We study the behaviour of the solutions to a dynamic evolution problem for a viscoelastic model with long memory, when the rate of change of the data tends to zero. We prove that a suitably rescaled version of the solutions converges to the solution of the corresponding stationary problem.

SN - 1424-9294 UR - https://doi.org/10.1007/s00032-021-00343-w JO - Milan Journal of Mathematics ER - TY - JOUR T1 - A numerical study of the jerky crack growth in elastoplastic materials with localized plasticity JF - Journal of Convex Analysis Y1 - 2020 A1 - Gianni Dal Maso A1 - Luca Heltai UR - https://arxiv.org/abs/2004.12705 N1 - To appear ER - TY - RPRT T1 - On the Cauchy problem for the wave equation on time-dependent domains Y1 - 2018 A1 - Gianni Dal Maso A1 - Rodica Toader AB - We introduce a notion of solution to the wave equation on a suitable class of time-dependent domains and compare it with a previous de nition. We prove an existence result for the solution of the Cauchy problem and present some additional conditions which imply uniqueness. PB - SISSA UR - http://preprints.sissa.it/handle/1963/35314 U1 - 35622 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - Existence for elastodynamic Griffith fracture with a weak maximal dissipation condition Y1 - 2018 A1 - Gianni Dal Maso A1 - Cristopher J. Larsen A1 - Rodica Toader AB - We consider a model of elastodynamics with fracture evolution, based on energy-dissipation balance and a maximal dissipation condition. We prove an existence result in the case of planar elasticity with a free crack path, where the maximal dissipation condition is satisfied among suitably regular competitor cracks. UR - http://preprints.sissa.it/handle/1963/35308 U1 - 35616 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - A minimization approach to the wave equation on time-dependent domains Y1 - 2018 A1 - Gianni Dal Maso A1 - Lucia De Luca AB - We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time. PB - SISSA UR - http://preprints.sissa.it/handle/1963/35318 U1 - 35627 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - Stochastic homogenisation of free-discontinuity problems Y1 - 2018 A1 - Filippo Cagnetti A1 - Gianni Dal Maso A1 - Lucia Scardia A1 - Caterina Ida Zeppieri AB - In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas. UR - http://preprints.sissa.it/handle/1963/35309 U1 - 35617 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - Transmission conditions obtained by homogenisation Y1 - 2018 A1 - Gianni Dal Maso A1 - Giovanni Franzina A1 - Davide Zucco AB - We study the asymptotic behaviour of solutions to variational problems in perforated domains with Neumann boundary conditions. We consider perforations that in the limit concentrate on a smooth manifold. We characterise the limits of the solutions and show that they solve a variational problem with a transmission condition across the manifold. This is expressed through a measure on the manifold, vanishing on sets of capacity zero. Then, we prove that every such measure can be obtained by homogenising suitable perforations. Eventually, we provide an asymptotic formula for this measure by using some auxiliary minimum problems. UR - http://preprints.sissa.it/handle/1963/35310 U1 - 35618 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - Gamma-Convergence of Free-discontinuity problems Y1 - 2017 A1 - Filippo Cagnetti A1 - Gianni Dal Maso A1 - Lucia Scardia A1 - Caterina Ida Zeppieri AB - We study the Gamma-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u. We obtain three main results: compactness with respect to Gamma-convergence, representation of the Gamma-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper. PB - SISSA UR - http://preprints.sissa.it/handle/1963/35276 U1 - 35583 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - Linearisation of multiwell energies Y1 - 2017 A1 - Roberto Alicandro A1 - Gianni Dal Maso A1 - Giuliano Lazzaroni A1 - Mariapia Palombaro AB - Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours. UR - http://preprints.sissa.it/handle/1963/35288 U1 - 35594 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation JF - Advances in Calculus of Variations Y1 - 2017 A1 - Gianni Dal Maso A1 - Gianluca Orlando A1 - Rodica Toader AB -We study the lower semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrized gradient. We introduce a suitable class of surface terms, which make the functional lower semicontinuous with respect to $L^1$ convergence.

PB - De Gruyter VL - 10 ER - TY - JOUR T1 - A lower semicontinuity result for a free discontinuity functional with a boundary term JF - Journal de Mathématiques Pures et Appliquées Y1 - 2017 A1 - Stefano Almi A1 - Gianni Dal Maso A1 - Rodica Toader AB -We study the lower semicontinuity in $GSBV^{p}(\Omega;\mathbb{R}^{m})$ of a free discontinuity functional $\mathcal{F}(u)$ that can be written as the sum of a crack term, depending only on the jump set $S_{u}$, and of a boundary term, depending on the trace of $u$ on $\partial\Omega$. We give sufficient conditions on the integrands for the lower semicontinuity of $\mathcal{F}$. Moreover, we prove a relaxation result, which shows that, if these conditions are not satisfied, the lower semicontinuous envelope of $\mathcal{F}$ can be represented by the sum of two integrals on $S_{u}$ and $\partial\Omega$, respectively.

