On the solvability of Dirichlet problem for the weighted p-Laplacian

Dominik Mielczarek; Jerzy Rydlewski; Ewa Szlachtowska

Displaying similar documents to “On the solvability of Dirichlet problem for the weighted p-Laplacian”

Characterization of associate spaces of weighted Lorentz spaces with applications

Amiran Gogatishvili, Luboš Pick, Filip Soudský (2014)

Studia Mathematica

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We characterize associate spaces of weighted Lorentz spaces GΓ(p,m,w) and present some applications of this result including necessary and sufficient conditions for a Sobolev-type embedding into $L^{\infty}$ .

Existence of solutions to the Poisson equation in $L_{p}$ -weighted spaces

Joanna Rencławowicz, Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

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We examine the Poisson equation with boundary conditions on a cylinder in a weighted space of $L_{p}$ , p≥ 3, type. The weight is a positive power of the distance from a distinguished plane. To prove the existence of solutions we use our result on existence in a weighted L₂ space.

Admissible functions for the Dirichlet space

Javad Mashreghi, Mahmood Shabankhah (2010)

Studia Mathematica

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Zero sets and uniqueness sets of the classical Dirichlet space are not completely characterized yet. We define the concept of admissible functions for the Dirichlet space and then apply them to obtain a new class of zero sets for . Then we discuss the relation between the zero sets of and those of $^{\infty}$ .

Essential norms of weighted composition operators on the space $ℋ^{\infty}$ of Dirichlet series

Pascal Lefèvre (2009)

Studia Mathematica

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We estimate the essential norm of a weighted composition operator relative to the class of Dunford-Pettis operators or the class of weakly compact operators, on the space $ℋ^{\infty}$ of Dirichlet series. As particular cases, we obtain the precise value of the generalized essential norm of a composition operator and of a multiplication operator.

L²-data Dirichlet problem for weighted form Laplacians

Wojciech Kozłowski (2008)

Colloquium Mathematicae

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We solve the L²-data Dirichlet boundary problem for a weighted form Laplacian in the unit Euclidean ball. The solution is given explicitly as a sum of four series.

On the Dirichlet problem associated with the Dunkl Laplacian

Mohamed Ben Chrouda (2016)

Annales Polonici Mathematici

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This paper deals with the questions of the existence and uniqueness of a solution to the Dirichlet problem associated with the Dunkl Laplacian $Δ_{k}$ as well as the hypoellipticity of $Δ_{k}$ on noninvariant open sets.

Embeddings of doubling weighted Besov spaces

Dorothee D. Haroske, Philipp Skandera (2014)

Banach Center Publications

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We study continuous embeddings of Besov spaces of type $B_{p, q}^{s} (ℝ ⁿ, w)$ , where s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞, and the weight w is doubling. This approach generalises recent results about embeddings of Muckenhoupt weighted Besov spaces. Our main argument relies on appropriate atomic decomposition techniques of such weighted spaces; here we benefit from earlier results by Bownik. In addition, we discuss some other related weight classes briefly and compare corresponding results.

Existence of solutions to the (rot,div)-system in L₂-weighted spaces

Wojciech M. Zajączkowski (2009)

Applicationes Mathematicae

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The existence of solutions to the elliptic problem rot v = w, div v = 0 in Ω ⊂ ℝ³, ${v \cdot n ̅ |}_{S} = 0$ , S = ∂Ω, in weighted Hilbert spaces is proved. It is assumed that Ω contains an axis L and the weight is a negative power of the distance to the axis. The main part of the proof is devoted to examining solutions in a neighbourhood of L. Their existence in Ω follows by regularization.

Existence of solutions to the nonstationary Stokes system in $H_{- μ}^{2, 1}$ , μ ∈ (0,1), in a domain with a distinguished axis. Part 1. Existence near the axis in 2d

W. M. Zajączkowski (2007)

Applicationes Mathematicae

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We consider the nonstationary Stokes system with slip boundary conditions in a bounded domain which contains some distinguished axis. We assume that the data functions belong to weighted Sobolev spaces with the weight equal to some power function of the distance to the axis. The aim is to prove the existence of solutions in corresponding weighted Sobolev spaces. The proof is divided into three parts. In the first, the existence in 2d in weighted spaces near the axis is shown. In the...

Existence of solutions to the (rot,div)-system in $L_{p}$ -weighted spaces

Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

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The existence of solutions to the elliptic problem rot v = w, div v = 0 in a bounded domain Ω ⊂ ℝ³, ${v \cdot n ̅ |}_{S} = 0$ , S = ∂Ω in weighted $L_{p}$ -Sobolev spaces is proved. It is assumed that an axis L crosses Ω and the weight is a negative power function of the distance to the axis. The main part of the proof is devoted to examining solutions of the problem in a neighbourhood of L. The existence in Ω follows from the technique of regularization.

