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.

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.

%B J. Eur. Math. Soc. (JEMS) 9 (2007) 219-252 %G en_US %U http://hdl.handle.net/1963/2037 %1 2159 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-09-03T16:39:37Z\\nNo. of bitstreams: 1\\n05-CNA-001.pdf: 331793 bytes, checksum: b6fd9f7cf79aacffa4d60dda74d183a6 (MD5) %R 10.4171/JEMS/78 %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 Report %D 2007 %T Time-dependent systems of generalized Young measures %A Gianni Dal Maso %A Antonio DeSimone %A Maria Giovanna Mora %A Massimiliano Morini %X 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. %B Netw. Heterog. Media 2 (2007) 1-36 %G en_US %U http://hdl.handle.net/1963/1795 %1 2749 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-04-04T13:16:42Z\\nNo. of bitstreams: 1\\n98M-2005.pdf: 339547 bytes, checksum: 695874ce74733dc0020a7e357f04cb1b (MD5) %0 Journal Article %J Arch. Ration. Mech. Anal. 171 (2004) 55-81 %D 2004 %T Higher order quasiconvexity reduces to quasiconvexity %A Gianni Dal Maso %A Irene Fonseca %A Giovanni Leoni %A Massimiliano Morini %X 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. %B Arch. Ration. Mech. Anal. 171 (2004) 55-81 %I Springer %G en_US %U http://hdl.handle.net/1963/2911 %1 1789 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-11T12:36:19Z\\nNo. of bitstreams: 1\\nmath.AP0305138.pdf: 272082 bytes, checksum: 245f93702444ac3eb1de7c86c1f83551 (MD5) %R 10.1007/s00205-003-0278-1 %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 %0 Journal Article %J Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002) 603-648 %D 2002 %T Global calibrations for the non-homogeneous Mumford-Shah functional %A Massimiliano Morini %X 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. %B Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002) 603-648 %I Scuola Normale Superiore di Pisa %G en_US %U http://hdl.handle.net/1963/3089 %1 1244 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-13T15:10:43Z\\nNo. of bitstreams: 1\\n0105141v1.pdf: 402689 bytes, checksum: 50ca626f4a7fc3ff76a466c506ca755b (MD5) %0 Thesis %D 2001 %T Free-discontinuity problems: calibration and approximation of solutions %A Massimiliano Morini %K Calibration of solutions %I SISSA %G en %U http://hdl.handle.net/1963/5398 %1 5223 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2012-02-08T08:33:00Z\\nNo. of bitstreams: 1\\nPhD_Morini_Massimiliano.pdf: 12298275 bytes, checksum: 4f513b32d38e8cca2297cce0ee09cfc6 (MD5) %0 Journal Article %J Ann. I. H. Poincare - An., 2001, 18, 403 %D 2001 %T Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set %A Maria Giovanna Mora %A Massimiliano Morini %B Ann. I. H. Poincare - An., 2001, 18, 403 %I SISSA Library %G en %U http://hdl.handle.net/1963/1479 %1 2684 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T13:02:59Z (GMT). No. of bitstreams: 1\\nmath.FA0006032.pdf: 333515 bytes, checksum: 2bace45bae580c519969228abb3d4db7 (MD5)\\n Previous issue date: 2000 %R 10.1016/S0294-1449(01)00075-0 %0 Journal Article %J Rend. Sem. Mat. Univ. Padova 104 (2000), 173--199 %D 2000 %T Functionals depending on curvatures with constraints %A Maria Giovanna Mora %A Massimiliano Morini %B Rend. Sem. Mat. Univ. Padova 104 (2000), 173--199 %I SISSA Library %G en %U http://hdl.handle.net/1963/1299 %1 3156 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:55:57Z (GMT). No. of bitstreams: 1\\nmath-FA0006031.pdf: 193412 bytes, checksum: 7d8c4469752cc70ae9c1a37a26a45acf (MD5)\\n Previous issue date: 1999 %0 Journal Article %J J. Math. Pures Appl. 79, 2 (2000) 141-162 %D 2000 %T Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets %A Gianni Dal Maso %A Maria Giovanna Mora %A Massimiliano Morini %B J. Math. Pures Appl. 79, 2 (2000) 141-162 %I SISSA Library %G en %U http://hdl.handle.net/1963/1261 %1 3194 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:55:27Z (GMT). No. of bitstreams: 1\\nmath-FA0006073.pdf: 195688 bytes, checksum: 199483eeec8fa727b6f84ce5270d3f32 (MD5)\\n Previous issue date: 1999 %R 10.1016/S0021-7824(99)00140-3