A1 Journal article (refereed)
The Nitsche phenomenon for weighted Dirichlet energy (2020)

Iwaniec, T., Onninen, J., & Radice, T. (2020). The Nitsche phenomenon for weighted Dirichlet energy. Advances in Calculus of Variations, 13 (3), 301-323. doi:10.1515/acv-2017-0060

JYU authors or editors

Publication details

All authors or editors: Iwaniec, Tadeusz; Onninen, Jani; Radice, Teresa

Journal or series: Advances in Calculus of Variations

ISSN: 1864-8258

eISSN: 1864-8266

Publication year: 2020

Volume: 13

Issue number: 3

Pages range: 301-323

Publisher: Walter de Gruyter GmbH

Publication country: Germany

Publication language: English

DOI: https://doi.org/10.1515/acv-2017-0060

Open Access: Publication channel is not openly available

Abstract

The present paper arose from recent studies of energy-minimal deformations of planar domains. We are concerned with the Dirichlet energy. In general the minimal mappings need not be homeomorphisms. In fact, a part of the domain near its boundary may collapse into the boundary of the target domain. In mathematical models of nonlinear elasticity this is interpreted as interpenetration of matter. We call such occurrence the Nitsche phenomenon, after Nitsche’s remarkable conjecture (now a theorem) about existence of harmonic homeomorphisms between annuli. Indeed the round annuli proved to be perfect choices to grasp the nuances of the problem. Several papers are devoted to a study of deformations of annuli. Because of rotational symmetry it seems likely that the Dirichlet energy-minimal deformations are radial maps. That is why we confine ourselves to radial minimal mappings. The novelty lies in the presence of a weight in the Dirichlet integral. We observe the Nitsche phenomenon in this case as well, see our main results Theorem and Theorem . However, the arguments require further considerations and new ingredients. One must overcome the inherent difficulties arising from discontinuity of the weight. The Lagrange–Euler equation is unavailable, because the outer variation violates the principle of none interpenetration of matter. Inner variation, on the other hand, leads to an equation that involves the derivative of the weight. But our weight function is only measurable which is the main challenge of the present paper.

Keywords: potential theory; calculus of variations

Free keywords: weighted Dirichlet energy; harmonic mappings; variational integrals

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