TY - JOUR T1 - Stable regular critical points of the Mumford-Shah functional are local minimizers JF - Annales de l'Institut Henri Poincare (C) Non Linear Analysis Y1 - 2015 A1 - Marco Bonacini A1 - Massimiliano Morini KW - Mumford-Shah functional AB -

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.

PB - SISSA VL - 32 UR - https://www.sciencedirect.com/science/article/pii/S0294144914000171 IS - 3 U1 - 6992 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Exact reconstruction of damaged color images using a total variation model JF - Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 1291-1331 Y1 - 2010 A1 - Irene Fonseca A1 - Giovanni Leoni A1 - Francesco Maggi A1 - Massimiliano Morini AB - 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. PB - Elsevier UR - http://hdl.handle.net/1963/4039 U1 - 363 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A higher order model for image restoration: the one dimensional case JF - SIAM J. Math. Anal. 40 (2009) 2351-2391 Y1 - 2009 A1 - Gianni Dal Maso A1 - Irene Fonseca A1 - Giovanni Leoni A1 - Massimiliano Morini AB - 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. UR - http://hdl.handle.net/1963/3174 U1 - 1127 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Globally stable quasistatic evolution in plasticity with softening JF - Netw. Heterog. Media 3 (2008) 567-614 Y1 - 2008 A1 - Gianni Dal Maso A1 - Antonio DeSimone A1 - Maria Giovanna Mora A1 - Massimiliano Morini AB - 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. UR - http://hdl.handle.net/1963/1965 U1 - 2228 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - A note on the differentiability of Lipschitz functions and the chain rule in Sobolev spaces Y1 - 2008 A1 - Massimiliano Morini UR - http://hdl.handle.net/1963/2654 U1 - 1469 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A second order minimality condition for the Mumford-Shah functional JF - Calc. Var. Partial Differential Equations 33 (2008) 37-74 Y1 - 2008 A1 - Filippo Cagnetti A1 - Maria Giovanna Mora A1 - Massimiliano Morini AB - 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. UR - http://hdl.handle.net/1963/1955 U1 - 2318 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A vanishing viscosity approach to quasistatic evolution in plasticity with softening JF - Arch. Ration. Mech. Anal. 189 (2008) 469-544 Y1 - 2008 A1 - Gianni Dal Maso A1 - Antonio DeSimone A1 - Maria Giovanna Mora A1 - Massimiliano Morini AB - 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. UR - http://hdl.handle.net/1963/1844 U1 - 2373 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results JF - Arch. Ration. Mech. Anal. 186 (2007) 477-537 Y1 - 2007 A1 - Irene Fonseca A1 - Nicola Fusco A1 - Giovanni Leoni A1 - Massimiliano Morini AB - 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. UR - http://hdl.handle.net/1963/2350 U1 - 1666 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Necessary and sufficient conditions for the chainrule in W1,1loc(RN;Rd) and BVloc(RN;Rd) JF - J. Eur. Math. Soc. (JEMS) 9 (2007) 219-252 Y1 - 2007 A1 - Giovanni Leoni A1 - Massimiliano Morini AB -

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.

UR - http://hdl.handle.net/1963/2037 U1 - 2159 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Surfactants in Foam Stability: A Phase-Field Model JF - Arch. Rational Mech. Anal. 183 (2007) 411-456 Y1 - 2007 A1 - Irene Fonseca A1 - Massimiliano Morini A1 - Valeriy Slastikov AB - 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. UR - http://hdl.handle.net/1963/2035 U1 - 2161 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Time-dependent systems of generalized Young measures Y1 - 2007 A1 - Gianni Dal Maso A1 - Antonio DeSimone A1 - Maria Giovanna Mora A1 - Massimiliano Morini AB - 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. JF - Netw. Heterog. Media 2 (2007) 1-36 UR - http://hdl.handle.net/1963/1795 U1 - 2749 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Higher order quasiconvexity reduces to quasiconvexity JF - Arch. Ration. Mech. Anal. 171 (2004) 55-81 Y1 - 2004 A1 - Gianni Dal Maso A1 - Irene Fonseca A1 - Giovanni Leoni A1 - Massimiliano Morini AB - 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. PB - Springer UR - http://hdl.handle.net/1963/2911 U1 - 1789 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Sequences of Singularly Perturbed Functionals Generating Free-Discontinuity Problems JF - SIAM J. Math. Anal. 35 (2003) 759-805 Y1 - 2003 A1 - Massimiliano Morini AB - 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. PB - SIAM UR - http://hdl.handle.net/1963/3071 U1 - 1262 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Global calibrations for the non-homogeneous Mumford-Shah functional JF - Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002) 603-648 Y1 - 2002 A1 - Massimiliano Morini AB - 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. PB - Scuola Normale Superiore di Pisa UR - http://hdl.handle.net/1963/3089 U1 - 1244 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - THES T1 - Free-discontinuity problems: calibration and approximation of solutions Y1 - 2001 A1 - Massimiliano Morini KW - Calibration of solutions PB - SISSA UR - http://hdl.handle.net/1963/5398 U1 - 5223 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set JF - Ann. I. H. Poincare - An., 2001, 18, 403 Y1 - 2001 A1 - Maria Giovanna Mora A1 - Massimiliano Morini PB - SISSA Library UR - http://hdl.handle.net/1963/1479 U1 - 2684 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Functionals depending on curvatures with constraints JF - Rend. Sem. Mat. Univ. Padova 104 (2000), 173--199 Y1 - 2000 A1 - Maria Giovanna Mora A1 - Massimiliano Morini PB - SISSA Library UR - http://hdl.handle.net/1963/1299 U1 - 3156 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets JF - J. Math. Pures Appl. 79, 2 (2000) 141-162 Y1 - 2000 A1 - Gianni Dal Maso A1 - Maria Giovanna Mora A1 - Massimiliano Morini PB - SISSA Library UR - http://hdl.handle.net/1963/1261 U1 - 3194 U2 - Mathematics U3 - Functional Analysis and Applications ER -