VL - 108 UR - http://hdl.handle.net/20.500.11767/15979 IS - 6 U1 - 34731 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Existence and uniqueness of dynamic evolutions for a peeling test in dimension one JF - Journal of Differential Equations Y1 - 2016 A1 - Gianni Dal Maso A1 - Giuliano Lazzaroni A1 - Lorenzo Nardini KW - Dynamic debonding KW - Dynamic energy release rate KW - Dynamic fracture KW - Griffith's criterion KW - Maximum dissipation principle KW - Wave equation in time-dependent domains AB -In this paper we present a one-dimensional model of a dynamic peeling test for a thin film, where the wave equation is coupled with a Griffith criterion for the propagation of the debonding front. Our main results provide existence and uniqueness for the solution to this coupled problem under different assumptions on the data.

VL - 261 UR - http://www.sciencedirect.com/science/article/pii/S0022039616301772 ER - TY - JOUR T1 - Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case JF - Calculus of Variations and Partial Differential Equations Y1 - 2016 A1 - Gianni Dal Maso A1 - Gianluca Orlando A1 - Rodica Toader AB -We study the asymptotic behavior of a variational model for damaged elasto-plastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter $\varepsilon$, which forces damage concentration on regions of codimension one. We determine the $\Gamma$-limit as $\varepsilon$ tends to zero and show that it contains an energy term involving the crack opening.

VL - 55 UR - https://doi.org/10.1007/s00526-016-0981-z ER - TY - RPRT T1 - A model for the quasistatic growth of cracks with fractional dimension Y1 - 2016 A1 - Gianni Dal Maso A1 - Marco Morandotti AB - We study a variational model for the quasistatic growth of cracks with fractional dimension in brittle materials. We give a minimal set of properties of the collection of admissible cracks which ensure the existence of a quasistatic evolution. Both the antiplane and the planar cases are treated. UR - http://urania.sissa.it/xmlui/handle/1963/35175 U1 - 35459 U2 - Mathematics ER - TY - RPRT T1 - Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition Y1 - 2015 A1 - Gianni Dal Maso A1 - Cristopher J. Larsen A1 - Rodica Toader AB - There are very few existence results for fracture evolution, outside of globally minimizing quasi-static evolutions. Dynamic evolutions are particularly problematic, due to the difficulty of showing energy balance, as well as of showing that solutions obey a maximal dissipation condition, or some similar condition that prevents stationary cracks from always being solutions. Here we introduce a new weak maximal dissipation condition and show that it is compatible with cracks constrained to grow smoothly on a smooth curve. In particular, we show existence of dynamic fracture evolutions satisfying this maximal dissipation condition, subject to the above smoothness constraints, and exhibit explicit examples to show that this maximal dissipation principle can indeed rule out stationary cracks as solutions. UR - http://urania.sissa.it/xmlui/handle/1963/35045 U1 - 35277 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data Y1 - 2015 A1 - Gianni Dal Maso A1 - Ilaria Lucardesi AB - Given a bounded open set $\Omega \subset \mathbb R^d$ with Lipschitz boundary and an increasing family $\Gamma_t$, $t\in [0,T]$, of closed subsets of $\Omega$, we analyze the scalar wave equation $\ddot{u} - div (A \nabla u) = f$ in the time varying cracked domains $\Omega\setminus\Gamma_t$. Here we assume that the sets $\Gamma_t$ are contained into a prescribed $(d-1)$-manifold of class $C^2$. Our approach relies on a change of variables: recasting the problem on the reference configuration $\Omega\setminus \Gamma_0$, we are led to consider a hyperbolic problem of the form $\ddot{v} - div (B\nabla v) + a \cdot \nabla v - 2 b \cdot \nabla \dot{v} = g$ in $\Omega \setminus \Gamma_0$. Under suitable assumptions on the regularity of the change of variables that transforms $\Omega\setminus \Gamma_t$ into $\Omega\setminus \Gamma_0$, we prove existence and uniqueness of weak solutions for both formulations. Moreover, we provide an energy equality, which gives, as a by-product, the continuous dependence of the solutions with respect to the cracks. UR - http://urania.sissa.it/xmlui/handle/1963/34629 U1 - 34832 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Editorial Y1 - 2014 A1 - Ciro Ciliberto A1 - Gianni Dal Maso A1 - Pasquale Vetro PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34712 U1 - 34926 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length Y1 - 2014 A1 - Gianni Dal Maso A1 - Gianluca Orlando A1 - Rodica Toader KW - cracked domains, energy release rate, higher order derivatives, asymptotic expansion of solutions AB -We consider the weak solution of the Laplace equation in a planar domain with a straight crack, prescribing a homogeneous Neumann condition on the crack and a nonhomogeneous Dirichlet condition on the rest of the boundary. For every k we express the k-th derivative of the energy with respect to the crack length in terms of a finite number of coefficients of the asymptotic expansion of the solution near the crack tip and of a finite number of other parameters, which only depend on the shape of the domain.

