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Tadeusz Iwaniec

Publications and source records attributed to Tadeusz Iwaniec.

29 records · Page 2Linked to original sources
arXiv2011

An Essay on the Interpolation Theorem of Józef Marcinkiewicz - Polish Patriot

In memory of Polish mathematicians murdured by the Soviets and the Nazis. The total record of accomplishments of Marcinkiewicz in his short life, his talent, perceptions rich in concepts, and technical novelties, go far beyond my ability to give full play within the confines of one article. The importance of Marcinkiewicz's short paper is reflected in the myriad applications and generalizations which earns the right to be called Marcinkiewicz Interpolation Theory Marcinkiewicz interpolation theorem came after the celebrated convexity theorem of M. Riesz and his student G.O. Thorin. These fundamental works by M. Riesz, G.O. Thorin and J. Marcinkiewicz deal with estimates of the Lp-norms of an operator, knowing its behavior at the end-points of the interval of the exponents p, where the operator is still defined. There are, however, some subtle differences between the Riesz-Thorin and the Marcinkiewicz ideas. Marcinkiewicz approach can be adapted to nonlinear operators, this is what we demonstrate in the present paper.

math.AP↗
arXiv2010

Burkholder integrals, Morrey's problem and quasiconformal mappings

Inspired by Morrey's Problem (on rank-one convex functionals) and the Burkholder integrals (of his martingale theory) we find that the Burkholder functionals $B_p$, $p \ge 2$, are quasiconcave, when tested on deformations of identity $f\in Id + C^\infty_0(Ω)$ with $B_p(Df(x)) \ge 0$ pointwise, or equivalently, deformations such that $|Df|^2 \leq \frac{p}{p-2} J_f$. In particular, this holds in explicit neighbourhoods of the identity map. Among the many immediate consequences, this gives the strongest possible $L^p$- estimates for the gradient of a principal solution to the Beltrami equation $\f_{\bar{z}} = μ(z) f_z$, for any $p$ in the critical interval $2 \leq p \leq 1+1/\|μ_f\|_\infty$. Examples of local maxima lacking symmetry manifest the intricate nature of the problem.

math.CA↗
arXiv2010

Existence of energy-minimal diffeomorphisms between doubly connected domains

The paper establishes the existence of homeomorphisms between two planar domains that minimize the Dirichlet energy. Specifically, among all homeomorphisms f : R -> R* between bounded doubly connected domains such that Mod (R) < Mod (R*) there exists, unique up to conformal authomorphisms of R, an energy-minimal diffeomorphism. No boundary conditions are imposed on f. Although any energy-minimal diffeomorphism is harmonic, our results underline the major difference between the existence of harmonic diffeomorphisms and the existence of the energy-minimal diffeomorphisms. The existence of globally invertible energy-minimal mappings is of primary pursuit in the mathematical models of nonlinear elasticity and is also of interest in computer graphics.

math.CV↗
arXiv2010

Hopf differentials and smoothing Sobolev homeomorphisms

We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space $W^{1,2}$ and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential is harmonic.

math.CV↗
arXiv2010

Doubly connected minimal surfaces and extremal harmonic mappings

The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Bjorling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial slope.

math.DG↗
arXiv2010

Harmonic mapping problem and affine capacity

The Harmonic Mapping Problem asks when there exists a harmonic homeomorphism between two given domains. It arises in the theory of minimal surfaces and in calculus of variations, specifically in hyperelasticity theory. We investigate this problem for doubly connected domains in the plane, where it already presents considerable challenge and leads to several interesting open questions.

math.CV↗
arXiv2009

The Nitsche conjecture

The conjecture in question concerns the existence of a harmonic homeomorphism between circular annuli A(r,R) and A(r*,R*), and is motivated in part by the existence problem for doubly-connected minimal surfaces with prescribed boundary. In 1962 J.C.C. Nitsche observed that the image annulus cannot be too thin, but it can be arbitrarily thick (even a punctured disk). Then he conjectured that for such a mapping to exist we must have the following inequality, now known as the Nitsche bound: R*/r* is greater than or equal to (R/r+r/R)/2. In this paper we give an affirmative answer to his conjecture. As a corollary, we find that among all minimal graphs over given annulus the upper slab of catenoid has the greatest conformal modulus.

math.CV↗
arXiv2009

Harmonic mappings of an annulus, Nitsche conjecture and its generalizations

As long ago as 1962 Nitsche conjectured that a harmonic homeomorphism $h \colon A(r,R) \to A(r_*, R_*)$ between planar annuli exists if and only if $\frac{R_*}{r_*} \ge {1/2} (\frac{R}{r} + \frac{r}{R})$. We prove this conjecture when the domain annulus is not too wide; explicitly, when $\log \frac{R}{r} \le {3/2}$. For general $A(r,R)$ the conjecture is proved under additional assumption that either $h$ or its normal derivative have vanishing average on the inner boundary circle. This is the case for the critical Nitsche mapping which yields equality in the above inequality. The Nitsche mapping represents so-called free evolution of circles of the annulus $A(r,R)$. It will be shown on the other hand that forced harmonic evolution results in greater ratio $\frac{R_*}{r_*}$. To this end, we introduce the underlying differential operators for the circular means of the forced evolution and use them to obtain sharp lower bounds of $\frac{R_*}{r_*}$.

math.CV↗
arXiv2008

Dynamics of quasiconformal fields

A uniqueness theorem is established for autonomous systems of ODEs, $\dot{x}=f(x)$, where $f$ is a Sobolev vector field with additional geometric structure, such as delta-monoticity or reduced quasiconformality. Specifically, through every non-critical point of $f$ there passes a unique integral curve.

math.CA↗