BV functions in Hilbert spaces
We study the basic theory of BV functions in a Hilbert space X endowed with a (not necessarily Gaussian) probability measure ν . We present necessary and sufficient conditions in order that a function u ∈ L p ( X , ν ) is of bounded variation. We also discuss the De Giorgi approach to BV functions t...
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Veröffentlicht in: | Mathematische annalen 2021-12, Vol.381 (3-4), p.1653-1722 |
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Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
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Zusammenfassung: | We study the basic theory of
BV
functions in a Hilbert space
X
endowed with a (not necessarily Gaussian) probability measure
ν
. We present necessary and sufficient conditions in order that a function
u
∈
L
p
(
X
,
ν
)
is of bounded variation. We also discuss the De Giorgi approach to
BV
functions through the behavior as
t
→
0
of
∫
X
‖
∇
T
(
t
)
u
‖
d
ν
, for a smoothing semigroup
T
(
t
). Particular attention is devoted to the case where
u
is the indicator function of a sublevel set
{
x
:
g
(
x
)
<
r
}
of a real Borel function
g
. We give several examples, for different measures
ν
such as weighted Gaussian measures, infinite products of non Gaussian measures, and invariant measures of some stochastic PDEs such as reaction-diffusion equations and Burgers equation. |
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ISSN: | 0025-5831 1432-1807 |
DOI: | 10.1007/s00208-020-02037-x |