PB - SISSA UR - http://hdl.handle.net/1963/7271 U1 - 7316 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - New results on Gamma-limits of integral functionals Y1 - 2014 A1 - Nadia Ansini A1 - Gianni Dal Maso A1 - Caterina Ida Zeppieri KW - Gamma-convergence PB - Elsevier UR - http://hdl.handle.net/1963/5880 U1 - 5745 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Quasi-static crack growth in hydraulic fracture JF - Nonlinear Analysis Y1 - 2014 A1 - Stefano Almi A1 - Gianni Dal Maso A1 - Rodica Toader AB -We present a variational model for the quasi-static crack growth in hydraulic fracture in the framework of the energy formulation of rate-independent processes. The cracks are assumed to lie on a prescribed plane and to satisfy a very weak regularity assumption.

PB - Elsevier VL - 109 UR - http://hdl.handle.net/20.500.11767/17350 IS - Nov U1 - 34741 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Quasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes JF - Journal of Dynamics and Differential Equations Y1 - 2014 A1 - Gianni Dal Maso A1 - Riccardo Scala AB -We introduce a model of dynamic visco-elasto-plastic evolution in the linearly elastic regime and prove an existence and uniqueness result. Then we study the limit of (a rescaled version of) the solutions when the data vary slowly. We prove that they converge, up to a subsequence, to a quasistatic evolution in perfect plasticity.

VL - 26 UR - https://doi.org/10.1007/s10884-014-9409-7 ER - 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 - RPRT T1 - Attainment results for nematic elastomers Y1 - 2013 A1 - Virginia Agostiniani A1 - Gianni Dal Maso A1 - Antonio DeSimone AB - We consider a class of non-quasiconvex frame indifferent energy densities which includes Ogden-type energy densities for nematic elastomers. For the corresponding geometrically linear problem we provide an explicit minimizer of the energy functional satisfying a nontrivial boundary condition. Other attainment results, both for the nonlinear and the linearized model, are obtained by using the theory of convex integration introduced by Mueller and Sverak in the context of crystalline solids. PB - SISSA UR - http://hdl.handle.net/1963/7174 U1 - 7201 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Fracture models as Gamma-limits of damage models JF - Communications on Pure and Applied Analysis 12 (2013) 1657-1686 Y1 - 2013 A1 - Gianni Dal Maso A1 - Flaviana Iurlano AB -We analyze the asymptotic behavior of a variational model for damaged elastic materials. This model depends on two small parameters, which govern the width of the damaged regions and the minimum elasticity constant attained in the damaged regions. When these parameters tend to zero, we find that the corresponding functionals Gamma-converge to a functional related to fracture mechanics. The corresponding problem is brittle or cohesive, depending on the asymptotic ratio of the two parameters.

PB - American Institute of Mathematical Sciences UR - http://hdl.handle.net/1963/4225 U1 - 3952 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - One-dimensional swimmers in viscous fluids: dynamics, controllability, and existence of optimal controls Y1 - 2013 A1 - Gianni Dal Maso A1 - Antonio DeSimone A1 - Marco Morandotti AB -In this paper we study a mathematical model of one-dimensional swimmers performing a planar motion while fully immersed in a viscous fluid. The swimmers are assumed to be of small size, and all inertial effects are neglected. Hydrodynamic interactions are treated in a simplified way, using the local drag approximation of resistive force theory. We prove existence and uniqueness of the solution of the equations of motion driven by shape changes of the swimmer. Moreover, we prove a controllability result showing that given any pair of initial and final states, there exists a history of shape changes such that the resulting motion takes the swimmer from the initial to the final state. We give a constructive proof, based on the composition of elementary maneuvers (straightening and its inverse, rotation, translation), each of which represents the solution of an interesting motion planning problem. Finally, we prove the existence of solutions for the optimal control problem of finding, among the histories of shape changes taking the swimmer from an initial to a final state, the one of minimal energetic cost.

