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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