Variational Inequalities

Abstract

The minimum problem we mentioned in the introduction to Chapter 2 can be generalized as follows:

$$\begin{gathered} \min imize\quad F\left( v \right) \equiv \frac{1}{2}\int_{\Omega } {\left( {{{\left| {\Delta v} \right|}^{2}} + {v^{2}}} \right)} dx - \int_{\Omega } {fvdx} \hfill \\ over\;a\;convex\;subset\;K\;of\;H_{0}^{1}\left( {\Omega \cup \Gamma } \right) \hfill \\ \end{gathered} $$

[with f ∈ L ² (Ω), Γ of class C ¹]. If u is a solution to this problem, for any choice of v in 𝕂 the function ℱ(u + λ(v — u)) of λ ∈ [0, 1] must attain its minimum at λ = 0; hence, u must satisfy the condition

$$u \in K,\quad \frac{d}{{d\lambda }}F\left( {u + \lambda \left( {v - u} \right)} \right)\left| {_{{\lambda = 0}} \geqslant 0\quad for\;v \in K,} \right. $$

which amounts to

$$u \in K,\quad a\left( {u,v - u} \right) \geqslant \int_{\Omega } {f\left( {v - u} \right)} dx\quad for\;v \in K $$

((4.1))

[where a(u, v) denotes the symmetric bilinear form (math)]. Vice versa, a solution of (4.1) necessarily minimizes ℱ(v) over 𝕂 (see Lemma 4.1 below). These simple observations are sufficient to introduce the content of the present chapter.