PB - SISSA UR - http://hdl.handle.net/1963/6467 U1 - 6412 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Gamma-convergence and H-convergence of linear elliptic operators JF - Journal de Mathématiques Pures et Appliquées, Available online 12 September 2012 Y1 - 2012 A1 - Nadia Ansini A1 - Gianni Dal Maso A1 - Caterina Ida Zeppieri KW - Linear elliptic operators PB - Elsevier UR - http://hdl.handle.net/1963/5878 U1 - 5746 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Linear elasticity obtained from finite elasticity by Gamma-convergence under weak coerciveness conditions JF - Ann. Inst. H. Poincare Anal. Non Lineaire Y1 - 2012 A1 - Virginia Agostiniani A1 - Gianni Dal Maso A1 - Antonio DeSimone KW - Nonlinear elasticity AB -The energy functional of linear elasticity is obtained as G-limit of suitable rescalings of the energies of finite elasticity...

PB - Gauthier-Villars;Elsevier VL - 29 UR - http://hdl.handle.net/1963/4267 U1 - 3996 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution JF - Calculus of variations and partial differential equations 44 (2012) 495-541 Y1 - 2012 A1 - Gianni Dal Maso A1 - Antonio DeSimone A1 - Francesco Solombrino AB -Cam-Clay plasticity is a well established model for the description of the mechanics of fine grained soils. As solutions can develop discontinuities in time, a weak notion of solution, in terms of a rescaled time s , has been proposed in [8] to give a meaning to this discontinuous evolution. In this paper we first prove that this rescaled evolution satisfies the flow-rule for the rate of plastic strain, in a suitable measure-theoretical sense. In the second part of the paper we consider the behavior of the evolution in terms of the original time variable t . We prove that the unrescaled solution satisfies an energy-dissipation balance and an evolution law for the internal variable, which can be expressed in terms of integrals depending only on the original time. Both these integral identities contain terms concentrated on the jump times, whose size can only be determined by looking at the rescaled formulation.

PB - Springer UR - http://hdl.handle.net/1963/3900 U1 - 809 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Crack growth with non-interpenetration : a simplified proof for the pure Neumann problem JF - Discrete and Continuous Dynamical Systems - Series A 31 (2011) 1219-1231 Y1 - 2011 A1 - Gianni Dal Maso A1 - Giuliano Lazzaroni AB - We present a recent existence result concerning the quasi-static evolution of cracks in hyperelastic brittle materials, in the frame-work of finite elasticity with non-interpenetration. In particular, here we consider the problem where no Dirichlet conditions are imposed, the boundary is traction-free, and the body is subject only to time-dependent volume forces. This allows us to present the main ideas of the proof in a simpler way, avoiding some of the technicalities needed in the general case, studied in. PB - American Institute of Mathematical Sciences UR - http://hdl.handle.net/1963/3801 U1 - 526 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Ennio De Giorgi and Γ-convergence JF - Discrete and Continuous Dynamical Systems - Series A 31 (2011) 1017-1021 Y1 - 2011 A1 - Gianni Dal Maso AB - Γ-convergence was introduced by Ennio De Giorgi in a series of papers published between 1975 and 1983. In the same years he developed many applications of this tool to a great variety of asymptotic problems in the calculus of variations and in the theory of partial differential equations. PB - American Institute of Mathematical Sciences UR - http://hdl.handle.net/1963/5308 U1 - 5138 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - An Existence and Uniqueness Result for the Motion of Self-Propelled Microswimmers JF - SIAM J. Math. Anal. Y1 - 2011 A1 - Gianni Dal Maso A1 - Antonio DeSimone A1 - Marco Morandotti AB -We present an analytical framework to study the motion of micro-swimmers in a viscous fluid. Our main result is that, under very mild regularity assumptions, the change of shape determines uniquely the motion of the swimmer. We assume that the Reynolds number is very small, so that the velocity field of the surrounding, infinite fluid is governed by the Stokes system and all inertial effects can be neglected. Moreover, we enforce the self propulsion constraint (no external forces and torques). Therefore, Newton\\\'s equations of motion reduce to the vanishing of the viscous drag force and torque acting on the body. By exploiting an integral representation of viscous force and torque, the equations of motion can be reduced to a system of six ordinary differential equations. Variational techniques are used to prove the boundedness and measurability of its coefficients, so that classical results on ordinary differential equations can be invoked to prove existence and uniqueness of the solution.

PB - Society for Industrial and Applied Mathematics VL - 43 UR - http://hdl.handle.net/1963/3894 U1 - 815 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Existence for wave equations on domains with arbitrary growing cracks JF - Rend. Lincei Mat. Appl. 22 (2011) 387-408 Y1 - 2011 A1 - Gianni Dal Maso A1 - Cristopher J. Larsen KW - Wave equation AB - In this paper we formulate and study scalar wave equations on domains with arbitrary growing cracks. This includes a zero Neumann condition on the crack sets, and the only assumptions on these sets are that they have bounded surface measure and are growing in the sense of set inclusion. In particular, they may be dense, so the weak formulations must fall outside of the usual weak formulations using Sobolev spaces. We study both damped and undamped equations, showing existence and, for the damped equation, uniqueness and energy conservation. PB - European Mathematical Society UR - http://hdl.handle.net/1963/4284 U1 - 4015 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Generalised functions of bounded deformation JF - J. Eur. Math. Soc. (JEMS), to appear Y1 - 2011 A1 - Gianni Dal Maso KW - free discontinuity problems, special functions of bounded deformation, jump set, rec- tifiability, slicing, approximate differentiability AB -We introduce the space GBD of generalized functions of bounded deformation and study the structure properties of these functions: the rectifiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for GBD, which leads to a compactness result for the space GSBD of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation of some variational problems arising in fracture mechanics in the framework of linearized elasticity.

