@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.
In this paper we prove necessary and sufficient conditions for the validity of the classical chain rule in Sobolev spaces and in the space of functions of bounded variation.
}, doi = {10.4171/JEMS/78}, url = {http://hdl.handle.net/1963/2037}, author = {Giovanni Leoni 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 {2007, title = {Time-dependent systems of generalized Young measures}, number = {SISSA;98/2005/M}, year = {2007}, abstract = {In this paper some new tools for the study of evolution problems in the framework of Young measures are introduced. A suitable notion of time-dependent system of generalized Young measures is defined, which allows to extend the classical notions of total variation and absolute continuity with respect to time, as well as the notion of time derivative. The main results are a Helly type theorem for sequences of systems of generalized Young measures and a theorem about the existence of the time derivative for systems with bounded variation with respect to time.}, url = {http://hdl.handle.net/1963/1795}, author = {Gianni Dal Maso and Antonio DeSimone and Maria Giovanna Mora and Massimiliano Morini} } @article {2004, title = {Higher order quasiconvexity reduces to quasiconvexity}, journal = {Arch. Ration. Mech. Anal. 171 (2004) 55-81}, number = {SISSA;41/2003/M}, year = {2004}, publisher = {Springer}, abstract = {In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly 2-quasiconvex function with p-growth at infinity, p>1, is the restriction to symmetric matrices of a 1-quasiconvex function with the same growth. As a consequence, lower semicontinuity results for second-order variational problems are deduced as corollaries of well-known first order theorems.}, doi = {10.1007/s00205-003-0278-1}, url = {http://hdl.handle.net/1963/2911}, author = {Gianni Dal Maso and Irene Fonseca and Giovanni Leoni and Massimiliano Morini} } @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} } @article {2002, title = {Global calibrations for the non-homogeneous Mumford-Shah functional}, journal = {Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002) 603-648}, number = {SISSA;41/2001/M}, year = {2002}, publisher = {Scuola Normale Superiore di Pisa}, abstract = {Using a calibration method we prove that, if $\\\\Gamma\\\\subset \\\\Omega$ is a closed regular hypersurface and if the function $g$ is discontinuous along $\\\\Gamma$ and regular outside, then the function $u_{\\\\beta}$ which solves $$ \\\\begin{cases} \\\\Delta u_{\\\\beta}=\\\\beta(u_{\\\\beta}-g)\& \\\\text{in $\\\\Omega\\\\setminus\\\\Gamma$} \\\\partial_{\\\\nu} u_{\\\\beta}=0 \& \\\\text{on $\\\\partial\\\\Omega\\\\cup\\\\Gamma$} \\\\end{cases} $$ is in turn discontinuous along $\\\\Gamma$ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional $$ \\\\int_{\\\\Omega\\\\setminus S_u}|\\\\nabla u|^2 dx +{\\\\cal H}^{n-1}(S_u)+\\\\beta\\\\int_{\\\\Omega\\\\setminus S_u}(u-g)^2 dx, $$ over $SBV(\\\\Omega)$, for $\\\\beta$ large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.}, url = {http://hdl.handle.net/1963/3089}, author = {Massimiliano Morini} } @mastersthesis {2001, title = {Free-discontinuity problems: calibration and approximation of solutions}, year = {2001}, school = {SISSA}, keywords = {Calibration of solutions}, url = {http://hdl.handle.net/1963/5398}, author = {Massimiliano Morini} } @article {2001, title = {Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set}, journal = {Ann. I. H. Poincare - An., 2001, 18, 403}, number = {SISSA;21/00/M}, year = {2001}, publisher = {SISSA Library}, doi = {10.1016/S0294-1449(01)00075-0}, url = {http://hdl.handle.net/1963/1479}, author = {Maria Giovanna Mora and Massimiliano Morini} } @article {2000, title = {Functionals depending on curvatures with constraints}, journal = {Rend. Sem. Mat. Univ. Padova 104 (2000), 173--199}, number = {SISSA;85/99/M}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1299}, author = {Maria Giovanna Mora and Massimiliano Morini} } @article {2000, title = {Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets}, journal = {J. Math. Pures Appl. 79, 2 (2000) 141-162}, number = {SISSA;47/99/M}, year = {2000}, publisher = {SISSA Library}, doi = {10.1016/S0021-7824(99)00140-3}, url = {http://hdl.handle.net/1963/1261}, author = {Gianni Dal Maso and Maria Giovanna Mora and Massimiliano Morini} }