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Relaxation of Functionals in the Space of Vector-Valued Functions of Bounded HessianFeb 08 2018In this paper it is shown that if $\Omega \subset \mathbb{R}^N$ is an open, bounded Lipschitz set, and if $f: \Omega \times \mathbb{R}^{d \times N \times N} \rightarrow [0, \infty)$ is a continuous function with $f(x, \cdot)$ of linear growth for all ... More

Multiphase mean curvature flows with high mobility contrasts: a phase-field approach, with applications to nanowiresNov 10 2017The structure of many multiphase systems is governed by an energy that penalizes the area of interfaces between phases weighted by surface tension coefficients. However, interface evolution laws depend also on interface mobility coefficients. Having in ... More

Continuum limit and stochastic homogenization of discrete ferromagnetic thin filmsDec 08 2016We study the discrete-to-continuum limit of ferromagnetic spin systems when the lattice spacing tends to zero. We assume that the atoms are part of a (maybe) non-periodic lattice close to a flat set in a lower dimensional space, typically a plate in three ... More

Optimal design of mixtures of ferromagnetic interactionsOct 20 2016We provide a general framework for the optimal design of surface energies on networks. We prove sharp bounds for the homogenization of discrete systems describing mixtures of ferromagnetic interactions by constructing optimal microgeometries, and we prove ... More

Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjectureOct 12 2016We review some classical results and more recent insights about the regularity theory for local minimizers of the Mumford and Shah energy and their connections with the Mumford and Shah conjecture. We discuss in details the links among the latter, the ... More

Global existence results for viscoplasticity at finite strainSep 28 2016We study a model for rate-dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of global-in-time solutions to the related PDE system. We reveal its underlying structure ... More

A simple phase-field approximation of the Steiner problem in dimension twoSep 02 2016In this paper we consider the branched transportation problem in 2D associated with a cost per unit length of the form $1 + \alpha m$ where $m$ denotes the amount of transported mass and $\alpha > 0$ is a fixed parameter (notice that the limit case $\alpha ... More

Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2DJun 17 2016Controlling the growth of material damage is an important engineering task with plenty of real world applications. In this paper we approach this topic from the mathematical point of view by investigating an optimal boundary control problem for a damage ... More

Viscous corrections of the Time Incremental Minimization Scheme and Visco-Energetic Solutions to Rate-Independent Evolution ProblemsJun 10 2016We propose the new notion of Visco-Energetic solutions to rate-independent systems $(X,\mathcal E,\mathsf d)$ driven by a time dependent energy $\mathcal E$ and a dissipation quasi-distance $\mathsf d$ in a general metric-topological space $X$. As for ... More

The Dirichlet problem for p-harmonic functions with respect to arbitrary compactificationsApr 29 2016We study the Dirichlet problem for p-harmonic functions on metric spaces with respect to arbitrary compactifications. A particular focus is on the Perron method, and as a new approach to the invariance problem we introduce Sobolev-Perron solutions. We ... More

Regularity for Free Interface Variational Problems in a General Class of GradientsMar 03 2016We present a way to study a wide class of optimal design problems with a perimeter penalization. More precisely, we address existence and regularity properties of saddle points to energies of the form \[ (u,A) \quad \mapsto \quad \int_{\Omega} fu \, \text{d}x ... More

Regularity for free interface variational problems in a general class of gradientsMar 03 2016Oct 28 2016We present a way to study a wide class of optimal design problems with a perimeter penalization. More precisely, we address existence and regularity properties of saddle points of energies of the form $$ (u,A) \quad \mapsto \quad \int_\Omega 2fu \; \text{d}x ... More

The Perron method for $p$-harmonic functions in unbounded sets in $\mathbf{R}^n$ and metric spacesApr 25 2015Sep 16 2016The Perron method for solving the Dirichlet problem for $p$-harmonic functions is extended to unbounded open sets in the setting of a complete metric space with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. The upper and lower ... More

Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energiesApr 17 2015Oct 02 2015In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz submanifolds in ... More

A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic mediaFeb 10 2015Sep 15 2016In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with nonhomogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori ... More

Gamma convergence of a family of surface--director bending energies with small tiltJan 12 2015We prove a Gamma-convergence result for a family of bending energies defined on smooth surfaces in $\mathbb{R}^3$ equipped with a director field. The energies strongly penalize the deviation of the director from the surface unit normal and control the ... More

Domain formation in magnetic polymer composites: an approach via stochastic homogenizationNov 17 2014Jun 21 2015We study the magnetic energy of magnetic polymer composite materials as the average distance between magnetic particles vanishes. We model the position of these particles in the polymeric matrix as a stochastic lattice scaled by a small parameter $\varepsilon$ ... More

The Cartan, Choquet and Kellogg properties for the fine topology on metric spacesOct 20 2014We prove the Cartan and Choquet properties for the fine topology on a complete metric space equipped with a doubling measure supporting a $p$-Poincar\'e inequality, $1 < p< \infty$. We apply these key tools to establish a fine version of the Kellogg property, ... More

Optimal bounds for periodic mixtures of ferromagnetic interactionsJun 27 2014In this paper we give optimal bounds for the homogenization of periodic Ising systems when the coefficients may take two given values in given proportions.

Second Order Asymptotic Development for the Anisotropic Cahn-Hilliard FunctionalJun 18 2014The asymptotic behavior of an anisotropic Cahn-Hilliard functional with prescribed mass and Dirichlet boundary condition is studied when the parameter epsilon that determines the width of the transition layers tends to zero. The double-well potential ... More

Variational analysis of a mesoscale model for bilayer membranesFeb 26 2014We present an asymptotic analysis of a mesoscale energy for bilayer membranes that has been introduced and analyzed in two space dimensions by the second and third author (Arch. Ration. Mech. Anal. 193, 2009). The energy is both non-local and non-convex. ... More

Shape Optimization Problems for Metric GraphsDec 13 2013We consider the shape optimization problem $$\min\big\{{\mathcal E}(\Gamma)\ :\ \Gamma\in{\mathcal A},\ {\mathcal H}^1(\Gamma)=l\ \big\},$$ where ${\mathcal H}^1$ is the one-dimensional Hausdorff measure and ${\mathcal A}$ is an admissible class of one-dimensional ... More

The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in Rn and metric spacesNov 23 2013We study the obstacle problem for unbounded sets in a proper metric measure space supporting a (p,p)-Poincare inequality. We prove that there exists a unique solution. We also prove that if the measure is doubling and the obstacle is continuous, then ... More

The weak Cartan property for the p-fine topology on metric spacesOct 30 2013We study the p-fine topology on complete metric spaces equipped with a doubling measure supporting a p-Poincare inequality, 1 < p< oo. We establish a weak Cartan property, which yields characterizations of the p-thinness and the p-fine continuity, and ... More

Asymptotic spectral analysis in semiconductor nanowire heterostructuresSep 16 2013Mathematical settings in which heterogeneous structures affect electron transport through a tube-shaped quantum waveguide are studied, highlighting the interaction between heterogeneities and geometric parameters like curvature and torsion. First, the ... More

Cascade of minimizers for a nonlocal isoperimetric problem in thin domainsSep 03 2013Apr 11 2014For $\Omega_\e=(0,\e)\times (0,1)$ a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by \[ \inf_u E^{\gamma}_{\Omega_\e}(u)\] where \[ E^{\gamma}_{\Omega_\e}(u):= P_{\Omega_\e}(\{u(x)=1\})+\gamma\int_{\Omega_\e}\abs{\nabla{v}}^2\,dx ... More

