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

Wojciech M. Zajączkowski

Displaying similar documents to “Existence and regularity of solutions of some elliptic system in domains with edges”

The $L^{2}$ $\bar{\partial}$ -Cauchy problem on weakly $q$ -pseudoconvex domains in Stein manifolds

Sayed Saber (2015)

Czechoslovak Mathematical Journal

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Let $X$ be a Stein manifold of complex dimension $n \geq 2$ and $Ω ⋐ X$ be a relatively compact domain with $C^{2}$ smooth boundary in $X$ . Assume that $Ω$ is a weakly $q$ -pseudoconvex domain in $X$ . The purpose of this paper is to establish sufficient conditions for the closed range of $\bar{\partial}$ on $Ω$ . Moreover, we study the $\bar{\partial}$ -problem on $Ω$ . Specifically, we use the modified weight function method to study the weighted $\bar{\partial}$ -problem with exact support in $Ω$ . Our method relies on the $L^{2}$ -estimates by Hörmander (1965) and by Kohn (1973). ...

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.

On the paradox of two $R_{1}, R_{2}$ domains in Schwarzschild exterior solution

A. Zięba (1972)

Colloquium Mathematicae

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Pluriharmonic extension in proper image domains

Rafał Czyż (2009)

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 $Ω_{π}$ be the image of D under the proper holomorphic map π. We characterize those continuous functions $f : \partial Ω_{π} \to ℝ$ that can be extended to a real-valued pluriharmonic function in $Ω_{π}$ .

New characterizations for weighted composition operator from Zygmund type spaces to Bloch type spaces

Xin-Cui Guo, Ze-Hua Zhou (2015)

Czechoslovak Mathematical Journal

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Let $u$ be a holomorphic function and $ϕ$ a holomorphic self-map of the open unit disk $𝔻$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $u C_{ϕ}$ from Zygmund type spaces to Bloch type spaces in $𝔻$ in terms of $u$ , $ϕ$ , their derivatives, and $ϕ^{n}$ , the $n$ -th power of $ϕ$ . Moreover, we obtain some similar estimates for the essential norms of the operators $u C_{ϕ}$ , from which sufficient and necessary conditions of compactness of $u C_{ϕ}$ follows immediately. ...

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...

On n-circled $ℋ^{\infty}$ -domains of holomorphy

Marek Jarnicki, Peter Pflug (1997)

Annales Polonici Mathematici

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We present various characterizations of n-circled domains of holomorphy $G \subset ℂ^{n}$ with respect to some subspaces of $ℋ^{\infty} (G)$ .

Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight functions

Honghui Yin, Zuodong Yang (2011)

Annales Polonici Mathematici

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Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ $- Δ_{p} u + {| u |}^{p - 2} u = f_{1 λ ₁} {(x) | u |}^{q - 2} u + 2 α / (α + β) g_{μ} {| u |}^{α - 2} u {| v |}^{β}$ , x ∈ Ω, ⎨ $- Δ_{p} v + {| v |}^{p - 2} v = f_{2 λ ₂} {(x) | v |}^{q - 2} v + 2 β / (α + β) g_{μ} {| u |}^{α} {| v |}^{β - 2} v$ , x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and $Ω \subset ℝ^{N}$ is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and $f_{i λ_{i}} (x) = λ_{i} f_{i +} (x) + f_{i -} (x)$ (i = 1,2) are sign-changing functions, where $f_{i \pm} (x) = m a x \pm f_{i} (x), 0$ , $g_{μ} (x) = a (x) + μ b (x)$ , and $Δ_{p} u = {d i v (| \nabla u |}^{p - 2} \nabla u)$ denotes the p-Laplace operator. We use variational methods.

A domain whose envelope of holomorphy is not a domain

Edgar Lee Stout (2006)

Annales Polonici Mathematici

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We construct a domain of holomorphy in $ℂ^{N}$ , N≥ 2, whose envelope of holomorphy is not diffeomorphic to a domain in $ℂ^{N}$ .

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.

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.

When $C_{p} (X)$ is domain representable

William Fleissner, Lynne Yengulalp (2013)

Fundamenta Mathematicae

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Let M be a metrizable group. Let G be a dense subgroup of $M^{X}$ . We prove that if G is domain representable, then $G = M^{X}$ . The following corollaries answer open questions. If X is completely regular and $C_{p} (X)$ is domain representable, then X is discrete. If X is zero-dimensional, T₂, and $C_{p} (X,)$ is subcompact, then X is discrete.

Curved thin domains and parabolic equations

M. Prizzi, M. Rinaldi, K. P. Rybakowski (2002)

Studia Mathematica

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Consider the family uₜ = Δu + G(u), t > 0, $x \in Ω_{ε}$ , $\partial_{ν_{ε}} u = 0$ , t > 0, $x \in \partial Ω_{ε}$ , $(E_{ε})$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set $Ω_{ε}$ is a thin domain in $ℝ^{l}$ , possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of $ℝ^{l}$ . If G is dissipative, then equation $(E_{ε})$ has a global attractor $_{ε}$ . We identify a “limit” equation for the family $(E_{ε})$ , prove convergence of trajectories and establish an upper semicontinuity result for the family $_{ε}$ as ε → 0⁺. ...

