Related papers: Curl and gradient of divergence operators in Space…

The properties of the vortex and the gradient of divergence operators ( $ \text{rot}$ and $\nabla \text{div}$ ) are studied in the space $ \mathbf {L}_2 (G) $ in a bounded domain $ G \subset \textrm {R}^3 $ with a smooth boundary $ \Gamma$…

The author studies the structure of space $ \mathbf {L} _ {2} (G) $ of vector-valued functions that are square integrable in a bounded connected domain $ G $ of the three-dimensional space with a smooth boundary and the role of gradient…

The author studies structure of space $\mathbf{L}_{2}(G)$ of vectors - functions, which are integrable with a square of the module on the bounded domain $G $of three-dimensional space with smooth boundary, and role of the gradient of…

In a bounded domain $G$ with smooth border studied boundary value and spectral problems for operators of the rotor (vortex) and the gradient of the divergence $+\lambda\,I$ in the Sobolev spaces. For $\lambda\neq 0$ these operators are…

The properties of curl and gradient of divergence operators in the domain $G$ of three-dimensional space are described. The self-conjugacy of these operators in the subspaces $\mathbf{L}_{2}(G) $ and the basis property of the system of…

In this work, the divergence and curl operators are obtained using the coordinate free non rigid basis formulation of differential geometry. Although the authors have attempted to keep the presentation self-contained as much as possible,…

In different branches of physics, we frequently deal with vector del operator ($\vec{\nabla}$). This del operator is generally used to find curl or divergence of a vector function or gradient of a scalar function. In many important cases,…

Rollings of reductive homogeneous spaces are investigated. More precisely, for a reductive homogeneous space $G / H$ with reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$, we consider rollings of $\mathfrak{m}$ over…

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms, whose coefficients lie in critical…

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with…

In this paper we study under what boundary conditions the inequality $$\|\nabla\omega\|_{L^2(\Omega)}^2\leq C\left(\|{\rm curl}\omega\|_{L^2(\Omega)}^2+ \|{\rm div}\omega\|_{L^2(\Omega)}^2+\|\omega\|_{L^2(\Omega)}^2\right) $$ holds true. It…

We propose a definition of the curl of a vector field X on a finite simple graph as the projection of X onto the orthogonal complement of circulation-free vector fields, where a vector field is circulation-free provided its line integral…

Building on arXiv:1902.03807, this paper develops a unifying study on the boundedness properties of several representative classes of hybrid operators, i.e. operators that enjoy both zero and non-zero curvature features. Specifically, via…

We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions its eigenvalues are…

In the last decades, many mathematicians have studied the {\em curl operator} on compact (both with or without empty boundary) three-manifolds, mainly the behaviour of its spectrum and some iso\-pe\-ri\-me\-tric problems associated with it.…

In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space $(M_g, \omega_{WP})$ of curves with genus $g>1$ has…

Let $k$ be an algebraically closed field of characteristic 0 and let $A$ be a finitely generated $k$-algebra that is a domain whose Gelfand-Kirillov dimension is in $[2,3)$. We show that if $A$ has a nonzero locally nilpotent derivation…

In this work, we investigate the system formed by the equations $\text{div } \vec w=g_0$ and $\text{curl } \vec w=\vec g$ in bounded star-shaped domains of $\mathbb{R}^3$. A Helmholtz-type decomposition theorem is established based on a…

In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional…

Equipped with the L^2-distortion distance, the space "X" of all metric measure spaces (X,d,m) is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of…