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Minimum action solutions of some vector field equations

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Abstract

The system of equations studied in this paper is −Δu i =g i(u) on ℝd,d≧2, withu:ℝd→ℝn andg i(u)=∂G/∂u i . Associated with this system is the action,S(u)=ε{1/2|∇u|2−G(u)}. Under appropriate conditions onG (which differ ford=2 andd≧3) it is proved that the system has a solution,u ≢0, of finite action and that this solution also minimizes the action within the class {v is a solution,v has finite action,v ≢0}.

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Authors and Affiliations

  1. Département de Mathématiques, Université Paris VI, 4, Place Jussieu, F-75230, Paris, Cédex 05, France

    Haim Brezis

  2. Departments of Mathematics and Physics, Princeton University, P.O. Box 708, 08544, Princeton, NJ, USA

    Elliott H. Lieb

Authors
  1. Haim Brezis
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  2. Elliott H. Lieb
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Additional information

Communicated by A. Jaffe

Work partially supported by U.S. National Science Foundation Grant PHY-81-16101-A02

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Brezis, H., Lieb, E.H. Minimum action solutions of some vector field equations. Commun.Math. Phys. 96, 97–113 (1984). https://doi.org/10.1007/BF01217349

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  • DOI: https://doi.org/10.1007/BF01217349

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