Minimization Techniques: Compact Problems
Part of the
book series (UTX)
Throughout this chapter we show how techniques based on minimization arguments can be used to establish existence results for various types of problems.
Our aim is not to describe the most general results, but to give a series of examples, and to show how simple techniques can be refined to treat more complex cases.
We start from sublinear problems, for which the energy functionals are bounded from below. Direct minimization arguments, based on convexity and coercivity allow one to establish rather easily the existence of a solution.
Next we turn to superlinear problems. For these types of nonlinearities direct minimization does not work anymore: the corresponding energy functionals are generally unbounded from below. We then present some methods of constrained minimization, where one restricts the functional to a subset of functions on which it is bounded from below, and tries to establish the existence of a minimum point. Special care must then be employed to show that the constrained minimum is truly a critical point of the unconstrained functional.
KeywordsWeak Solution Nontrivial Solution Trivial Solution Nonlinear Eigenvalue Problem Bibliographical Note
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