Closure and commutability results for Gamma-limits and the geometric linearization and homogenization of multi-well energy functionalsAug 05 2013Under a suitable notion of equivalence of integral densities we prove a $\Gamma$-closure theorem for integral functionals: The limit of a sequence of $\Gamma$-convergent families of such functionals is again a $\Gamma$-convergent family. Its $\Gamma$-limit ... More

Compactness of special functions of bounded higher variationOct 24 2012Given an open set \Omega\subset\R^m and n>1, we introduce the new spaces GB_nV(\Omega) of Generalized functions of bounded higher variation and GSB_nV(\Omega) of Generalized special functions of bounded higher variation that generalize, respectively, ... More

The variational capacity with respect to nonopen sets in metric spacesOct 01 2012We pursue a systematic treatment of the variational capacity on metric spaces and give full proofs of its basic properties. A novelty is that we study it with respect to nonopen sets, which is important for Dirichlet and obstacle problems on nonopen sets, ... More

Obstacle and Dirichlet problems on arbitrary nonopen sets, and fine topologyAug 24 2012We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain Adams' criterion for the solubility of ... More

C^{1,1} regularity for degenerate elliptic obstacle problemsJun 05 2012Jan 05 2016The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, ... More

Maximum principles for boundary-degenerate second-order linear elliptic differential operatorsApr 30 2012Sep 12 2013We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary ... More

Evolution models for mass transportation problemsApr 07 2012We present a survey on several mass transportation problems, in which a given mass dynamically moves from an initial configuration to a final one. The approach we consider is the one introduced by Benamou and Brenier in [5], where a suitable cost functional ... More

Gamma-expansion for a 1D Confined Lennard-Jones model with point defectMar 19 2012Apr 13 2012We compute a rigorous asymptotic expansion of the energy of a point defect in a 1D chain of atoms with second neighbour interactions. We propose the Confined Lennard-Jones model for interatomic interactions, where it is assumed that nearest neighbour ... More

Korn's second inequality and geometric rigidity with mixed growth conditionsMar 06 2012Geometric rigidity states that a gradient field which is $L^p$-close to the set of proper rotations is necessarily $L^p$-close to a fixed rotation, and is one key estimate in nonlinear elasticity. In several applications, as for example in the theory ... More

Boundary-degenerate elliptic operators and Holder continuity for solutions to variational equations and inequalitiesOct 25 2011Mar 09 2016The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the ... More

Existence, uniqueness, and global regularity for degenerate elliptic obstacle problems in mathematical financeSep 06 2011The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the ... More

On the commutability of homogenization and linearization in finite elasticityNov 16 2010May 16 2011We study non-convex elastic energy functionals associated to (spatially) periodic, frame indifferent energy densities with a single non-degenerate energy well at SO(n). Under the assumption that the energy density admits a quadratic Taylor expansion at ... More

Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica FunctionalSep 29 2010We establish some new results about the $\Gamma$-limit, with respect to the $L^1$-topology, of two different (but related) phase-field approximations of the so-called Euler's Elastica Bending Energy for curves in the plane.

Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measuresAug 12 2010Jan 15 2011We establish a general weak* lower semicontinuity result in the space $\BD(\Omega)$ of functions of bounded deformation for functionals of the form $$\Fcal(u) := \int_\Omega f \bigl(x, \Ecal u \bigr) \dd x + \int_\Omega f^\infty \Bigl(x, \frac{\di E^s ... More

Phase transitions and minimal hypersurfaces in hyperbolic space]Feb 10 2010The purpose of this paper is to investigate the Cahn-Hillard approximation for entire minimal hypersurfaces in the hyperbolic space. Combining comparison principles with minimization and blow-up arguments, we prove existence results for entire local minimizers ... More

Higher-order phase transitions with line-tension effectNov 09 2009The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. ... More

Lower semicontinuous functionals for Almgren's multiple valued functionsOct 30 2009Mar 17 2011We consider general integral functionals on the Sobolev spaces of multiple valued functions, introduced by Almgren. We characterize the semicontinuous ones and recover earlier results of Mattila as a particular case. Moreover, we answer positively to ... More

Approximation of the Helfrich's functional via Diffuse InterfacesOct 29 2009Oct 30 2009We give a rigorous proof of the approximability of the so-called Helfrich's functional via diffuse interfaces, under a constraint on the ratio between the bending rigidity and the Gauss-rigidity.

