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research
Global calibrations for the non-homogeneous Mumford-Shah functional
Authors
Massimiliano Morini
Publication date
1 January 2001
Publisher
View
on
arXiv
Abstract
Using a calibration method we prove that, if
Γ
⊂
Ω
\Gamma\subset \Omega
Γ
⊂
Ω
is a closed regular hypersurface and if the function
g
g
g
is discontinuous along
Γ
\Gamma
Γ
and regular outside, then the function
u
β
u_{\beta}
u
β
​
which solves
{
Δ
u
β
=
β
(
u
β
−
g
)
inÂ
Ω
∖
Γ
∂
ν
u
β
=
0
onÂ
∂
Ω
∪
Γ
\begin{cases} \Delta u_{\beta}=\beta(u_{\beta}-g)& \text{in $\Omega\setminus\Gamma$} \partial_{\nu} u_{\beta}=0 & \text{on $\partial\Omega\cup\Gamma$} \end{cases}
{
Δ
u
β
​
=
β
(
u
β
​
−
g
)
​
in Ω
∖
Γ
∂
ν
​
u
β
​
=
0
​
onÂ
∂
Ω
∪
Γ
​
is in turn discontinuous along
Γ
\Gamma
Γ
and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional
∫
Ω
∖
S
u
∣
∇
u
∣
2
d
x
+
H
n
−
1
(
S
u
)
+
β
∫
Ω
∖
S
u
(
u
−
g
)
2
d
x
,
\int_{\Omega\setminus S_u}|\nabla u|^2 dx +{\cal H}^{n-1}(S_u)+\beta\int_{\Omega\setminus S_u}(u-g)^2 dx,
∫
Ω
∖
S
u
​
​
∣∇
u
∣
2
d
x
+
H
n
−
1
(
S
u
​
)
+
β
∫
Ω
∖
S
u
​
​
(
u
−
g
)
2
d
x
,
over
S
B
V
(
Ω
)
SBV(\Omega)
SB
V
(
Ω
)
, for
β
\beta
β
large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.Comment: 33 page
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Archivio istituzionale della Ricerca - Università degli Studi di Parma
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Numérisation de Documents Anciens Mathématiques
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