On quasi-conformal self-mappings of the unit disc and elliptic PDEs in the plane

Kalaj, David

doi:10.1017/s0308210511000862

Cited by 2 publications

(6 citation statements)

References 25 publications

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“…Furthermore, it was also shown in the same paper that if p = 1, then f is absolutely continuous on the boundary of B 2 . In a certain sense, the results from [21] optimize the results of the Kalaj, Mateljević, Pavlović, Partyka, Sakan, Astala, Manojlović [16,17,19,18,27,28,29,30,14,15,5], since it does not assume that the mapping is harmonic, neither its weak Laplacian is bounded. Furthermore, the two-dimensional result by Kalaj and Saksman in [21] has been extended by Kalaj and Zlatičanin in [22] to a higher-dimensional case.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 89%

Quasiconformal solutions to elliptic partial differential equations

Kalaj

2023

Ann. Fenn. Math.

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In this paper, we assume that \(G\) and \(\Omega\) are two Jordan domains in \(\mathbb{R}^n\) with \(\mathcal{C}^2\) boundaries, where \(n\ge 2\), and prove that every quasiconformal mapping \(f\in\mathcal{W}^{2,1+\epsilon}_{\mathrm{loc}}\) of \(G\) onto \(\Omega\), satisfying the elliptic partial differential inequality \(|L_ A[f]|\lesssim (\|Df\|^2+|g|)\), with \(g\in\mathcal{L}^p(G)\), where \(p>n\), is Lipschitz continuous. The result is sharp since for \(p=n\), the mapping \(f\) is not necessarily Lipschitz continuous. This extends several results for harmonic quasiconformal mappings.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 89%

Quasiconformal solutions to elliptic partial differential equations

Kalaj

2023

Ann. Fenn. Math.

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“…Proof Similar to the proof of Theorem 1.1, we let

Δ f (z) = g (z), z \in D,

where

g false(z false) = l false(z false) (| D f false(z false) {false|}^{2} + 1)

and

{| | l | |}_{\infty} ⩽ \max false{a, b false}

. By [12, Theorem 1.1], there is constant

C (a, b, K) > 0

, such that

| D f | ⩽ C (a, b, K)

holds for any

z \in D

. Let

N false(a, b, K false) : = \max false{a, b false} false(C^{2} (a, b, K) + 1 false) < \min \{\frac{24}{false(8 K + 3 false) π}, \frac{12}{13 π}\} .

From the proof of Theorem 1.1 and inequality (31), we have

\begin{matrix} true \\ left | f_{z} false(z false) | - | f_{\bar{z}} false(z false) | ⩾ false(1 - k false) | f_{z} false(z false) | <...> \end{matrix}

…”

Section: The Proof Of Corollary 14mentioning

confidence: 99%

“…By assumption, we have

g \in C (D)

. Hence, we get

{| | g | |}_{\infty} ⩽ \max false{a, b false} \cdot false(C^{2} (a, b, K) + 1 false) : = N false(a, b, K false)

by [12, Theorem 1.1]. In the following discussions, we assume

N false(a, b, K false) < \min \{\frac{24}{false(8 K + 3 false) π}, \frac{12}{13 π}\} .

It is well known that a K ‐quasiconformal self‐mapping f of

double-struckD

has a homeomorphic extension

f^{*}

to the closure

\bar{D}

.…”

Section: The Proof Of Theorem 11mentioning

confidence: 99%

“…Proof By [12, Theorem 1.1], there is constant

C (a, b, K) > 0

, such that

| D f | ⩽ C (a, b, K)

holds for any

z \in D

. Now, let

Δ f (z) = g (z), z \in D,

where

g false(z false) = l false(z false) (| D f false(z false) {false|}^{2} + 1)

and

{| | l | |}_{\infty} ⩽ \max false{a, b false};

one can simply define

l false(z false) : = Δ f false(z false) \cdot {(| D f false(z false) |}^{2} {+ 1)}^{- 1}, z \in D

.…”

Section: The Proof Of Theorem 11mentioning

confidence: 99%

“…By assumption, we have g ∈ C(D). Hence, we get ||g|| ∞ max{a, b} • (C 2 (a, b, K ) + 1) := N (a, b, K ) by [12,Theorem 1.1]. In the following discussions, we assume…”

mentioning

confidence: 99%

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