$Γ$-convergence of some super quadratic functionals with singular weightsMar 05 2009We study the $\Gamma$-convergence of the following functional ($p>2$) $$ F_{\epsilon}(u):=\epsilon^{p-2}\int_{\Omega}|Du|^p d(x,\partial \Omega)^{a}dx+\frac{1}{\epsilon^{\frac{p-2}{p-1}}}\int_{\Omega}W(u) d(x,\partial \Omega)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\epsilon}}\int_{\partial\Omega}V(Tu)d\mathcal{H}^2, ... More

Stripe patterns in a model for block copolymersFeb 16 2009We consider a pattern-forming system in two space dimensions defined by an energy G_e. The functional G_e models strong phase separation in AB diblock copolymer melts, and patterns are represented by {0,1}-valued functions; the values 0 and 1 correspond ... More

Homogenization of variational problems in manifold valued BV-spacesApr 03 2008Nov 04 2008This paper extends the result of \cite{BM} on the homogenization of integral functionals with linear growth defined for Sobolev maps taking values in a given manifold. Through a $\Gamma$-convergence analysis, we identify the homogenized energy in the ... More

Homogenization of variational problems in manifold valued Sobolev spacesDec 11 2007Apr 22 2008Homogenization of integral functionals is studied under the constraint that admissible maps have to take their values into a given smooth manifold. The notion of tangential homogenization is defined by analogy with the tangential quasiconvexity introduced ... More

Dimensional reduction for energies with linear growth involving the bending momentNov 05 2007Apr 24 2008A $\Gamma$-convergence analysis is used to perform a 3D-2D dimension reduction of variational problems with linear growth. The adopted scaling gives rise to a nonlinear membrane model which, because of the presence of higher order external loadings inducing ... More

Lower semicontinuity of quasiconvex bulk energies in SBV and integral representation in dimension reductionDec 07 2006Nov 07 2007A result of Larsen concerning the structure of the approximate gradient of certain sequences of functions with Bounded Variation is used to present a short proof of Ambrosio's lower semicontinuity theorem for quasiconvex bulk energies in $SBV$. It enables ... More

Multiscale nonconvex relaxation and application to thin filmsApr 26 2006$\Gamma$-convergence techniques are used to give a characterization of the behavior of a family of heterogeneous multiple scale integral functionals. Periodicity, standard growth conditions and nonconvexity are assumed whereas a stronger uniform continuity ... More

3D-2D analysis of a thin film with periodic microstructureApr 26 2006The purpose of this article is to study the behavior of a heterogeneous thin film whose microstructure oscillates on a scale that is comparable to that of the thickness of the domain. The argument is based on a 3D-2D dimensional reduction through a $\Gamma$-convergence ... More

Quasistatic evolution of a brittle thin filmApr 26 2006This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution ... More

Quasistatic evolution problems for linearly elastic - perfectly plastic materialsDec 10 2004The problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate-independent processes. Existence of solutions is proved through the use of incremental variational problems ... More

A lower semicontinuity result for some integral functionals in the space SBDJun 30 2003The purpose of this paper is to study the lower semicontinuity with respect to the strong $L^1$-convergence, of some integral functionals defined in the space SBD of special functions with bounded deformation. Precisely, let $U$ be a bounded open subset ... More

A note on the integral representation of functionals in the space SBD(O)Apr 27 2001In this paper we study the integral representation in the space SBD(O) of special functions with bounded deformation of some L^1-norm lower semicontinuous functionals invariant with respect to rigid motions.