Spectral properties of weighted composition operators on the Bloch and Dirichlet spaces

Ted Eklund, Mikael Lindström, Paweł Mleczko (2016)

Studia Mathematica

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The spectra of invertible weighted composition operators $u C_{φ}$ on the Bloch and Dirichlet spaces are studied. In the Bloch case we obtain a complete description of the spectrum when φ is a parabolic or elliptic automorphism of the unit disc. In the case of a hyperbolic automorphism φ, exact expressions for the spectral radii of invertible weighted composition operators acting on the Bloch and Dirichlet spaces are derived.

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.

Singularities in Muckenhoupt weighted function spaces

Dorothee D. Haroske (2008)

Banach Center Publications

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We study weighted function spaces of Lebesgue, Besov and Triebel-Lizorkin type where the weight function belongs to some Muckenhoupt $_{p}$ class. The singularities of functions in these spaces are characterised by means of envelope functions.

The weak Dirichlet and Neumann problem for the Laplacian in $L^{q}$ for bounded and exterior domains. Applications

Simader, Christian G.

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Natural boundary value problems for weighted form laplacians

Wojciech Kozłowski, Antoni Pierzchalski (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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The four natural boundary problems for the weighted form Laplacians $L = a d δ + b δ d, a, b > 0$ acting on polynomial differential forms in the $n$ -dimensional Euclidean ball are solved explicitly. Moreover, an algebraic algorithm for generating a solution from the boundary data is given in each case.

The spectral decomposition of weighted shifts and the $A^{p}$ condition

Earl Berkson, T. A. Gillespie (1990)

Colloquium Mathematicae

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On Dirichlet type spaces on the unit ball of $ℂ^{n}$

Małgorzata Michalska (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we discuss characterizations of Dirichlet type spaces on the unit ball of $ℂ^{n}$ obtained by P. Hu and W. Zhang [2], and S. Li [4].

Monotonicity of generalized weighted mean values

Alfred Witkowski (2004)

Colloquium Mathematicae

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The author gives a new simple proof of monotonicity of the generalized extended mean values $M (r, s) = \leq {((\int f^{s} d μ) / (\int f^{r} d μ))}^{1 / (s - r)}$ introduced by F. Qi.

Weighted inequalities for gradients on non-smooth domains

Caroline Sweezy, J. Michael Wilson

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We prove sufficiency of conditions on pairs of measures μ and ν, defined respectively on the interior and the boundary of a bounded Lipschitz domain Ω in d-dimensional Euclidean space, which ensure that, if u is the solution of the Dirichlet problem. Δu = 0 in Ω, ${u |}_{\partial Ω} = f$ , with f belonging to a reasonable test class, then $(\int_{Ω} {| \nabla u |}^{q} {d μ)}^{1 / q} \leq (\int_{\partial Ω} {| f |}^{p} {d ν)}^{1 / p}$ , where 1 < p ≤ q < ∞ and q ≥ 2. Our sufficiency conditions resemble those found by Wheeden and Wilson for the Dirichlet problem on $ℝ ₊^{d + 1}$ . As in that case we attack the...

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 $A_{q}$ .

The representation of multi-hypergraphs by set intersections

Stanisław Bylka, Jan Komar (2007)

Discussiones Mathematicae Graph Theory

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This paper deals with weighted set systems (V,,q), where V is a set of indices, $\subset 2^{V}$ and the weight q is a nonnegative integer function on . The basic idea of the paper is to apply weighted set systems to formulate restrictions on intersections. It is of interest to know whether a weighted set system can be represented by set intersections. An intersection representation of (V,,q) is defined to be an indexed family $R = {(R_{v})}_{v \in V}$ of subsets of a set S such that $| ⋂_{v \in E} R_{v} | = q (E)$ for each E ∈ . A necessary condition...

On the Banach-Stone problem

Jyh-Shyang Jeang, Ngai-Ching Wong (2003)

Studia Mathematica

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Let X and Y be locally compact Hausdorff spaces, let E and F be Banach spaces, and let T be a linear isometry from C₀(X,E) into C₀(Y,F). We provide three new answers to the Banach-Stone problem: (1) T can always be written as a generalized weighted composition operator if and only if F is strictly convex; (2) if T is onto then T can be written as a weighted composition operator in a weak sense; and (3) if T is onto and F does not contain a copy of $ℓ ₂^{\infty}$ then T can be written as a weighted...

Weighted composition operators between weighted Banach spaces of holomorphic functions and weighted Bloch type space

Elke Wolf (2009)

Annales Polonici Mathematici

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Let ϕ: → and ψ: → ℂ be analytic maps. They induce a weighted composition operator $ψ C_{ϕ}$ acting between weighted Banach spaces of holomorphic functions and weighted Bloch type spaces. Under some assumptions on the weights we give a necessary as well as a sufficient condition for such an operator to be bounded resp. compact.