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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

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

Nonlinear analysis and quasiconformal mappings from the perspective of PDEs

93%

Iwaniec T.

Banach Center Publications

1999

tom 48

nr 1

119-140

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

Squeezing the Sierpinski sponge

59%

Iwaniec T. , Martin G.

Studia Mathematica

2002

tom 149

nr 2

133-145

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".

Interpolation theorem for the p-harmonic transform

59%

D'Onofrio L. , Iwaniec T.

Studia Mathematica

2003

tom 159

nr 3

373-390

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.

New and old function spaces in the theory of PDEs and nonlinear analysis

59%

Iwaniec T. , Sbordone C.

Banach Center Publications

2004

tom 64

nr 1

85-104

Projections onto gradient fields and $L^{p}$-estimates for degenerated elliptic operators

41%

Iwaniec T.

Studia Mathematica

1982-1983

tom 75

nr 3

293-312