Relaxed Energies for Harmonic Maps

  • F. Bethuel
  • H. Brezis
  • J. M. Coron
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)

Abstract

Let
$$Omega \subset {\mathbb{R}^3}$$
be an open bounded set such that
$$ \partial \Omega $$
is smooth. Set
$$ {H^1}\left( {\Omega ;{S^2}} \right) = \left\{ {u \in {H^1}\left( {\Omega ;{\mathbb{R}^3}} \right);|u\left( x \right)| = 1 a.e.} \right\} $$
and
$$ H_\varphi ^1\left( {\Omega ;{S^2}} \right) = \left\{ {u \in {H^1}\left( {\Omega ;{S^2}} \right);u = \varphi on \partial \Omega } \right\}, $$
where
$$ \varphi :\partial \Omega \to {S^2} $$
is a given boundary data.

Keywords

Manifold 

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Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • F. Bethuel
    • 1
  • H. Brezis
    • 2
    • 3
  • J. M. Coron
    • 4
  1. 1.CERMA, ENPCLa Courtine Noisy le GrandFrance
  2. 2.Département de MathématiquesUniversité P. et M. CurieParis Cedex 05France
  3. 3.Department of MathematicsRutgers University, Hill CenterNew BrunswickUSA
  4. 4.Département de MathématiquesUniversité Paris-SudOrsay CedexFrance

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