TY - JOUR
T1 - Second Order Asymptotic Development for the Anisotropic Cahn-Hilliard Functional
Y1 - 2014
A1 - Gianni Dal Maso
A1 - Irene Fonseca
A1 - Giovanni Leoni
KW - Gamma-convergence, Cahn-Hilliard functional, phase transitions
AB - The asymptotic behavior of an anisotropic Cahn-Hilliard functional with prescribed mass and Dirichlet boundary condition is studied when the parameter $\varepsilon$ that determines the width of the transition layers tends to zero. The double-well potential is assumed to be even and equal to $|s-1|^\beta$ near $s=1$, with $1<\beta<2$. The first order term in the asymptotic development by $\Gamma$-convergence is well-known, and is related to a suitable anisotropic perimeter of the interface. Here it is shown that, under these assumptions, the second order term is zero, which gives an estimate on the rate of convergence of the minimum values.
PB - SISSA
UR - http://hdl.handle.net/1963/7390
N1 - This article is composed if 33 pages and recorded in PDF format
U1 - 7439
U2 - Mathematics
U4 - 1
U5 - MAT/05 ANALISI MATEMATICA
ER -
TY - JOUR
T1 - Analytical validation of a continuum model for epitaxial growth with elasticity on vicinal surfaces
Y1 - 2013
A1 - Gianni Dal Maso
A1 - Irene Fonseca
A1 - Giovanni Leoni
KW - singular nonlinear parabolic equations, Hilbert transform, thin films
AB - In this paper it is shown existence of weak solutions of a variational inequality derived from the continuum model introduced by Xiang [7, formula (3.62)] (see also the work of Xiang and E [8] and Xu and Xiang [9]) to describe the self-organization of terraces and steps driven by misfit elasticity between a film and a substrate in heteroepitaxial growth. This model is obtained as a continuum limit of discrete theories of Duport, Politi, and Villain [3] and Tersoff, Phang, Zhang, and Lagally[6].
PB - Springer
UR - http://hdl.handle.net/1963/7245
U1 - 7284
U2 - Mathematics
U4 - 1
U5 - MAT/05 ANALISI MATEMATICA
ER -
TY - JOUR
T1 - Singular perturbation models in phase transitions for second order materials
JF - Indiana Univ. Math. J. 60 (2011) 367-409
Y1 - 2011
A1 - Milena Chermisi
A1 - Gianni Dal Maso
A1 - Irene Fonseca
A1 - Giovanni Leoni
AB - A variational model proposed in the physics literature to describe the onset of pattern formation in two-component bilayer membranes and amphiphilic monolayers leads to the analysis of a Ginzburg-Landau type energy with a negative term depending on the first derivative of the phase function. Scaling arguments motivate the study of the family of second order singular perturbed energies Fe having a negative term depending on the first derivative of the phase function. Here, the asymptotic behavior of {Fe} is studied using G-convergence techniques. In particular, compactness results and an integral representation of the limit energy are obtained.
PB - Indiana University
UR - http://hdl.handle.net/1963/3858
U1 - 851
U2 - Mathematics
U3 - Functional Analysis and Applications
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 - Nonlocal character of the reduced theory of thin films with higher order perturbations
JF - Adv. Calc. Var. 3 (2010) 287-319
Y1 - 2010
A1 - Gianni Dal Maso
A1 - Irene Fonseca
A1 - Giovanni Leoni
UR - http://hdl.handle.net/1963/3754
U1 - 563
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 - 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 - 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 -