Variational and topological methods have proved to be powerful tools in the resolution of concrete nonlinear problems appearing in many disciplines where classical methods may fail. This is the case in particular for Critical Point Theory, which has become very successful in recent decades. Here, the aim of critical point theory is to look for critical points of a functional on a manifold, not necessarily in the finite dimensional setting. In general cases, one is interested not only on minima or maxima (that can be found by more direct arguments), but also on saddle points whose existence is usually proved by min-max procedures.

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Variational Problems: A Brief Retrospective

  • Yanheng Ding,
  • Tian Xu

摘要

Variational and topological methods have proved to be powerful tools in the resolution of concrete nonlinear problems appearing in many disciplines where classical methods may fail. This is the case in particular for Critical Point Theory, which has become very successful in recent decades. Here, the aim of critical point theory is to look for critical points of a functional on a manifold, not necessarily in the finite dimensional setting. In general cases, one is interested not only on minima or maxima (that can be found by more direct arguments), but also on saddle points whose existence is usually proved by min-max procedures.