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

100%
Iwaniec T.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Preliminaries........................................................................................................ 5 1. Auxiliary results......................................................................................................... 13 2. The second order equations.................................................................................. 14 3. Some properties of Sobolev and Besov spaces................................................ 20 4. Classes $Λ^α(G, H)$, 0 < a ≤ 1............................................................................ 21 5. The case of Lipschitz characteristics................................................................... 26 6. Existence of second partial derivatives and its consequences...................... 29 7. Local boundedness of the Jacobian.................................................................... 33 8. Smoothness.............................................................................................................. 41 References.................................................................................................................... 44
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Nonlinear analysis and quasiconformal mappings from the perspective of PDEs

95%
Iwaniec T.
Banach Center Publications
|
1999
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tom 48
|
nr 1
119-140
EN
Contents Introduction 119 1. Quasiregular mappings 120 2. The Beltrami equation 121 3. The Beltrami-Dirac equation 122 4. A quest for compactness 124 5. Sharp $L^p$-estimates versus variational integrals 125 6. Very weak solutions 128 7. Nonlinear commutators 129 8. Jacobians and wedge products 131 9. Degree formulas 134 References 136
3
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An essay on the Interpolation Theorem of Józef Marcinkiewicz - Polish patriot

95%
Iwaniec T.
Banach Center Publications
|
2011
|
tom 95
|
nr 1
55-74
4
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Squeezing the Sierpinski sponge

61%
Iwaniec T., Martin G.
Studia Mathematica
|
2002
|
tom 149
|
nr 2
133-145
EN
We give an example relating to the regularity properties of mappings with finite distortion. This example suggests conditions to be imposed on the distortion function in order to avoid "cavitation in measure".
5
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New and old function spaces in the theory of PDEs and nonlinear analysis

61%
Iwaniec T., Sbordone C.
Banach Center Publications
|
2004
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tom 64
|
nr 1
85-104
6
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Interpolation theorem for the p-harmonic transform

61%
D'Onofrio L., Iwaniec T.
Studia Mathematica
|
2003
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tom 159
|
nr 3
373-390
EN
We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces $ℒ^{s}(ℝⁿ)$ arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation $div|∇u|^{p-2∇} u = div 𝔣$. In this example the p-harmonic transform is essentially inverse to $div(|∇|^{p-2}∇)$. To every vector field $𝔣 ∈ ℒ^{q}(ℝⁿ,ℝⁿ)$ our operator $ℋ_{p}$ assigns the gradient of the solution, $ℋ_{p}𝔣 = ∇u ∈ ℒ^{p}(ℝⁿ,ℝⁿ)$. The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our arguments require substantial innovations as compared with the classical interpolation theory of Riesz, Thorin and Marcinkiewicz. The subject is largely motivated by recent developments in geometric function theory.
7
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Projections onto gradient fields and $L^{p}$-estimates for degenerated elliptic operators

42%
Iwaniec T.
Studia Mathematica
|
1982-1983
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tom 75
|
nr 3
293-312
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