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In this paper, we study certain inequalities and a related result for weighted Sobolev spaces on Hölder-α domains, where the weights are powers of the distance to the boundary. We obtain results regarding the divergence equation's solvability, and the improved Poincaré, the fractional Poincaré, and the Korn inequalities. The proofs are based on a local-to-global argument that involves a kind of atomic decomposition of functions and the validity of a weighted discrete Hardy-type inequality on trees. The novelty of our approach lies in the use of this weighted discrete Hardy inequality and a sufficient condition that allows us to study the weights of our interest. As a consequence, the assumptions on the weight exponents that appear in our results are weaker than those in the literature. [ABSTRACT FROM AUTHOR]

We consider the boundary value problem associated with the divergence operator on a bounded regular subset of R n , with homogeneous Dirichlet boundary condition. We prove the existence of a classical solution under slight assumptions on the datum. [ABSTRACT FROM AUTHOR]

In the following note, we focus on the problem of existence of continuous solutions vanishing at infinity to the equation div v = f for f ∈ L n (ℝ n ) and satisfying an estimate of the type || v ||∞ ⩽ C || f || n for any f ∈ L n (ℝ n ), where C > 0 is related to the constant appearing in the Sobolev–Gagliardo–Nirenberg inequality for functions with bounded variation (BV functions). [ABSTRACT FROM AUTHOR]

Let Ω ⊂ Rn be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, Hpr (Ω)and Hpz (Ω), and Hardy-Sobolev spaces, H1,pr (Ω) and H1,pz,0 (Ω) on , for p ∈ ( n/n+1, 1]. The authors establish grand maximal function characterizations of these spaces. As applications, the authors obtain some div-curl lemmas in these settings and, when is a bounded Lipschitz domain, the authors prove that the divergence equation div u = f for f ∈ Hpz (Ω) is solvable in H1,pz,0 (Ω) with suitable regularity estimates.

This article continues a previous study (Int. J. Quantum Chem.114, 1645 (2014)) in which is presented a theory that discusses the possibility to induce a novel field – to be called molecular field – via the interaction of an external electro-magnetic (EM) field and a molecular system. Assuming the molecular system is made up of two coupled adiabatic states the theory leads from three time-space curl equations and one time-space divergence equation to a set of decoupled wave equations usually encountered for fields. In the present study, wave equations are derived for an external field having two features: (1) the field intensity is relativelyhigh enough; (2) the duration of the interaction isshort enough. For this situation, the study reveals that the just mentioned interaction creates two fields that coexist within a molecule: one is just a scalar EM field essentially identical to the external EM field and the other, a vectorial field, produced by the non-adiabatic coupling terms. In addition, we mention that the wave velocities related to these two fields are identical to the wave velocity, c, of the external EM field. [ABSTRACT FROM PUBLISHER]

Divergence equation is considered in a thin-tube structure, connected finite union of thin finite cylinders (in the case of two dimensions, respectively, thin rectangles) where the ratio of the diameter and the heights of the cylinders is the small parameter. Various norms of the solution of divergence equation are estimated over the norms of the right-hand side. The exact dependence of constants in these estimates on the small parameteris obtained. [ABSTRACT FROM PUBLISHER]

In this paper, we study the weighted Korn inequality on some irregular domains, e.g., s -John domains and domains satisfying quasihyperbolic boundary conditions. Examples regarding sharpness of the Korn inequality on these domains are presented. Moreover, we show that Korn inequalities imply certain Poincaré inequality. [ABSTRACT FROM AUTHOR]

When an external, time-dependent field interacts with a molecular system various phenomena may take place. However, concentrating on a region close enough to a point of conical intersection, we find that this external field builds up a field similar to an electromagnetic field formed on the one hand by Field-Dressed nonadiabatic coupling terms which are reminiscent of the Maxwell-Lorentz Vector potentials, and on the other hand via a scalar potential formed by the dipole-interaction with an external field. In this article, we show that this new field, to be termed Molecular Field, is characterized by several spatial and space-time Field-Dressed Curl equations and one, single, space-time Field-Dressed Divergence equation. These equations are then shown to yield, just as in the general theory of electromagnetism, the corresponding Field-Dressed Wave Equations. This achievement could be materialized employing the (1,2) antisymmetric matrix elements of any of the 2×2 dimensional Field-Dressed nonadiabatic coupling matrices. © 2014 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

Divergence and vorticity are well known to be geometrically invariant quantities in that their mathematical forms are independent of the orientation of the coordinate axes. Various other functions of the elements of the horizontal velocity gradient tensor are invariants in the same sense: examples are the resultant deformation and the determinant and Frobenius norm of the tensor. A brief account of these quadratic invariants is given, including expressions relating them to divergence and vorticity and to one another, and noting their occurrence in the divergence equation. Assuming shallow-water dynamics with background rotation, time-evolution equations for the resultant deformation and the other quadratic invariants are derived and compared. None rivals the vorticity and potential vorticity equations for compactness, but each may be written quite concisely in terms of familiar quantities. Corresponding time-evolution equations under quasi-geostrophic shallow-water dynamics are also derived, and lead to a simple prognostic equation for the ageostrophic vorticity. [ABSTRACT FROM AUTHOR]

Abstract: Let be a bounded simply connected domain. We show that, for a fixed (every) , the divergence equation is solvable in for every , if and only if is a John domain, if and only if the weighted Poincaré inequality holds for some (every) . This gives a positive answer to a question raised by Russ (2013) in the case of bounded simply connected domains. In higher dimensions similar results are proved under some additional assumptions on the domain in question. [Copyright &y& Elsevier]

We consider the equation $$\text{ div}\,\mathbb{Y }=f$$, with $$f$$ a zero average function on the torus $$\mathbb{T }^d$$. In their seminal paper, Bourgain and Brezis [J Am Math Soc 16(2):393-426, (electronic)] proved the existence of a solution $$\mathbb{Y }\in W^{1,d}\cap L^\infty $$ for a datum $$f\in L^d$$. We extend their result to the critical Sobolev spaces $$W^{s,p}$$ with $$(s+1)p=d$$ and $$p\ge 2$$. More generally, we prove a similar result in the scale of Triebel-Lizorkin spaces. We also consider the equation $$\text{ div} \,\mathbb{Y }=f$$ in a bounded domain $$\varOmega $$ subject to zero Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

Abstract: We aim at extending the existence theory for the equation in a bounded or exterior domain with homogeneous Dirichlet boundary conditions, to a class of solutions which need not have a trace at the boundary. Typically, the weak solutions that we shall consider will belong to some Besov space with . After generalizing the notion of a solution for this equation, we propose an explicit construction by means of the classical Bogovskiĭ formula. This construction enables us to keep track of a “marginal” information about the trace of solutions. In particular, it ensures that the trace is zero if f is smooth enough. We expect our approach to be of interest for the study of rough solutions to systems of fluid mechanics. [Copyright &y& Elsevier]

We study the solvability of the divergence equation in weighted spaces and Lebesgue spaces with variable exponents, where the weights are so called Muckenhoupt weights. The question of constructing divergence free test functions, which can be used for problems arising in fluid dynamics, is also addressed. The approach is based on an explicit representation formula for solutions of the divergence equation due to Bogovskiĭ and the theory of singular integral operators. The developed methods are used to prove an existence result for fluids which satisfy a p(·)-growth condition. [ABSTRACT FROM AUTHOR]