A Note on Div-Curl Lemma

Gala, Sadek

Displaying similar documents to “A Note on Div-Curl Lemma”

Estimates for differential operators of vector analysis involving $L^{1}$ -norm

Vladimir G. Maz'ya (2010)

Journal of the European Mathematical Society

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Atomic decomposition on Hardy-Sobolev spaces

Yong-Kum Cho, Joonil Kim (2006)

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As a natural extension of $L^{p}$ Sobolev spaces, we consider Hardy-Sobolev spaces and establish an atomic decomposition theorem, analogous to the atomic decomposition characterization of Hardy spaces. As an application, we deduce several embedding results for Hardy-Sobolev spaces.

Hardy and Hardy-Sobolev Spaces on Strongly Lipschitz Domains and Some Applications

Xiaming Chen, Renjin Jiang, Dachun Yang (2016)

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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...

Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^{1}$

A. Fiorenza, C. Sbordone (1998)

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We prove an existence and uniqueness theorem for the elliptic Dirichlet problem for the equation div a(x,∇u) = f in a planar domain Ω. Here $f \in L^{1} (Ω)$ and the solution belongs to the so-called grand Sobolev space $W_{0}^{1, 2)} (Ω)$ . This is the proper space when the right hand side is assumed to be only $L^{1}$ -integrable. In particular, we obtain the exponential integrability of the solution, which in the linear case was previously proved by Brezis-Merle and Chanillo-Li.

A new characterization of the Sobolev space

Piotr Hajłasz (2003)

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The purpose of this paper is to provide a new characterization of the Sobolev space $W^{1, 1} (ℝ ⁿ)$ . We also show a new proof of the characterization of the Sobolev space $W^{1, p} (ℝ ⁿ)$ , 1 ≤ p < ∞, in terms of Poincaré inequalities.

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Jean Bourgain, Haïm Brezis (2007)

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We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension $n$ , with data in $L^{1}$ . We also present related results concerning differential forms with coefficients in the limiting Sobolev space $W^{1, n}$ .

An inclusion operator in Hardy spaces on the unit ball in $C^{N}$

M. Jevtić (1988)

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Factorization of sequences in discrete Hardy spaces

Santiago Boza (2012)

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The purpose of this paper is to obtain a discrete version for the Hardy spaces $H^{p} (ℤ)$ of the weak factorization results obtained for the real Hardy spaces $H^{p} (ℝ ⁿ)$ by Coifman, Rochberg and Weiss for p > n/(n+1), and by Miyachi for p ≤ n/(n+1). It represents an extension, in the one-dimensional case, of the corresponding result by A. Uchiyama who obtained a factorization theorem in the general context of spaces X of homogeneous type, but with some restrictions on the measure that exclude the case...

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Joseph A. Cima, Angeliki Kazas, Michael I. Stessin (2003)

Studia Mathematica

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With φ an inner function and $M_{φ}$ the multiplication operator on a given Hardy space it is known that for any given function f in the Hardy space we may use the Wold decomposition to obtain a factorization of the given f (not the Riesz factorization). This new factorization has been shown to be useful in the study of commutants of Toeplitz operators. We study the smoothness of each factor of this factorization. We show in some cases that the factors lie in the same Hardy space (or smoothness...

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Estimates for the Hardy-Littlewood maximal function on the Heisenberg group

Jacek Zienkiewicz (2005)

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We prove the dimension free estimates of the $L^{p} \to L^{p}$ , 1< p ≤ ∞, norms of the Hardy-Littlewood maximal operator related to the optimal control balls on the Heisenberg group ℍⁿ.

On limiting embeddings of Besov spaces

V. I. Kolyada, A. K. Lerner (2005)

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We investigate the classical embedding $B_{p, θ}^{s} \subset B_{q, θ}^{s - n (1 / p - 1 / q)}$ . The sharp asymptotic behaviour as s → 1 of the operator norm of this embedding is found. In particular, our result yields a refinement of the Bourgain, Brezis and Mironescu theorem concerning an analogous problem for the Sobolev-type embedding. We also give a different, elementary proof of the latter theorem.

Extending Hardy fields by non- $^{\infty}$ -germs

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Annales Polonici Mathematici

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For a large class of Hardy fields their extensions containing non- $^{\infty}$ -germs are constructed. Hardy fields composed of only non- $^{\infty}$ -germs, apart from constants, are also considered.

On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion

Piotr Mucha, Wojciech Zajączkowski (2000)

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The local-in-time existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion is proved. We show the existence of solutions with lowest possible regularity for this problem such that $u \in W_{r}^{2, 1} ({\tilde{Ω}}^{T})$ with r>3. The existence is proved by the method of successive approximations where the solvability of the Cauchy-Neumann problem for the Stokes system is applied. We have to underline that in the $L_{p}$ -approach the Lagrangian coordinates must be used....

Doubly commuting submodules of the Hardy module over polydiscs

Jaydeb Sarkar, Amol Sasane, Brett D. Wick (2013)

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In this note we establish a vector-valued version of Beurling’s theorem (the Lax-Halmos theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the “weak” completion problem in $H^{\infty} (ⁿ)$ .

Stationary states for a two-dimensional singular Schrödinger equation

Paolo Caldiroli, Roberta Musina (2001)

Bollettino dell'Unione Matematica Italiana

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In questo articolo studiamo problemi di Dirichlet singolari, lineari e semilineari, della forma ${|x|}^{2} Δ u = f (u)$ in $Ω$ , $u = 0$ su $\partial Ω$ , dove $Ω$ è un dominio in $R^{2}$ e $f (u) = λ u$ o $f (u) = λ u + {|u|}^{p - 2} u$ con $p > 2$ (o nonlinearità più generali). In tali problemi bidimensionali emergono alcune difficoltà a causa della non validità della disuguaglianza di Hardy in $R^{2}$ e a causa delle invarianze dell'equazione $- {|x|}^{2} Δ u = f (u)$ . Pertanto opportune condizioni su $λ$ e $Ω$ sono necessarie al fine di garantire l'esistenza di una soluzione positiva. Per esempio, se $Γ_{0}$ è una curva...

$H^{p}$ Sobolev spaces

Robert S. Strichartz (1990)

Colloquium Mathematicae

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Hardy-Sobolev Inequalities for Hessian Integrals

Nunzia Gavitone (2007)

Bollettino dell'Unione Matematica Italiana

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Using appropriate symmetrization arguments, we prove the Hardy-Sobolev type inequalities for Hessian Integrals which extend the classical results, well known for Sobolev functions. For such inequalities the value of the best constant is given. Finally we give an improvement of these inequalities by adding a second term that, involves another singular weight which is a suitable negative power of $\log(|x|)$ .

The isoperimetric inequality in the Hardy class $H^{1}$

M. Mateljević (1979)

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Sharp embedding results for spaces of smooth functions with power weights

Martin Meyries, Mark Veraar (2012)

Studia Mathematica

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We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on $ℝ^{d}$ , equipped with power weights $w (x) = {| x |}^{γ}$ , γ > -d. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.