Regularity theorems for solutions of partial differential equations for quasiconformal mappings in several dimensions

Tadeusz Iwaniec

Displaying similar documents to “Regularity theorems for solutions of partial differential equations for quasiconformal mappings in several dimensions”

Loewner chains and quasiconformal extension of holomorphic mappings

Hidetaka Hamada, Gabriela Kohr (2003)

Annales Polonici Mathematici

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Let f(z,t) be a Loewner chain on the Euclidean unit ball B in ℂⁿ. Assume that f(z) = f(z,0) is quasiconformal. We give a sufficient condition for f to extend to a quasiconformal homeomorphism of $ℝ^{2 n}$ onto itself.

Quasiconformal mappings and exponentially integrable functions

Fernando Farroni, Raffaella Giova (2011)

Studia Mathematica

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We prove that a K-quasiconformal mapping f:ℝ² → ℝ² which maps the unit disk onto itself preserves the space EXP() of exponentially integrable functions over , in the sense that u ∈ EXP() if and only if $u \circ f^{- 1} \in E X P ()$ . Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate $1 / (1 + K l o g K) \leq (| | u \circ f^{- 1} {| |}_{E X P ()} {) / (| | u | |}_{E X P ()}) \leq 1 + K l o g K$ for every u ∈ EXP(). Similarly, we consider the distance from $L^{\infty}$ in EXP and we prove that if f: Ω → Ω’ is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then $1 / K \leq (d i s t_{E X P (f (G))} (u \circ f^{- 1}, L^{\infty} (f (G)))) / (d i s t_{E X P (f (G))} (u, L^{\infty} (G))) \leq K$ for every u ∈ EXP(). We also...

Composition operator and Sobolev-Lorentz spaces $W L^{n, q}$

Stanislav Hencl, Luděk Kleprlík, Jan Malý (2014)

Studia Mathematica

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Let Ω,Ω’ ⊂ ℝⁿ be domains and let f: Ω → Ω’ be a homeomorphism. We show that if the composition operator $T_{f} : u \mapsto u \circ f$ maps the Sobolev-Lorentz space $W L^{n, q} (Ω^{'})$ to $W L^{n, q} (Ω)$ for some q ≠ n then f must be a locally bilipschitz mapping.

Quasihomographies in the theory of Teichmüller spaces

Zając Józef

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CONTENTSIntroduction............................................................................................................................5 I. Special functions of quasiconformal theory.....................................................................10 1. Introduction.................................................................................................................10 2. The distortion function $Φ_{K}$ .....................................................................................11 3....

Uniform convergence of the generalized Bieberbach polynomials in regions with zero angles

F. G. Abdullayev (2001)

Czechoslovak Mathematical Journal

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Let $C$ be the extended complex plane; $G \subset C$ a finite Jordan with $0 \in G$ ; $w = ϕ (z)$ the conformal mapping of $G$ onto the disk $B (0; ρ_{0}) : = \}w |w| < ρ_{0}\{$ normalized by $ϕ (0) = 0$ and $ϕ^{'} (0) = 1$ . Let us set $ϕ_{p} (z) : = \int_{0}^{z} {[ϕ^{'} (ζ)]}^{2 / p} d ζ$ , and let $π_{n, p} (z)$ be the generalized Bieberbach polynomial of degree $n$ for the pair $(G, 0)$ , which minimizes the integral $\underset{G}{\int \int} {|ϕ_{p}^{'} (z) - P_{n}^{'} (z)|}^{p} d σ_{z}$ in the class of all polynomials of degree not exceeding $\leq n$ with $P_{n} (0) = 0$ , $P_{n}^{'} (0) = 1$ . In this paper we study the uniform convergence of the generalized Bieberbach polynomials $π_{n, p} (z)$ to $ϕ_{p} (z)$ on $\bar{G}$ with interior and exterior zero angles and determine its dependence...

The generalized Neumann-Poincaré operator and its spectrum

Partyka Dariusz

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CONTENTSIntroduction..........................................................................................................................................................................5Preliminaries. Complex harmonic functions..........................................................................................................................7I. Spectral values and eigenvalues of a Jordan curve........................................................................................................19 1.1....

A new characterization of the Sobolev space

Piotr Hajłasz (2003)

Studia Mathematica

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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.

On spaces $L^{p (x)}$ and $W^{k, p (x)}$

Ondrej Kováčik, Jiří Rákosník (1991)

Czechoslovak Mathematical Journal

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The Bohr-Pál theorem and the Sobolev space $W ₂^{1 / 2}$

Vladimir Lebedev (2015)

Studia Mathematica

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The well-known Bohr-Pál theorem asserts that for every continuous real-valued function f on the circle there exists a change of variable, i.e., a homeomorphism h of onto itself, such that the Fourier series of the superposition f ∘ h converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings f into the Sobolev space $W ₂^{1 / 2} ()$ . This refined version of the Bohr-Pál theorem does not extend to complex-valued functions. We show that if...

