Tadeusz Iwaniec

Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48)

This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.

Quasiregular mappings in even dimensions

O. Introduction 1. Notation 2. Some exterior algebra 3. Differential forms in Lnm (i2, A n) 4. Differential systems for quasiregular mappings 5. Liouville Theorem in even dimensions 6. Hodge theory in LP(R n) 7. The Beltrami equation in even dimensions 8. The Beurling-Ahlfors operator 9. Regularity theorems for quasiregular mappings 10. The Caccioppoli type estimate 11. Removability theorems for quasiregular mappings 12. Some examples Appendix: The 4-dimensional case

Integral estimates for null Lagrangians

Sobolev spaces of differential forms are studied, Lp-projections onto exact forms are introduced as a tool to obtain integral estimates for null Lagrangians. New results on compensated compactness are given and mean-coercive variational integrals are found. Existence of minima of certain mean-coercive functionals is established.

On mappings with integrable dilatation

A factorization of Stoilows type has been obtained for mappings in R2 with integrable dilatation. 0. INTRODUCTION For Q a domain in Rn (an open and connected set), we consider a mapping -f Rn of the Sobolev class W1, (Qn IR1n) with nonnegative Jacobian, J(x, f) > 0 a.e. We say that f has finite dilatation if (0. 1) IDf(x)ln 1. J(x, f) If J(x, f) = 0, then Df(x) = 0, and in this case we put K(x, f) = 1 a.e. Therefore the dilatation function K(, f): Q -* [1, oo) is defined almost everywhere in Q. A mapping f E W 1(Q, n En2 ) is said to be K-quasi-regular, 1 < K < oo, if K(x, f) < K a.e. If, in addition, f is a homeomorphism, we say that f is K-quasi-conformal. A well-known result in the theory of quasi-regular mappings [Re] states that if K( , f) E LO(Q2), then f is either constant or an open mapping. In two Received by the editors September 10, 1991. 1991 Mathematics Subject Classification. Primary 30C62.

Beltrami operators in the plane

We determine optimal Lp-properties for the solutions of the general nonlinear elliptic system in the plane of the form fz = H(z, fz), h ∈ L(C), where H is a measurable function satisfying |H(z, w1)−H(z, w2)| ≤ k|w1−w2| and k is a constant k < 1. We will also establish the precise invertibility and spectral properties in Lp(C) for the operators I − Tμ, I − μT, and T − μ, where T is the Beurling transform. These operators are basic in the theory of quasiconformal mappings and in linear and nonlinear elliptic partial differential equations in two dimensions. In particular, we prove invertibility in Lp(C) whenever 1 + ‖μ‖∞ < p < 1 + 1/‖μ‖∞. We also prove related results with applications to the regularity of weakly quasiconformal mappings.

Mappings of BMO–bounded distortion

This paper can be viewed as a sequel to the work [9] where the theory of mappings of BMO–bounded distortion is developed, largely in even dimensions, using singular integral operators and recent developments in the theory of higher integrability of Jacobians in Hardy–Orlicz spaces. In this paper we continue this theme refining and extending some of our earlier work as well as obtaining results in new directions. The planar case was studied earlier by G. David [4]. In particular he obtained existence theorems, modulus of continuity estimates and bounds on area distortion for mappings of BMO–distortion (in fact, in slightly more generality). We obtain similar results in all even dimensions. One of our main new results here is the extension of the classical theorem of Painleve concerning removable singularties for bounded analytic functions to the class of mappings of BMO bounded distortion. The setting of the plane is of particular interest and somewhat more can be said here because of the existence theorem, or “the measurable Riemann mapping theorem”, which is not available in higher dimensions. We give a construction to show our results are qualitatively optimal. Another surprising fact is that there are domains which support no bounded quasiregular mappings, but admit

Extremal mappings of finite distortion

The theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where one finds concrete applications in non-linear elasticity and the calculus of variations. In this paper we initiate the study of extremal problems for mappings with finite distortion and extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead of its supremum norm. For instance, we show the following. Suppose that

Nonlinear Hodge Theory on Manifolds with Boundary(

f_o

RIESZ TRANSFORMS AND ELLIPTIC PDES WITH VMO COEFFICIENTS

is a homeomorphism of the circle with

The Gehring Lemma

f_{o}^{-1} \in {\cal W}^{1/2, 2}