%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 Mumford–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 No. of bitstreams: 1 bonmor.pdf: 602550 bytes, checksum: 45e935bdd017828bd56281b8e33a3d3a (MD5) %& 533 %R 10.1016/j.anihpc.2014.01.006 %0 Journal Article %J Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 1291-1331 %D 2010 %T Exact reconstruction of damaged color images using a total variation model %A Irene Fonseca %A Giovanni Leoni %A Francesco Maggi %A Massimiliano Morini %X In this paper the reconstruction of damaged piecewice constant color images is studied using a RGB total variation based model for colorization/inpainting. In particular, it is shown that when color is known in a uniformly distributed region, then reconstruction is possible with maximal fidelity. %B Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 1291-1331 %I Elsevier %G en_US %U http://hdl.handle.net/1963/4039 %1 363 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-09-06T09:54:56Z\\nNo. of bitstreams: 1\\nflmm.pdf: 434739 bytes, checksum: f315d8807349e077c524fe14a1619a56 (MD5) %R 10.1016/j.anihpc.2010.06.004 %0 Journal Article %J SIAM J. Math. Anal. 40 (2009) 2351-2391 %D 2009 %T A higher order model for image restoration: the one dimensional case %A Gianni Dal Maso %A Irene Fonseca %A Giovanni Leoni %A Massimiliano Morini %X The higher order total variation-based model for image restoration proposed by Chan, Marquina, and Mulet in [6] is analyzed in one dimension. A suitable functional framework in which the minimization problem is well posed is being proposed and it is proved analytically that the\\nhigher order regularizing term prevents the occurrence of the staircase effect. The generalized version of the model considered here includes, as particular cases, some curvature dependent functionals. %B SIAM J. Math. Anal. 40 (2009) 2351-2391 %G en_US %U http://hdl.handle.net/1963/3174 %1 1127 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-23T07:52:52Z\\nNo. of bitstreams: 1\\nDM-Fon-Leo-Mor-08-preprint.pdf: 336946 bytes, checksum: 32db893a2b928f559b6744296e1d4f2c (MD5) %R 10.1137/070697823 %0 Journal Article %J Netw. Heterog. Media 3 (2008) 567-614 %D 2008 %T Globally stable quasistatic evolution in plasticity with softening %A Gianni Dal Maso %A Antonio DeSimone %A Maria Giovanna Mora %A Massimiliano Morini %X We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each\\ntime interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response. %B Netw. Heterog. Media 3 (2008) 567-614 %G en_US %U http://hdl.handle.net/1963/1965 %1 2228 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-05-15T11:03:21Z\\nNo. of bitstreams: 1\\nDM-DeS-Mor-Mor-05-3-final.pdf: 395475 bytes, checksum: 5789494df44986207d416d5d1ea15d22 (MD5) %0 Report %D 2008 %T A note on the differentiability of Lipschitz functions and the chain rule in Sobolev spaces %A Massimiliano Morini %G en_US %U http://hdl.handle.net/1963/2654 %1 1469 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-05-12T11:42:20Z\\nNo. of bitstreams: 1\\nchainmor.pdf: 232603 bytes, checksum: 2798c276d92d0d30c48523748c30a163 (MD5) %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. Ration. Mech. Anal. 189 (2008) 469-544 %D 2008 %T A vanishing viscosity approach to quasistatic evolution in plasticity with softening %A Gianni Dal Maso %A Antonio DeSimone %A Maria Giovanna Mora %A Massimiliano Morini %X We deal with quasistatic evolution problems in plasticity with softening, in the framework of small strain associative elastoplasticity. The presence of a nonconvex term due to the softening phenomenon requires a nontrivial extension of the variational framework for rate-independent problems to the case of a nonconvex energy functional. We argue that, in this case, the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view. Thus, we analize a different selection criterion for the solutions of the quasistatic evolution problem, based on a viscous approximation. This leads to a generalized formulation in terms of Young measures, developed in the first part of the paper. In the second part we apply our approach to some concrete examples. %B Arch. Ration. Mech. Anal. 189 (2008) 469-544 %G en_US %U http://hdl.handle.net/1963/1844 %1 2373 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-07-21T11:03:58Z\\nNo. of bitstreams: 1\\nmath.AP-0606718.pdf: 563956 bytes, checksum: 80e8ac8a9120c15b452698af3abc77d1 (MD5) %R 10.1007/s00205-008-0117-5 %0 Journal Article %J Arch. Ration. Mech. Anal. 186 (2007) 477-537 %D 2007 %T Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results %A Irene Fonseca %A Nicola Fusco %A Giovanni Leoni %A Massimiliano Morini %X Strained epitaxial films grown on a relatively thick substrate are considered in the context of plane linear elasticity. The total free energy of the system is assumed to be the sum of the energy of the free surface of the film and the strain energy. Because of the lattice mismatch between film and substrate, flat configurations are in general energetically unfavorable and a corrugated or islanded morphology is the preferred growth mode of the strained film. After specifying the functional setup in which the existence problem can be properly framed, a study of the qualitative properties of the solutions is undertaken. New regularity results for volume-constrained local minimizers of the total free energy are established, leading, as a byproduct, to a rigorous proof of the zero-contact-angle condition between islands and wetting layers. %B Arch. Ration. Mech. Anal. 186 (2007) 477-537 %G en_US %U http://hdl.handle.net/1963/2350 %1 1666 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-11-06T11:41:16Z\\nNo. of bitstreams: 1\\nwetting-december-30-06.pdf: 1519775 bytes, checksum: 40f435ddce013b1017d5c3e2369fa7c8 (MD5) %R 10.1007/s00205-007-0082-4 %0 Journal Article %J J. Eur. Math. Soc. (JEMS) 9 (2007) 219-252 %D 2007 %T Necessary and sufficient conditions for the chainrule in W1,1loc(RN;Rd) and BVloc(RN;Rd) %A Giovanni Leoni %A Massimiliano Morini %X

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