Diploma thesis as pdf file - Johannes Kepler University, Linz

2.1 The central path 15
Then the function θ(t) = η(t)−η(0)
t
from below by 0. Hence there exists
Further we have
∫
0 ≤ η ′ (0) =
It also follows
is monotone nondecreasing and it’s bounded
η ′ η(t) − η(0)
(0) = lim
≥ 0, t ∈ (0,1).
t→0+ t
Ω
φ(v t ) − φ(v 0 )
t
φ(v t ) − φ(0)
∇u κ ∇(ū − u κ )dω + 〈 f ,ū − u κ 〉 + κ lim
.
t→0 t
∫
= κ t −1 (ln(v 0 − ψ) − ln(v t − ψ))dω.
Ω
As t → 0, the integrand converges to u κ−ū
u κ −ψ a.e. in Ω. Further, on the set {u κ ≥
ū} we have
0 ≤ t −1 (ln(v 0 − ψ) − ln(v t − ψ)) ≤ u κ − ū
v t − ψ ≤ u κ − ψ
v t − ψ ≤ 1
1 −t ,
and hence, by theorem of dominated convergence, we obtain
∫
∫
lim t −1 u κ − ū
(ln(v 0 − ψ) − ln(v t − ψ))dω =
t→0+ {u κ ≥ū}
{u κ ≥ū} u κ − ψ dω.
On the other hand, on the set u κ < ū we have
t −1 (ln(v 0 − ψ) − ln(v t − ψ)) ↘ u κ − ū
u κ − ψ
< 0, for t → 0 + .
Since
∫
lim
t→0+ {u κ