Variational Boundary-Value Problems, Monotone Operators, and Variational Inequalities

Abstract

If K(u) is a differentiable functional on a Banach space U, and if P: U → U′ is its gradient, we have shown that the abstract problem of finding u ∈ U such that MATH

$$ {\left\langle {P\left( {\text{u}} \right),\,\eta } \right\rangle _U} = 0\,\,\,\,\,\,\,\,\,\forall \eta \in U $$

(6.1)

is equivalent to finding critical points of K(u). The classical variational method is, therefore, an indirect method; instead of solving directly the weak problem (6.1), we seek special points in the domain of an associated functional.