Diploma thesis as pdf file - Johannes Kepler University, Linz

2.1 The central path 13
By virtue of lower semicontinuity of F(u):
Then it holds that
so u = ū is the solution to the problem.
lim F(u n) ≥ F(ū).
n→∞
F(ū) ≤ inf
u∈V F(u),
Let now u 1 ,u 2 ∈ V be two solutions to the problem. Then 1 2 (u 1 + u 2 ) ∈ V. If
F(u) is strictly convex function, then
( )
1
F
2 (u 1 + u 2 )
< min
u∈V F(u),
so only one solution exists for the strictly convex F(u).
We make an assumption (A1):
∃ũ ∈ H 1 0 (Ω) :
∫
− ln(ũ − ψ)dω < ∞.
Ω
Theorem 2.1.3 With the assumption (A1) for each κ > 0 the problem (2.1.1) has
a unique solution u κ .
Proof Let κ > 0 be an arbitrary fixed number. We apply theorem 2.1.2 with
V = H0 1(Ω) and F(u) = J κ(u). J κ (u) is the sum of the strictly convex continuous
∫
function J(u) and the function φ(u) = − ln(u − ψ)dω. It’s clear that φ(u) is a
convex function and consequently J κ (u) is strictly convex. Using the inequality
lnz ≤ z for z > 0 we obtain
∫
φ(u) ≥ − max{u − ψ,0} ≥ −‖u − ψ‖ L 2 (Ω) µ(Ω) 1 2 .
Ω
Since ‖u − ψ‖ ≥ ‖u − ψ‖ L 2 (Ω), together with the assumption (A1) it follows that
φ and hence also J κ is proper. Moreover, we obtain
J κ (u) ≥ c‖u‖ 2 − ‖ f ‖‖u‖ − κ‖u − ψ‖µ(Ω) 1 2
Ω