Free Minima for Convex Functionals, Ritz Method and the Gradient Method

In this chapter we show the intimate connection between the convexity of the functional F and the monotonicity of the operator F′ which fully corresponds to the known connection in the case of real functions F: ℝ → ℝ. In this way we obtain an approach to the theory of monotone operators F′ by means of convex minimum problems. In contrast to general minimum problems, convex minimum problems have a number of crucial advantages: (i) According to the main theorem and its variants in Sections 38.3 and 38.5, there result simple existence propositions in reflexive B-spaces. (ii) By Theorem 38.C, it follows from the strict convexity of F that the minimum point is unique. (iii) Local minima are always global minima. (iv) The Euler equation F′(u) = 0, where u ∈ int D(F), is not only a necessary condition but also a sufficient condition for a free local minimum of F at u. (v) One has productive approximation methods at one’s disposal in the Ritz and gradient methods.