Archive for Rational Mechanics and Analysis

, Volume 158, Issue 2, pp 155–171 | Cite as

Univalent σ-Harmonic Mappings

  • Giovanni Alessandrini
  • Vincenzo Nesi


We study mappings from ℝ2 into ℝ2 whose components are weak solutions to the elliptic equation in divergence form, div (σ∇u)= 0, which we call σ-harmonic mappings. We prove sufficient conditions for the univalence, i.e., injectivity, of such mappings. Moreover we prove local bounds in BMO on the logarithm of the Jacobian determinant of such univalent mappings, thus obtaining the a.e. nonvanishing of their Jacobian. In particular, our results apply to σ-harmonic mapping associated with any periodic structure and therefore they play an important role in homogenization.


Weak Solution Elliptic Equation Divergence Form Univalent Mapping Periodic Structure 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Giovanni Alessandrini
    • 1
  • Vincenzo Nesi
    • 2
  1. 1.Dipartimento di Scienze Matematiche¶Universitá di Trieste¶Via A. Valerio 12/1, 34127 Trieste, Italy¶e-mail: alessang@univ.trieste.itIT
  2. 2.Dipartimento di Matematica¶Universitá di Roma, La Sapienza¶P. le A. Moro 2, 00185 Rome, Italy¶e-mail: nesi@mat.uniroma1.itIT

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