On the equation x”(t) = F(t, x(t)) in the Sobolev space $H^{1} (R)$

Piotr Fijałkowski (1991)

Annales Polonici Mathematici

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On sets of confluent and related mappings in the space $Y^{X}$

T. Maćkowiak (1976)

Colloquium Mathematicae

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On a result by Clunie and Sheil-Small

Dariusz Partyka, Ken-ichi Sakan (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping $F$ in the unit disk $𝔻$ , if $F (𝔻)$ is a convex domain, then the inequality $| G (z_{2}) - G (z_{1}) | < | H (z_{2}) - H (z_{1}) |$ holds for all distinct points $z_{1}, z_{2} \in 𝔻$ . Here $H$ and $G$ are holomorphic mappings in $𝔻$ determined by $F = H + \bar{G}$ , up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain $Ω$ in $ℂ$ and improve it provided $F$ is additionally a quasiconformal mapping...

On a Sobolev type inequality and its applications

Witold Bednorz (2006)

Studia Mathematica

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Assume ||·|| is a norm on ℝⁿ and ||·||⁎ its dual. Consider the closed ball $T : = B_{| | \cdot | |} (0, r)$ , r > 0. Suppose φ is an Orlicz function and ψ its conjugate. We prove that for arbitrary A,B > 0 and for each Lipschitz function f on T, $s u p_{s, t \in T} | f (s) - f (t) | \leq 6 A B (\int_{0}^{r} ψ (1 / A ε^{n - 1}) ε^{n - 1} d ε + 1 / (n | B_{| | \cdot | |} (0, 1) |) \int_{T} φ (1 / B | | \nabla f (u) | | ⁎) d u)$ , where |·| is the Lebesgue measure on ℝⁿ. This is a strengthening of the Sobolev inequality obtained by M. Talagrand. We use this inequality to state, for a given concave, strictly increasing function η: ℝ₊ → ℝ with η(0) = 0, a necessary and sufficient condition on...

Optimal embeddings of critical Sobolev-Lorentz-Zygmund spaces

Hidemitsu Wadade (2014)

Studia Mathematica

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We establish the embedding of the critical Sobolev-Lorentz-Zygmund space $H_{p, q, λ ₁, . . ., λ ₘ}^{n / p} (ℝ ⁿ)$ into the generalized Morrey space $ℳ_{Φ, r} (ℝ ⁿ)$ with an optimal Young function Φ. As an application, we obtain the almost Lipschitz continuity for functions in $H_{p, q, λ ₁, . . ., λ ₘ}^{n / p + 1} (ℝ ⁿ)$ . O’Neil’s inequality and its reverse play an essential role in the proofs of the main theorems.

Exponent of growth of polynomial mappings of $C^{2}$ into $C^{2}$

Jacek Chadzyński, Tadeusz Krasiński (1988)

Banach Center Publications

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Regularity for minimizers of non-autonomous non-quadratic functionals in the case $1<p<2$ : an a priori estimate

Andrea Gentile (2018)

Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche

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We establish an a priori estimate for the second derivatives of local minimizers of integral functionals of the form $\mathcal{F}(\nu,\Omega)=\int_{\Omega}f(x,D\nu(x))\,dx$ with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the $x$ variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

Pointwise regularity associated with function spaces and multifractal analysis

Stéphane Jaffard (2006)

Banach Center Publications

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The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) f has a given pointwise regularity exponent H. This notion has many variants depending on the global hypotheses made on f; if f locally belongs to a Banach space E, then a family of pointwise regularity spaces $C_{E}^{α} (x ₀)$ are constructed, leading to a notion of pointwise regularity with respect to E; the case $E = L^{\infty}$ corresponds to the usual Hölder regularity,...

On a theorem of Lindelof

Vladimir Ya. Gutlyanskii, Olli Martio, Vladimir Ryazanov (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We give a quasiconformal version of the proof for the classical Lindelof theorem: Let $f$ map the unit disk $𝔻$ conformally onto the inner domain of a Jordan curve $𝒞$ : Then $𝒞$ is smooth if and only if arg $f^{'} (z)$ has a continuous extension to $\bar{𝔻}$ . Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.

Sobolev-Besov spaces of measurable functions

Hans Triebel (2010)

Studia Mathematica

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The paper deals with spaces $L_{p}^{s} (ℝ ⁿ)$ of Sobolev type where s > 0, 0 < p ≤ ∞, and their relations to corresponding spaces $B_{p, q}^{s} (ℝ ⁿ)$ of Besov type where s > 0, 0 < p ≤ ∞, 0 < q ≤ ∞, in terms of embedding and real interpolation.