Page 82 - IJOCTA-15-3

C. D’Apice et .al. / IJOCTA, Vol.15, No.3, pp.449-463 (2025)
if φ ∈ BV (Ω), and and setting
!
2
Λ ε,τ (φ) = ∞, if φ ∈ L (Ω) \ BV (Ω). f
V 1 φ 0 = − f − c 0 log
ε,τ ε,τ,0 0
c
ε,τ,0
1
Taking into account the fact that Ψ ∈ C (R), × δ ε (l 1 − φ 0 − τ)
loc ε,τ
it can be shown that !
m−1
X 0 f

Φ ε φ 0 ε,τ + λh = Φ ε φ 0 ε,τ + f − c ε,τ,j log 0
j=1 c ε,τ,j

+ λD φ Φ ε φ 0 [h] + r(h, λ), 0 0
ε,τ
× δ ε (φ ε,τ − l j − τ)H ε (−φ ε,τ + l j+1 + τ)
2
for any h ∈ L (Ω), where, as λ → 0,|r(h, λ)| = m−1 !
2
o(|λ|), and D φ Φ ε (φ 0 ε,τ ) : L (Ω) → R is a linear − X f − c 0 log f
continuous functional with the following represen- ε,τ,j c 0 ε,τ,j
j=1
tation:
× δ ε (−φ 0 + l j+1 + τ)H ε (φ 0 − l j − τ)
Z ε,τ ε,τ
1 h ′ 0
0

D φ Φ ε φ ε,τ [h] = − Ψ (l 0 − φ ε,τ )h dx f
ε Ω + f − c 0 log δ ε (φ 0 − l m − τ)
Z ε,τ,m c 0 ε,τ
i ε,τ,m
′
+ Ψ (φ 0 ε,τ − l m+1 )h dx . (23) h 0 1−τ 0
Ω + (φ ε,τ − l m+1 ) , if φ ε,τ − l m+1 > 0,
δ−1 0, otherwise,
′
Here, Ψ (z) = δz , if z ≥ 0, . 0 1−τ 0 i
0, if z < 0 (l 0 − φ ε,τ ) , if l 0 − φ ε,τ > 0, 2 − τ
− ,
0, otherwise, ε
We also observe that the right-hand side in (28)
(23) can be represented as follows:
J (w, φ) =
1 Z ′ 0 ′ 0 0  Z
Ψ (φ − l m+1 ) − Ψ (l 0 − φ ) [φ − φ ] dx
f
ε Ω ε,τ ε,τ ε,τ  L (w) |M Dφ| if φ ∈ BV (Ω),
ε
2 − τ Z 0  Ω 2
= [φ − φ ε,τ ] +∞ if φ ∈ L (Ω) \ BV (Ω),
ε Ω (29)
0 1−τ 0
(φ − l m+1 ) , if φ − l m+1 > 0, m
× ε,τ ε,τ dx X
′
0, otherwise V 2 φ 0 ε,τ = α δ (φ 0 ε,τ − l j + τ), (30)
ε
j=1
Z
2 − τ
− [φ − φ 0 ε,τ ] we see that inequality (25) leads to the relation
ε Ω
Z Z
0 1−τ 0
f
(l 0 − φ ) , if l 0 − φ > 0, V 1 φ 0 φ dx + V 2 φ 0 φ |M Dφ 0 |
× ε,τ ε,τ dx. ε,τ ε,τ ε ε,τ
0, otherwise Ω Ω
i
h
(24) + J (w, φ) − J (w, φ 0 ≥ o(1)h,
ε,τ
)
w=φ
2
∀ φ ∈ L (Ω), (31)
Since (c 0 ε,τ , φ 0 ) is an optimal pair to the ap- 0
ε,τ
L (Ω)
proximated minimization problem (8)–(12), it fol- where h = φ−φ ε,τ , and o(1) → 0 as ∥h∥ 2 → 0.
lows from representation (19)–(22) that
J ε,τ (c 0 , φ) − J ε,τ (c 0 , φ 0 ) ≥ 0 Hence,
ε,τ ε,τ ε,τ (25)
2
∀ φ ∈ L (Ω). 0
− V 1 φ ε,τ ∈
Taking into account that, for each φ ∈ BV (Ω), Z 0 f 0
ε
Λ ε,τ (φ) − Λ ε,τ φ 0 ∂ V 2 φ ε,τ φ |M Dφ ε,τ | + J (w, φ) .
ε,τ Ω 0
Z w=φ ε,τ
f

= L(φ) − L(φ 0 ε,τ ) |M Dφ 0 ε,τ | Since
ε
Ω Z
Z Z 0 f 0
∂ V 2 φ φ |M Dφ | + J (w, φ)
f
f
0
+ L(φ)|M Dφ| − L(φ)|M Dφ ε,τ | Ω ε,τ ε ε,τ
ε
ε
Ω Ω Z
(26) 0 f 0
= ∂ V 2 φ ε,τ φ |M Dφ ε,τ |
ε
with Ω
+ ∂J (w, φ)
 
m
X
L(φ) =  ε + α δ ε (φ − l j + τ)  , (27) by the Moreau–Rockafellar Theorem, it follows
j=1 that the Euler–Lagrange equation for φ 0 takes
ε,τ
454