Diploma thesis as pdf file - Johannes Kepler University, Linz

1.3 Alternative equivalent formulations 7
Proof Let u ∈ K be the solution to the minimization problem (1.1.1). Then for
t ∈ [0,1] : u+t(v−u) ∈ K ∀v ∈ K. Function defined by φ(t) = J(u+t(v−u)), t ∈
[0,1] attains its minimum at the point t = 0, i.e.
φ(0) ≤ φ(t) ∀t ∈ [0,1].
Then
∫
φ(t) − φ(0)
0 ≤ lim
= ∇u∇(v − u)dω − 〈 f ,v − u〉.
t→0+ t
Ω
∫
Let u ∈ K be such that ∇u∇(v − u)dω ≥ 〈 f ,v − u〉 ∀v ∈ K. Then for any
Ω
v ∈ K it holds :
∫
J(v)−J(u) = φ(1)−φ(0) = ∇u∇(v−u)dω −〈 f ,v−u〉+ 1 ∫
Ω 2
The next theorem states about the well-posedness of the problem:
Ω
|∇(v−u)| 2 dω ≥ 0.
Theorem 1.3.3 There exist unique solution to the problem (1.3.1). In addition,
the mapping f → u is Lipschitz, that is, if u 1 ,u 2 are solutions to the problem
(1.3.1) corresponding to f 1 , f 2 ∈ H −1 , then
where L > 0 constant.
‖u 1 − u 2 ‖ ≤ L‖ f 1 − f 2 ‖ H −1, (1.1)
Proof Existence of the unique solution results from the previous discussions,
namely, from the equivalence of the variational inequality (1.3.1) to the minimization
problem (1.1.1).
We demonstrate validity of (1.1). We set v = u 2 in the variational inequality
for the solution u 1 and v = u 1 in the inequality for u 2 . Upon adding we obtain:
∫
|∇(u 1 − u 2 )| 2 dω ≤ 〈 f 1 − f 2 ,u 1 − u 2 〉.
Ω
From the coercitivity of the form a(u,v) = ∫ Ω ∇u∇vdω, it follows that
C(Ω)‖u 1 − u 2 ‖ 2 ≤ 〈 f 1 − f 2 ,u 1 − u 2 〉 ≤ ‖ f 1 − f 2 ‖ H −1‖u 1 − u 2 ‖.