Introduction to Minimax Methods
This chapter is an introduction to a broad class of methods that have been shown to be extremely useful in a variety of contexts.
We confine ourselves to the simplest cases, but we try to motivate the ideas involved in the construction of the main tools, so that the interested reader can turn to the study of more complex problems with a minimum of background.
In the preceding chapters we have mainly looked for critical points of a functional as minimum points, either on the whole space, or suitably restricting the functional to sets where minima could be shown to exist. In this chapter, on the contrary, we concentrate on the search of critical points that are not global minima, for example saddle points. The procedures to do this, called minimax methods, are quite elaborate, and we introduce the main steps gradually.
KeywordsWeak Solution Smale Condition Critical Point Theory Smale Sequence Mountain Pass Theorem
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