Weighted inequalities for gradients on non-smooth domains

Caroline Sweezy; J. Michael Wilson

Displaying similar documents to “Weighted inequalities for gradients on non-smooth domains”

A Littlewood-Paley type inequality with applications to the elliptic Dirichlet problem

Caroline Sweezy (2007)

Annales Polonici Mathematici

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Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to $L u = d i v \vec{f}$ in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the $L^{q} (Ω, d μ)$ norm of |∇u| is dominated by the $L^{p} (Ω, d v)$ norms of $d i v \vec{f}$ and $| \vec{f} |$ . If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.

Existence and regularity of solutions of some elliptic system in domains with edges

Wojciech M. Zajączkowski

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CONTENTS1. Introduction.......................................................................52. Notation and auxiliary results............................................93. Statement of the problem (1.1)-(1.3)..............................204. The problem (3.14).........................................................225. Auxiliary results in $D_{ϑ}$ ...............................................346. Existence of solutions of (3.14) in $H_{μ}^{k} (D_{ϑ})$ ............417. Green function................................................................528....

On some $L^{p}$ -estimates for solutions of elliptic equations in unbounded domains

Sara Monsurrò, Maria Transirico (2015)

Mathematica Bohemica

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In this review article we present an overview on some a priori estimates in $L^{p}$ , $p > 1$ , recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two $L^{p}$ -bounds, $p > 2$ , for...

Lipschitz continuity in Muckenhoupt 𝓐₁ weighted function spaces

Dorothee D. Haroske (2011)

Banach Center Publications

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We study continuity envelopes of function spaces $B_{p, q}^{s} (ℝ ⁿ, w)$ and $F_{p, q}^{s} (ℝ ⁿ, w)$ where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

Isometric composition operators on weighted Dirichlet space

Shi-An Han, Ze-Hua Zhou (2016)

Czechoslovak Mathematical Journal

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We investigate isometric composition operators on the weighted Dirichlet space $𝒟_{α}$ with standard weights ${(1 - | z |}^{2})^{α}$ , $α > - 1$ . The main technique used comes from Martín and Vukotić who completely characterized the isometric composition operators on the classical Dirichlet space $𝒟$ . We solve some of these but not in general. We also investigate the situation when $𝒟_{α}$ is equipped with another equivalent norm.

Solutions to the equation div u = f in weighted Sobolev spaces

Katrin Schumacher (2008)

Banach Center Publications

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We consider the problem div u = f in a bounded Lipschitz domain Ω, where f with $\int_{Ω} f = 0$ is given. It is shown that the solution u, constructed as in Bogovski’s approach in [1], fulfills estimates in the weighted Sobolev spaces $W_{w}^{k, q} (Ω)$ , where the weight function w is in the class of Muckenhoupt weights $A_{q}$ .

Weighted variable $L^{p}$ integral inequalities for the maximal operator on non-increasing functions

C. J. Neugebauer (2009)

Studia Mathematica

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Let $B_{p}$ be the Ariõ-Muckenhoupt weight class which controls the weighted $L^{p}$ -norm inequalities for the Hardy operator on non-increasing functions. We replace the constant p by a function p(x) and examine the associated $L^{p (x)}$ -norm inequalities of the Hardy operator.

Existence and nonexistence of solutions for a quasilinear elliptic system

Qin Li, Zuodong Yang (2015)

Annales Polonici Mathematici

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By a sub-super solution argument, we study the existence of positive solutions for the system ⎧ $- Δ_{p} u = a ₁ (x) F ₁ (x, u, v)$ in Ω, ⎪ $- Δ_{q} v = a ₂ (x) F ₂ (x, u, v)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in $ℝ^{N}$ with smooth boundary or $Ω = ℝ^{N}$ . A nonexistence result is obtained for radially symmetric solutions.

Finite element variational crimes in the case of semiregular elements

Alexander Ženíšek (1996)

Applications of Mathematics

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The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain $Ω$ whose boundary $\partial Ω$ is formed by two circles $Γ_{1}$ , $Γ_{2}$ with the same center $S_{0}$ and radii $R_{1}$ , $R_{2} = R_{1} + ϱ$ , where $ϱ ≪ R_{1}$ . On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for $u = 0$ are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus...