PB - SISSA UR - http://hdl.handle.net/1963/6374 U1 - 6309 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling JF - Calculus of Variations and Partial Differential Equations 40 (2011) 125-181 Y1 - 2011 A1 - Gianni Dal Maso A1 - Antonio DeSimone A1 - Francesco Solombrino KW - Cam-Clay plasticity AB -Cam-Clay nonassociative plasticity exhibits both hardening and softening behaviour, depending on the loading. For many initial data the classical formulation of the quasistatic evolution problem has no smooth solution. We propose here a notion of generalized solution, based on a viscoplastic approximation. To study the limit of the viscoplastic evolutions we rescale time, in such a way that the plastic strain is uniformly Lipschitz with respect to the rescaled time. The limit of these rescaled solutions, as the viscosity parameter tends to zero, is characterized through an energy-dissipation balance, that can be written in a natural way using the rescaled time. As shown in [4] and [6], the proposed solution may be discontinuous with respect to the original time. Our formulation allows to compute the amount of viscous dissipation occurring instantaneously at each discontinuity time.

PB - Springer UR - http://hdl.handle.net/1963/3670 U1 - 635 U2 - Mathematics U3 - Functional Analysis and Applications 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 - Homogenization of fiber reinforced brittle material: the intermediate case JF - Adv. Calc. Var. 3 (2010) 345-370 Y1 - 2010 A1 - Gianni Dal Maso A1 - Caterina Ida Zeppieri AB - We derive a cohesive fracture model by homogenizing a periodic composite material whose microstructure is characterized by the presence of brittle inclusions in a reticulated unbreakable elastic structure. PB - Walter de Gruyter UR - http://hdl.handle.net/1963/3607 U1 - 694 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 - Quasistatic crack growth in elasto-plastic materials: the two-dimensional case JF - Arch. Ration. Mech. Anal. 196 (2010) 867-906 Y1 - 2010 A1 - Gianni Dal Maso A1 - Rodica Toader AB - We study a variational model for the quasistatic evolution of elasto-plastic materials with cracks in the case of planar small strain associative elasto-plasticity. UR - http://hdl.handle.net/1963/2964 U1 - 1736 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Quasistatic crack growth in finite elasticity with non-interpenetration JF - Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 257-290 Y1 - 2010 A1 - Gianni Dal Maso A1 - Giuliano Lazzaroni AB -We present a variational model to study the quasistatic growth of brittle cracks in hyperelastic materials, in the framework of finite elasticity, taking\\ninto account the non-interpenetration condition.

UR - http://hdl.handle.net/1963/3397 U1 - 935 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case JF - Netw. Heterog. Media 5 (2010) 97-132 Y1 - 2010 A1 - Gianni Dal Maso A1 - Francesco Solombrino KW - Cam-Clay plasticity AB -We study the spatially uniform case of the problem of quasistatic evolution in small strain nonassociative elastoplasticity (Cam-Clay model). Through the introdution of a viscous approximation, the problem reduces to determine the limit behavior of the solutions of a singularly perturbed system of ODE\\\'s in a finite dimensional Banach space. Depending on the sign of two explicit scalar indicators, we see that the limit dynamics presents, under quite generic assumptions, the alternation of three possible regimes: the elastic regime, when the limit equation is just the equation of linearized elasticity, the slow dynamics, when the strain evolves smoothly on the yield surface and plastic flow is produced, and the fast dynamics, which may happen only in the softening regime, where\\nviscous solutions exhibit a jump across a heteroclinic orbit of an auxiliary system.

