Existence of the Minimum of the Functional F in the Space H _A . Generalized Solutions

Abstract

Consider the functional

$$Fu = \left( {Au, u} \right) - 2\left( {f, u} \right),\,\,\,\,\,u \in {D_A},$$

(11.1)

from Chap. 9 (p. 114). Under the assumption that the operator A is positive on the linear set D _A , we know, by Theorem 9.2, that if the equation Au = f has a solution¹) u₀ ∈ D _A , then the functional F assumes, for this element u₀, its minimal value among all the elementsu₀ ∈ D _A , and - conversely — if F assumes on D _A its minimal value for a certain element u₀ ∈ D _A , then this element is the solution of the equation Au = f in H. At the end of Chap. 9 we have noted that Theorem 9.2 has a conditional character since it assumes either the existence of the solution u ₀ ∈ D _A of the given equation or the existence of the element u₀ ∈ D _A minimizing the functional F on D _A , while the existence of such an element u₀ is a priori guaranteed in neither the first nor the second case.