The equation $\bar{\partial}u=f$ the intersection of pseudoconvex domains

Alessandro Perotti (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Viene studiata l'equazione $\bar{\partial}u=f$ per le forme regolari sulla chiusura dell'intersezione di $k$ domini pseudoconvessi. Si costruisce un operatore soluzione in forma integrale e sotto ipotesi opportune si ottengono stime della soluzione nelle norme $\mathbf{C}^{k}$ .

A remark on separate holomorphy

Marek Jarnicki, Peter Pflug (2006)

Studia Mathematica

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Let X be a Riemann domain over $ℂ^{k} \times ℂ^{ℓ}$ . If X is a domain of holomorphy with respect to a family ℱ ⊂(X), then there exists a pluripolar set $P \subset ℂ^{k}$ such that every slice $X_{a}$ of X with a∉ P is a region of holomorphy with respect to the family ${f |}_{X_{a}} : f \in ℱ$ .

Characterization of cycle domains via Kobayashi hyperbolicity

Gregor Fels, Alan Huckleberry (2005)

Bulletin de la Société Mathématique de France

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A real form $G$ of a complex semi-simple Lie group $G^{ℂ}$ has only finitely many orbits in any given $G^{ℂ}$ -flag manifold $Z = G^{ℂ} / Q$ . The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits $D$ generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of $D$ which, with very few well-understood exceptions, are parameterized by the Wolf cycle...

The method of rotation and Marcinkiewicz integrals on product domains

Jiecheng Chen, Dashan Fan, Yiming Ying (2002)

Studia Mathematica

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We give some rather weak sufficient condition for $L^{p}$ boundedness of the Marcinkiewicz integral operator $μ_{Ω}$ on the product spaces $ℝ ⁿ \times ℝ^{m}$ (1 < p < ∞), which improves and extends some known results.

Embeddings of Besov-Morrey spaces on bounded domains

Dorothee D. Haroske, Leszek Skrzypczak (2013)

Studia Mathematica

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We study embeddings of spaces of Besov-Morrey type, $i d_{Ω} :_{p ₁, u ₁, q ₁}^{s ₁} (Ω) ↪_{p ₂, u ₂, q ₂}^{s ₂} (Ω)$ , where $Ω \subset ℝ^{d}$ is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of $i d_{Ω}$ . This continues our earlier studies relating to the case of $ℝ^{d}$ . Moreover, we also characterise embeddings into the scale of $L_{p}$ spaces or into the space of bounded continuous functions.

A note on the weighted Khintchine-Groshev Theorem

Mumtaz Hussain, Tatiana Yusupova (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $W (m, n; \underset{̲}{ψ})$ denote the set of $ψ_{1}, ..., ψ_{n}$ –approximable points in $ℝ^{m n}$ . The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions $\underset{̲}{ψ}$ . Removing monotonicity from the Khintchine–Groshev theorem is attributed to different authors for different cases of $m$ and $n$ . It can not be removed for $m = n = 1$ as Duffin–Schaeffer provided the counter example. We deal with the only remaining case $m = 2$ and thereby remove all unnecessary conditions from the Khintchine–Groshev theorem. ...

On the potential theory of some systems of coupled PDEs

Abderrahim Aslimani, Imad El Ghazi, Mohamed El Kadiri, Sabah Haddad (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1} u = - μ_{1} v$ , $L_{2} v = - μ_{2} u$ , on a domain $D$ of $ℝ^{d}$ , where $μ_{1}$ and $μ_{2}$ are suitable measures on $D$ , and $L_{1}$ , $L_{2}$ are two second order linear differential elliptic operators on $D$ with coefficients of class $𝒞^{\infty}$ . We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_{1}$ and $L_{2}$ , and...

A weighted inequality for the Hardy operator involving suprema

Pavla Hofmanová (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $u$ be a weight on $(0, \infty)$ . Assume that $u$ is continuous on $(0, \infty)$ . Let the operator $S_{u}$ be given at measurable non-negative function $ϕ$ on $(0, \infty)$ by $S_{u} ϕ (t) = sup_{0 < τ \leq t} u (τ) ϕ (τ) .$ We characterize weights $v, w$ on $(0, \infty)$ for which there exists a positive constant $C$ such that the inequality ${(\int_{0}^{\infty} {[S_{u} ϕ (t)]}^{q} w (t) d t)}^{\frac{1}{q}} ≲ {(\int_{0}^{\infty} {[ϕ (t)]}^{p} v (t) d t)}^{\frac{1}{p}}$ holds for every $0 < p, q < \infty$ . Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.

A bifurcation theory for some nonlinear elliptic equations

Biagio Ricceri (2003)

Colloquium Mathematicae

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We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ( $P_{λ}$ ) ⎩ $u_{∣ \partial Ω} = 0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ( $P_{λ}$ ) admits a non-zero, non-negative strong solution $u_{λ} \in ⋂_{p \geq 2} W^{2, p} (Ω)$ such that $l i m_{λ \to 0 ⁺} | | u_{λ} {| |}_{W^{2, p} (Ω)} = 0$ for all p ≥ 2. Moreover, the function $λ \mapsto I_{λ} (u_{λ})$ is negative and decreasing in ]0,λ*[, where $I_{λ}$ is the energy functional related to ( $P_{λ}$ ). ...