In this section we present necessary and/or sufficient conditions for local extrema of real functionals. The most famous ones are the Euler and Lagrange necessary conditions and the Lagrange sufficient condition. We also present the brachistochrone problem, one of the oldest problems in the calculus of variations. We also discuss regularity of the point of a local extremum. The methods presented in this section are motivated by the equation f(x) = 0 (6.1.1) where f is a continuous real function defined in ℝ. The solution of this equation can be transformed to the problem of finding a local extremum of the integral F of f (i.e., F′(x) = f(x), x ∈ ℝ). Indeed, if there exists a point x0 ∈ ℝ at which the function F has its local extremum, then the derivative F′(x0) necessarily vanishes due to a familiar theorem of the first-semester calculus. The problem of finding solutions of (6.1.1) can be thus transformed to the problem of finding local extrema of the function F. On the other hand, one should keep in mind that the equation (6.1.1) may have a solution which is not a local extremum of F.
KeywordsBanach Space Weak Solution Variational Method Real Hilbert Space Local Extremum
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