The Nitsche phenomenon for weighted Dirichlet energy

Research output: Contribution to journalArticle

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 1.4 and Theorem 1.7. 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.

Original languageEnglish (US)
JournalAdvances in Calculus of Variations
DOIs
StateAccepted/In press - Mar 28 2018

Fingerprint

Ring or annulus
Dirichlet
Minimal Energy
Energy
Theorem
Dirichlet Integral
Nonlinear Elasticity
Rotational symmetry
Euler-Lagrange Equations
Euler equations
Violate
Weight Function
Elasticity
Discontinuity
Harmonic
Likely
Mathematical Model
Mathematical models
Derivatives
Derivative

Keywords

  • harmonic mappings
  • variational integrals
  • Weighted Dirichlet energy

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

The Nitsche phenomenon for weighted Dirichlet energy. / Iwaniec, Tadeusz; Onninen, Jani Kristian; Radice, Teresa.

In: Advances in Calculus of Variations, 28.03.2018.

Research output: Contribution to journalArticle

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