UR - http://hdl.handle.net/1963/3671 U1 - 634 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 - Homogenization of fiber reinforced brittle materials: the extremal cases JF - SIAM J. Math. Anal. 41 (2009) 1874-1889 Y1 - 2009 A1 - Marco Barchiesi A1 - Gianni Dal Maso AB - We analyze the behavior of a fragile material reinforced by a reticulated elastic unbreakable structure in the case of antiplane shear. The microscopic geometry of this material is described by means of two small parameters: the period $\\\\varepsilon$ of the grid and the ratio $\\\\delta$ between the thickness of the fibers and the period $\\\\varepsilon$. We show that the asymptotic behavior as $\\\\varepsilon\\\\to0^+$ and $\\\\delta\\\\to0^+$ depends dramatically on the relative size of these parameters. Indeed, in the two cases considered, i.e., $\\\\varepsilon\\\\ll\\\\delta$ and $\\\\varepsilon\\\\gg\\\\delta$, we obtain two different limit models: a perfectly elastic model and an elastic model with macroscopic cracks, respectively. PB - SIAM UR - http://hdl.handle.net/1963/2705 U1 - 1396 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Quasistatic evolution for Cam-Clay plasticity: examples of spatially homogeneous solutions JF - Math. Models Methods Appl. Sci. 19 (2009) 1643-1711 Y1 - 2009 A1 - Gianni Dal Maso A1 - Antonio DeSimone AB - We study a quasistatic evolution problem for Cam-Clay plasticity under a special loading program which leads to spatially homogeneous solutions. Under some initial conditions, the solutions exhibit a softening behaviour and time discontinuities.\\nThe behavior of the solutions at the jump times is studied by a viscous approximation. UR - http://hdl.handle.net/1963/3395 U1 - 937 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A variational model for quasistatic crack growth in nonlinear elasticity: some qualitative properties of the solutions JF - Boll. Unione Mat. Ital. (9) 2 (2009) 371-390 Y1 - 2009 A1 - Gianni Dal Maso A1 - Alessandro Giacomini A1 - Marcello Ponsiglione UR - http://hdl.handle.net/1963/2675 U1 - 1425 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Decomposition results for functions with bounded variation JF - Boll. Unione Mat. Ital. (9) 1 (2008) 497-505 Y1 - 2008 A1 - Gianni Dal Maso A1 - Rodica Toader PB - Unione Matematica Italiana UR - http://hdl.handle.net/1963/3535 U1 - 729 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 - JOUR T1 - Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics JF - Calc. Var. Partial Differential Equations 31 (2008) 137-145 Y1 - 2008 A1 - Gianni Dal Maso A1 - Adriana Garroni AB - In this note we consider a free discontinuity problem for a scalar function, whose energy depends also on the size of the jump. We prove that the gradient of every smooth local minimizer never exceeds a constant, determined only by the data of the problem. UR - http://hdl.handle.net/1963/1723 U1 - 2428 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 - On a notion of unilateral slope for the Mumford-Shah functional JF - NoDEA 13 (2007) 713-734 Y1 - 2007 A1 - Gianni Dal Maso A1 - Rodica Toader AB - In this paper we introduce a notion of unilateral slope for the Mumford-Shah functional, and provide an explicit formula in the case of smooth cracks. We show that the slope is not lower semicontinuous and study the corresponding relaxed functional. UR - http://hdl.handle.net/1963/2059 U1 - 2137 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Quasistatic crack growth for a cohesive zone model with prescribed crack path Y1 - 2007 A1 - Gianni Dal Maso A1 - Chiara Zanini AB - In this paper we study the quasistatic crack growth for a cohesive zone model. We assume that the crack path is prescribed and we study the time evolution of the crack in the framework of the variational theory of rate-independent processes. JF - Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 253-279 UR - http://hdl.handle.net/1963/1686 U1 - 2447 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Quasistatic evolution problems for pressure-sensitive plastic materials JF - Milan J. Math. 75 (2007) 117-134 Y1 - 2007 A1 - Gianni Dal Maso A1 - Alexey Demyanov A1 - Antonio DeSimone AB - We study quasistatic evolution problems for pressure-sensitive plastic materials in the context of small strain associative perfect plasticity. UR - http://hdl.handle.net/1963/1962 U1 - 2231 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 - Quasistatic evolution problems for linearly elastic-perfectly plastic materials JF - Arch. Ration. Mech. Anal. 180 (2006) 237-291 Y1 - 2006 A1 - Gianni Dal Maso A1 - Antonio DeSimone A1 - Maria Giovanna Mora AB - The 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 in spaces of functions with bounded deformation. This provides a new approximation result for the solutions of the quasistatic evolution problem, which are shown to be absolutely continuous in time. Four equivalent formulations of the problem in rate form are derived. A strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain. UR - http://hdl.handle.net/1963/2129 U1 - 2114 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Variational problems in fracture mechanics Y1 - 2006 