73 research outputs found
n-Harmonic mappings between annuli
The central theme of this paper is the variational analysis of homeomorphisms
h\colon \mathbb X \onto \mathbb Y between two given domains . We look for the extremal mappings in the
Sobolev space which minimize the energy
integral Because of the
natural connections with quasiconformal mappings this -harmonic alternative
to the classical Dirichlet integral (for planar domains) has drawn the
attention of researchers in Geometric Function Theory. Explicit analysis is
made here for a pair of concentric spherical annuli where many unexpected
phenomena about minimal -harmonic mappings are observed. The underlying
integration of nonlinear differential forms, called free Lagrangians, becomes
truly a work of art.Comment: 120 pages, 22 figure
An Essay on the Interpolation Theorem of J\'ozef 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
Dynamics of quasiconformal fields
A uniqueness theorem is established for autonomous systems of ODEs,
, where is a Sobolev vector field with additional geometric
structure, such as delta-monoticity or reduced quasiconformality. Specifically,
through every non-critical point of there passes a unique integral curve.Comment: 26 pages, 1 figur
Hopf differentials and smoothing Sobolev homeomorphisms
We prove that planar homeomorphisms can be approximated by diffeomorphisms in
the Sobolev space and in the Royden algebra. As an application, we
show that every discrete and open planar mapping with a holomorphic Hopf
differential is harmonic.Comment: 22 pages, 0 figure
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.Comment: 33 pages, 2 figures. Expanded introduction and references; added
discussion of doubly-connected minimal surface
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.Comment: 14 pages, 1 figur
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