Communications in Mathematical Physics

, Volume 96, Issue 1, pp 97–113 | Cite as

Minimum action solutions of some vector field equations

  • Haim Brezis
  • Elliott H. Lieb


The system of equations studied in this paper is −Δu i =g i (u) on ℝ d ,d≧2, withu:ℝ d →ℝ n andg i (u)=∂G/∂u i . Associated with this system is the action,S(u)=ε{1/2|∇u|2G(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}.


Neural Network Statistical Physic Complex System Vector Field Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Haim Brezis
    • 1
  • Elliott H. Lieb
    • 2
  1. 1.Département de MathématiquesUniversité Paris VIParis, Cédex 05France
  2. 2.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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