A1 - Gianni Dal Maso AB - We present some recent existence results for the variational model of crack growth in brittle materials proposed by Francfort and Marigo in 1998. These results, obtained in collaboration with Francfort and Toader, cover the case of arbitrary space dimension with a general quasiconvex bulk energy and with prescribed boundary deformations and applied loads. UR - http://hdl.handle.net/1963/1816 U1 - 2398 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Quasistatic Crack Growth in Nonlinear Elasticity JF - Arch. Ration. Mech. Anal. 176 (2005) 165-225 Y1 - 2005 A1 - Gianni Dal Maso A1 - Gilles A. Francfort A1 - Rodica Toader AB - In this paper, we prove a new existence result for a variational model of crack growth in brittle materials proposed in [15]. We consider the case of $n$-dimensional finite elasticity, for an arbitrary $n\\\\ge1$, with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads, both depending on time. UR - http://hdl.handle.net/1963/2293 U1 - 1723 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - ABST T1 - Solutions of Neumann problems in domains with cracks and applications to fracture mechanics Y1 - 2005 A1 - Gianni Dal Maso UR - http://hdl.handle.net/1963/1684 U1 - 79 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains JF - Ann. Inst. H. Poincaré. Anal. Non Linéaire 21 (2004), (4), p. 445-486. Y1 - 2004 A1 - Gianni Dal Maso A1 - Francois Murat AB - We consider a sequence of Dirichlet problems in varying domains (or, more generally, of relaxed Dirichlet problems involving measures in M_0) for second order linear elliptic operators in divergence form with varying matrices of coefficients. When the matrices H-converge to a matrix A^0, we prove that there exist a subsequence and a measure mu^0 in M_0 such that the limit problem is the relaxed Dirichlet problem corresponding to A^0 and mu^0. We also prove a corrector result which provides an explicit approximation of the solutions in the H^1-norm, and which is obtained by multiplying the corrector for the H-converging matrices by some special test function which depends both on the varying matrices and on the varying domains. PB - SISSA Library UR - http://hdl.handle.net/1963/1611 U1 - 2507 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 - Quasi-static evolution in brittle fracture: the case of bounded solutions JF - Quad. Mat. Dip. Mat. Seconda Univ. Napoli 14 (2004) 245-266 Y1 - 2004 A1 - Gianni Dal Maso A1 - Gilles A. Francfort A1 - Rodica Toader AB - The main steps of the proof of the existence result for the quasi-static evolution of cracks in brittle materials, obtained in [7] in the vector case and for a general quasiconvex elastic energy, are presented here under the simplifying assumption that the minimizing sequences involved in the problem are uniformly bounded in $L^\\\\infty$. UR - http://hdl.handle.net/1963/2229 U1 - 2015 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Autonomous integral functionals with discontinous nonconvex integrands: Lipschitz regularity of mimimizers, DuBois-Reymond necessary conditions and Hamilton-Jacobi equations JF - Applied Math.Optim. 48 (2003), no.1, p.39-66 Y1 - 2003 A1 - Gianni Dal Maso A1 - Helene Frankowska PB - SISSA Library UR - http://hdl.handle.net/1963/1625 U1 - 2493 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The calibration method for the Mumford-Shah functional and free-discontinuity problems JF - Calc. Var. Partial Differential Equations 16 (2003) 299-333 Y1 - 2003 A1 - Giovanni Alberti A1 - Guy Bouchitte A1 - Gianni Dal Maso AB - We present a minimality criterion for the Mumford-Shah functional, and more generally for non convex variational integrals on SBV which couple a surface and a bulk term. This method provides short and easy proofs for several minimality results. PB - Springer UR - http://hdl.handle.net/1963/3051 U1 - 1282 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A stability result for nonlinear Neumann problems under boundary variations JF - J.Math. Pures Appl. (9) 82 (2003) no.5 , 503 Y1 - 2003 A1 - Gianni Dal Maso A1 - Francois Ebobisse A1 - Marcello Ponsiglione AB - In this paper we study, in dimension two, the stability of the solutions of some nonlinear elliptic equations with Neumann boundary conditions, under perturbations of the domains in the Hausdorff complementary topology. PB - SISSA Library UR - http://hdl.handle.net/1963/1618 U1 - 2500 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Linearized elasticity as gamma-limit of finite elasticity JF - Set-Valued Anal. 10 (2002), p.165-183 Y1 - 2002 A1 - Gianni Dal Maso A1 - Matteo Negri A1 - Danilo Percivale PB - Springer UR - http://hdl.handle.net/1963/3052 U1 - 1281 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A model for the quasi-static growth of a brittle fracture: existence and approximation results JF - Math. Models Methods Appl. Sci., 12 (2002), no. 12, 1773 Y1 - 2002 A1 - Gianni Dal Maso A1 - Rodica Toader AB - We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith\\\'s criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough. PB - SISSA Library UR - http://hdl.handle.net/1963/1571 U1 - 2547 