Existence of a renormalized solution of nonlinear degenerate elliptic problems

Youssef Akdim, Chakir Allalou (2014)

Applicationes Mathematicae

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We study a general class of nonlinear elliptic problems associated with the differential inclusion $β (u) - d i v (a (x, D u) + F (u)) ∋ f$ in Ω where $f \in L^{\infty} (Ω)$ . The vector field a(·,·) is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in function spaces we prove existence of renormalized solutions for general $L^{\infty}$ -data.

Continuous pluriharmonic boundary values

Per Åhag, Rafał Czyż (2007)

Annales Polonici Mathematici

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Let $D_{j}$ be a bounded hyperconvex domain in $ℂ^{n_{j}}$ and set $D = D ₁ \times \dots \times D_{s}$ , j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

Hölder continuity of bounded generalized solutions for some degenerated quasilinear elliptic equations with natural growth terms

Salvatore Bonafede (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove the local Hölder continuity of bounded generalized solutions of the Dirichlet problem associated to the equation $\sum_{i = 1}^{m} \frac{\partial}{\partial x_{i}} a_{i} (x, u, \nabla u) - c_{0} {| u |}^{p - 2} u = f (x, u, \nabla u),$ assuming that the principal part of the equation satisfies the following degenerate ellipticity condition $λ (| u |) \sum_{i = 1}^{m} a_{i} (x, u, η) η_{i} \geq ν (x) {| η |}^{p},$ and the lower-order term $f$ has a natural growth with respect to $\nabla u$ .

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Suying Liu, Minghua Yang (2018)

Czechoslovak Mathematical Journal

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Let $L$ be a non-negative self-adjoint operator acting on $L^{2} (ℝ^{n})$ satisfying a pointwise Gaussian estimate for its heat kernel. Let $w$ be an $A_{r}$ weight on $ℝ^{n} \times ℝ^{n}$ , $1 < r < \infty$ . In this article we obtain a weighted atomic decomposition for the weighted Hardy space $H_{L, w}^{p} (ℝ^{n} \times ℝ^{n})$ , $0 < p \leq 1$ associated to $L$ . Based on the atomic decomposition, we show the dual relationship between $H_{L, w}^{1} (ℝ^{n} \times ℝ^{n})$ and ${BMO}_{L, w} (ℝ^{n} \times ℝ^{n})$ .

Moser-Trudinger and logarithmic HLS inequalities for systems

Itai Shafrir, Gershon Wolansky (2005)

Journal of the European Mathematical Society

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We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere $S^{2}$ , on a bounded domain $Ω \subset ℝ^{2}$ and on all of $ℝ^{2}$ . In some cases we also address the question of existence of minimizers.

Some weighted norm inequalities for a one-sided version of $g *_{λ}$

L. de Rosa, C. Segovia (2006)

Studia Mathematica

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We study the boundedness of the one-sided operator $g ⁺_{λ, φ}$ between the weighted spaces $L^{p} (M ¯ w)$ and $L^{p} (w)$ for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of $g ⁺_{λ, φ}$ . For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of $g ⁺_{λ, φ}$ from $L^{p} ({(M ¯)}^{[p / 2] + 1} w)$ to $L^{p} (w)$ , where ${(M ¯)}^{k}$ denotes the operator M¯ iterated k times.

Some Banach spaces of Dirichlet series

Maxime Bailleul, Pascal Lefèvre (2015)

Studia Mathematica

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The Hardy spaces of Dirichlet series, denoted by $^{p}$ (p ≥ 1), have been studied by Hedenmalm et al. (1997) when p = 2 and by Bayart (2002) in the general case. In this paper we study some $L^{p}$ -generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces, denoted $^{p}$ and $ℬ^{p}$ . Each could appear as a “natural” way to generalize the classical case of the unit disk. We recover classical properties of spaces of analytic functions: boundedness of point evaluation, embeddings...