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Perimeter, Mumford-Shah functionals, and doubling metric measures. (English) Zbl 1115.49034
Birindelli, Isabeau (ed.) et al., Proceedings of the workshop on second order subelliptic equations and applications, Cortona, Italy, June 16–22, 2003. Potenza: Università degli Studi della Basilicata, Dipartimento di Matematica, S.I.M. Lecture Notes of Seminario Interdisciplinare di Matematica 3, 27-37 (2004).
The authors study the problem of existence of minima for the Mumford-Shah functional defined on weighted $$BV$$ functions $\int_\Omega | \nabla u| ^2 \omega^{1-\frac{2}{n}}dx +\int_{S^\omega_u}\omega^{1-\frac{1}{n}}d{\mathcal H}^{n-1}+\int_\Omega | u-g| ^2dx.$ The framework is that of weighted spaces with $$\omega$$ a strong $$A_\infty$$ weight (as introduced by G. David and S. Semmes [Lect. Notes Pure Appl. Math. 122, 101–111 (1990; Zbl 0752.46014)]). The results of the present paper are essentially contained in the previous papers [Calc. Var. Partial Differ. Equ. 16, No. 3, 283–298 (2003; Zbl 1025.49028)] and [Proc. R. Soc. Edinb., Sect. A, Math. 135, No. 1, 1–23 (2005; Zbl 1172.26312)].
For the entire collection see [Zbl 1058.00010].
MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting 28A12 Contents, measures, outer measures, capacities