Mappings of Least Dirichlet Energy and their Hopf Differentials

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Abstract

The paper is concerned with mappings h: X onto→ between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in X) of the energy-minimal mappings is established within the H̄2(X, Y) of strong limits of homeomorphisms in the Sobolev space W1,2(X, Y), a result of considerable interest in the mathematical models of nonlinear elasticity. The inner variation of the independent variable in X leads to the Hopf differential hzhdz⊗dz and its trajectories. For a pair of doubly connected domains, in which X has finite conformal modulus, we establish the following principle: A mapping h ∈ H̄2(X, Y) is energy-minimal if and only if its Hopf-differential is analytic in X and real along ∂X. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of slits in X (cognate with cracks). Slits are triggered by points of concavity of Y. They originate from ∂X and advance along vertical trajectories of the Hopf differential toward X where they eventually terminate, so no crosscuts are created.

Original languageEnglish (US)
Pages (from-to)401-453
Number of pages53
JournalArchive for Rational Mechanics and Analysis
Volume209
Issue number2
DOIs
StatePublished - Aug 2013

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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