%0 Journal Article
%J Annales de l'Institut Henri Poincare (C) Non Linear Analysis
%D 2015
%T Stable regular critical points of the Mumford-Shah functional are local minimizers
%A Marco Bonacini
%A Massimiliano Morini
%K Mumford-Shah functional
%X In this paper it is shown that any regular critical point of the Mumfordâ€“Shah functional, with positive definite second variation, is an isolated local minimizer with respect to competitors which are sufficiently close in the $L^1$

-topology. A global minimality result in small tubular neighborhoods of the discontinuity set is also established.

%B Annales de l'Institut Henri Poincare (C) Non Linear Analysis
%I SISSA
%V 32
%P 533-570
%G en
%U https://www.sciencedirect.com/science/article/pii/S0294144914000171
%N 3
%1 6992
%2 Mathematics
%4 1
%# MAT/05 ANALISI MATEMATICA
%$ Submitted by Marco Bonacini (mbonacin@sissa.it) on 2013-07-25T12:49:07Z
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%& 533
%R 10.1016/j.anihpc.2014.01.006
%0 Journal Article
%J Calc. Var. Partial Differential Equations 33 (2008) 37-74
%D 2008
%T A second order minimality condition for the Mumford-Shah functional
%A Filippo Cagnetti
%A Maria Giovanna Mora
%A Massimiliano Morini
%X A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\\\\Gamma)$, $\\\\Gamma$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $\\\\Gamma$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.
%B Calc. Var. Partial Differential Equations 33 (2008) 37-74
%G en_US
%U http://hdl.handle.net/1963/1955
%1 2318
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-03-02T12:54:00Z\\nNo. of bitstreams: 1\\nCMM.pdf: 358759 bytes, checksum: c414c0080a17971ecba1251a5890b94f (MD5)
%R 10.1007/s00526-007-0152-3
%0 Journal Article
%J Arch. Rational Mech. Anal. 183 (2007) 411-456
%D 2007
%T Surfactants in Foam Stability: A Phase-Field Model
%A Irene Fonseca
%A Massimiliano Morini
%A Valeriy Slastikov
%X The role of surfactants in stabilizing the formation of bubbles in foams is studied using a phase-field model. The analysis is centered on a van der Walls-Cahn-Hilliard-type energy with an added term accounting for the interplay between the presence of a surfactant density and the creation of interfaces. In particular, it is concluded that the surfactant segregates to the interfaces, and that the prescriptionof the distribution of surfactant will dictate the locus of interfaces, what is in agreement with experimentation.
%B Arch. Rational Mech. Anal. 183 (2007) 411-456
%G en_US
%U http://hdl.handle.net/1963/2035
%1 2161
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-09-03T15:51:40Z\\nNo. of bitstreams: 1\\n05-CNA-012.pdf: 414998 bytes, checksum: 3c44841388edba53080d531beadad798 (MD5)
%R 10.1007/s00205-006-0012-x
%0 Journal Article
%J SIAM J. Math. Anal. 35 (2003) 759-805
%D 2003
%T Sequences of Singularly Perturbed Functionals Generating Free-Discontinuity Problems
%A Massimiliano Morini
%X We prove that a wide class of singularly perturbed functionals generates as $\\\\Gamma$-limit a functional related to a free-discontinuity problem. Several applications of the result are shown.
%B SIAM J. Math. Anal. 35 (2003) 759-805
%I SIAM
%G en_US
%U http://hdl.handle.net/1963/3071
%1 1262
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-10T10:24:32Z\\nNo. of bitstreams: 1\\nsingper.pdf: 523839 bytes, checksum: 9d1bcc064d165e9b964422c8acd396bb (MD5)
%R 10.1137/S0036141001395388