Michele Miranda (University of Lecce)
Special Functions of Bounded Variation in Doubling Metric Measure Spaces
Abstract.
In this lecture we present the theory of special functions of bounded
variation (characterised by a total variation measure which is the sum of
a
``volume'' energy and of a ``surface'' energy) to metric measure spaces
endowed with a Poincare' inequality. In this framework, which includes
all Carnot--Caratheodory spaces, we use and improve previous results in
[1], [2] and [5] to show the basic compactness
theorem of special functions of bounded variation. In a particular class
of ``isotropic'' spaces, which includes all Carnot groups of step 2 and
spaces induced by a continuous and strong A-infinity weight, we are able
to
show the lower semicontinuity of a Mumford-Shah type functional, extending
previous results by Song and Yang [6], Citti, Manfredini and
Sarti [4] in the Heisenberg group and by Franchi and Baldi [3]
in weighted spaces.
[1] Ambrosio, L. "Some fine properties of sets of finite perimeter in
Ahlfors-regular metric spaces"
Advances in Math. (2001), 51-67
[2] Ambrosio, L. "Fine properties of sets of finite perimeter in doubling
metric measure spaces"
Set Valued An. (2002), 111-128
[3] Bald, A. and Franchi, B. "Mumford-Shah type functionals associated
with doubling metric measures"
preprint.
[4] Citti, G., Manfredini, M. and Sarti, A. "A note on the Mumford-Shah in
Heisenberg space"
preprint.
[5] Miranda, M.Jr. "Functions of bounded variation on good metric spaces"
J. Math. Pures et Appl., 82, (2003), 975-1004
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