Elliptic Variational Inequalities - Computational Methods for Quantitative Finance - page 271

A.3 Theorems of Stampacchia and Lax-Milgram

The theorems of Stampacchia and Lax-Milgram are useful existence results for
stationary problems. Let

V · V and innerproduct

be a Hilbert space with a norm

· , · V .

Definition A.3.1 A bilinear form b( · , · ) : V × V → R is

(i) continuous , if there exists C 1 > 0 such that
∀ u, v ∈ V :| b(u, v) |≤ C 1 u V v V , (A.15)

(ii) coercive , if there exists C 2 > 0 such that
∀

2
V

u ∈ V : b(u, u) ≥ C 2 u . (A.16)

Theorem A.3.2 Let b( · , · ) : V × V → R be a continuous and coercive bilinear form

∈ V ∗ V ∅= K ⊂ V

on , be closed and convex . Then , for any there exists a unique

u ∈ K solution of the variational inequality

Find u ∈ K

such that
b(u, v − u) ≥ (v − u)

(A.17)

∀ v ∈ K .

The proof uses

Theorem A.3.3 (Banach's fixed point theorem) Let X be a complete metric space
and S : X → X a mapping with

d( S v 1 , S v 2 ) ≤ κd(v 1 ,v 2 )

∀ v 1 ,v 2 ∈ S (A.18)

for some κ< 1. Then , the problem
Find u

∈ X such that u = S u (A.19)

admits a unique solution .

∈ V ∗ , there exists, by Theorem A.2.1 , a unique Proof of Theorem A.3.2 Given

f ∈ V such that

∀ v ∈ V : (v) = f, v V .

V ∗ For every fixed u ∈ V ,themap v −→ b(u, v) is in and there is a unique repre-

sentative in V , denoted by Bu , such that b(u, v) = Bu,v V , ∀ v ∈ V . The operator

B : V → V is linear and, by ( A.15 ), ( A.16 ),

Bu V ≤ C 1 u V ∀ u ∈ V , (A.20)

2
V

Bu,u V ≥ C 2 u ∀ u ∈ V . (A.21)

Hence, ( A.17 ) may be rewritten as
Find u ∈ K such that

(A.22)

− V ≥ − V ∀ ∈ K

Bu,v u f, v u v .