@article {2013, title = {Stable regular critical points of the Mumford-Shah functional are local minimizers}, journal = {Annales de l{\textquoteright}Institut Henri Poincare (C) Non Linear Analysis}, volume = {32}, number = {SISSA preprint;SISSA 33/2013/MATE}, year = {2015}, pages = {533-570}, publisher = {SISSA}, chapter = {533}, abstract = {

In this paper it is shown that any regular critical point of the Mumford{\textendash}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.

}, keywords = {Mumford-Shah functional}, doi = {10.1016/j.anihpc.2014.01.006}, url = {https://www.sciencedirect.com/science/article/pii/S0294144914000171}, author = {Marco Bonacini and Massimiliano Morini} } @article {2008, title = {A second order minimality condition for the Mumford-Shah functional}, journal = {Calc. Var. Partial Differential Equations 33 (2008) 37-74}, number = {SISSA;82/2006/M}, year = {2008}, abstract = {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.}, doi = {10.1007/s00526-007-0152-3}, url = {http://hdl.handle.net/1963/1955}, author = {Filippo Cagnetti and Maria Giovanna Mora and Massimiliano Morini} } @article {2007, title = {Surfactants in Foam Stability: A Phase-Field Model}, journal = {Arch. Rational Mech. Anal. 183 (2007) 411-456}, year = {2007}, abstract = {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.}, doi = {10.1007/s00205-006-0012-x}, url = {http://hdl.handle.net/1963/2035}, author = {Irene Fonseca and Massimiliano Morini and Valeriy Slastikov} } @article {2003, title = {Sequences of Singularly Perturbed Functionals Generating Free-Discontinuity Problems}, journal = {SIAM J. Math. Anal. 35 (2003) 759-805}, number = {SISSA;60/2001/M}, year = {2003}, publisher = {SIAM}, abstract = {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.}, doi = {10.1137/S0036141001395388}, url = {http://hdl.handle.net/1963/3071}, author = {Massimiliano Morini} }