique, various lower estimates for $|f(z)|/|z|, f(0)=0,$ as $z\to 0,$ are derived under appropriate integral conditions on complex/directional dilatations. The sharpness of the above bounds is illustrated by several examples.

Strictly positive definite non-isotropic kernels on two-point homogeneous manifolds: the asymptotic approach

We present sufficient conditions for a family of positive definite kernels on a compact two-point homogeneous space to be strictly positive definite based on their expansion in eigenfunctions of the Laplace–Beltrami operator. We also present a characterisation of this kernel class. The family analyzed is a generalization of the isotropic kernels and the case of a real sphere is analyzed in details.

Twisted Hypersurfaces in Euclidean 5-Space

The twisted hypersurfaces x with the (0,0,0,0,1) rotating axis in five-dimensional Euclidean space E5 is considered. The fundamental forms, the Gauss map, and the shape operator of x are calculated. In E5, describing the curvatures by using the Cayley–Hamilton theorem, the curvatures of hypersurfaces x are obtained. The solutions of differential equations of the curvatures of the hypersurfaces are open problems. The umbilically and minimality conditions to the curvatures of x are determined. Additionally, the Laplace–Beltrami operator relation of x is given.