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.

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 -