Monotone Hopf-Harmonics
Year of publication
2020
Authors
Iwaniec, Tadeusz; Onninen, Jani
Abstract
We introduce the concept of monotone Hopf-harmonics in 2D as an alternative to harmonic homeomorphisms. Much of the foregoing is motivated by the principle of non-interpenetration of matter in the mathematical theory of Nonlinear Elasticity (NE). The question we are concerned with is whether or not a Dirichlet energy-minimal mapping between Jordan domains with a prescribed boundary homeomorphism remains injective in the domain. The classical theorem of Radó–Kneser–Choquet asserts that this is the case when the target domain is convex. An alternative way to deal with arbitrary target domains is to minimize the Dirichlet energy subject to only homeomorphisms and their limits. This leads to the so called Hopf–Laplace equation. Among its solutions (some rather surreal) are continuous monotone mappings of Sobolev class W1,2 loc , called monotone Hopf-harmonics. It is at the heart of the present paper to show that such solutions are correct generalizations of harmonic homeomorphisms and, in particular, are legitimate deformations of hyperelastic materials in the modern theory of NE. We make this clear by means of several examples.
Show moreOrganizations and authors
University of Jyväskylä
Onninen Jani
Publication type
Publication format
Article
Parent publication type
Journal
Article type
Original article
Audience
ScientificPeer-reviewed
Peer-ReviewedMINEDU's publication type classification code
A1 Journal article (refereed), original researchPublication channel information
Publisher
Volume
237
Issue
2
Pages
743-777
ISSN
Publication forum
Publication forum level
3
Open access
Open access in the publisher’s service
No
Self-archived
No
Other information
Fields of science
Mathematics
Keywords
[object Object],[object Object],[object Object]
Publication country
Germany
Internationality of the publisher
International
Language
English
International co-publication
Yes
Co-publication with a company
No
DOI
10.1007/s00205-020-01518-2
The publication is included in the Ministry of Education and Culture’s Publication data collection
Yes