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A model for the quasi-static growth of brittle fractures based on local minimization JF - Math.Models Methods Appl. Sci., 12 (2002) , p.1773-1800. Y1 - 2002 A1 - Gianni Dal Maso A1 - Rodica Toader AB - We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith\\\'s criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough. PB - SISSA Library UR - http://hdl.handle.net/1963/1621 U1 - 2497 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A Model for the Quasi-Static Growth of Brittle Fractures: Existence and Approximation Results JF - Arch. Ration. Mech. Anal. 162 (2002) 101-135 Y1 - 2002 A1 - Gianni Dal Maso A1 - Rodica Toader AB - We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith\\\'s theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution. PB - Springer UR - http://hdl.handle.net/1963/3056 U1 - 1277 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Gamma-limit of periodic obstacles JF - Acta Appl. Math., 2001, 65, 207-215 Y1 - 2001 A1 - Gianni Dal Maso A1 - Paola Trebeschi AB - We compute the Gamma-limit of a sequence obstacle functionals in the case of periodic obstacles. PB - SISSA Library UR - http://hdl.handle.net/1963/1495 U1 - 2668 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A monotonicity approach to nonlinear Dirichlet problems in perforated domains JF - Adv. Math. Sci. Appl. 11 (2001) 721-751 Y1 - 2001 A1 - Gianni Dal Maso A1 - Igor V. Skrypnik AB - We study the asymptotic behaviour of solutions to Dirichlet problems in perforated domains for nonlinear elliptic equations associated with monotone operators. The main difference with respect to the previous papers on this subject is that no uniformity is assumed in the monotonicity condition. Under a very general hypothesis on the holes of the domains, we construct a limit equation, which is satisfied by the weak limits of the solutions. The additional term in the limit problem depends only on the local behaviour of the holes, which can be expressed in terms of suitable nonlinear capacities associated with the monotone operator. PB - SISSA Library UR - http://hdl.handle.net/1963/1555 U1 - 2563 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Uniqueness of solutions to Hamilton-Jacobi equations arising in the Calculus of Variations JF - Optimal control and partial differential equations : in honour of professor Alain Bensoussan\\\'s 60th birthday / edited by José Luis Menaldi, Edmundo Rofman, and Agnès Sulem.,Amsterdam : IOS Press, 2001, p. 335-345 Y1 - 2001 A1 - Gianni Dal Maso A1 - Helene Frankowska AB - We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi equation associated with a Bolza problem of the Calculus of Variations, assuming that the Lagrangian is autonomous, continuous, superlinear, and satisfies the usual convexity hypothesis. Under the same assumptions we prove also the uniqueness, in a class of lower semicontinuous functions, of a slightly different notion of solution, where classical derivatives are replaced only by subdifferentials. These results follow from a new comparison theorem for lower semicontinuous viscosity supersolutions of the Hamilton-Jacobi equation, that is proved in the general case of lower semicontinuous Lagrangians. PB - SISSA Library UR - http://hdl.handle.net/1963/1515 U1 - 2648 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The Calibration Method for Free Discontinuity Problems JF - European Congress of Mathematics. Volume I : Barcelona, July 10-14, 2000 / Carles Casacuberta ... [et al.], editors. , Boston : Birkhauser, 2001, p. 317-326. Y1 - 2000 A1 - Gianni Dal Maso AB - The calibration method is used to identify some minimizers of the Mumford-Shah functional. The method is then extended to more general free discontinuity problems. PB - SISSA Library UR - http://hdl.handle.net/1963/1496 U1 - 2667 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 - TY - JOUR T1 - Value Functions for Bolza Problems with Discontinuous Lagrangians and Hamilton-Jacobi inequalities JF - ESAIM Control Optim. Calc. Var., 5 (2000), n. 5, p. 369-393. Y1 - 2000 A1 - Gianni Dal Maso A1 - Helene Frankowska PB - SISSA Library UR - http://hdl.handle.net/1963/1514 U1 - 2649 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Asymptotic behaviour of nonlinear elliptic higher order equations in perforated domains JF - Journal d\\\'Analyse Mathematique, Volume 79, 1999, Pages: 63-112 Y1 - 1999 A1 - Gianni Dal Maso A1 - Igor V. Skrypnik SN - 1618-1891 UR - http://hdl.handle.net/1963/6433 U1 - 6374 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - The calibration method for the Mumford-Shah functional JF - C. R. Acad. Sci. Paris Ser. I Math. 329 (1999), no. 3, 249-254 Y1 - 1999 A1 - Giovanni Alberti A1 - Guy Bouchitte A1 - Gianni Dal Maso AB - In this Note we adapt the calibration method to functionals of Mumford-Shah type, and provide a criterion (Theorem 1) to verify that a given function is energy minimizing. Among other applications, we use this criterion to show that certain triple-junction configurations are minimizing (Example 3). PB - Elsevier UR - http://hdl.handle.net/1963/1235 U1 - 2708 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Discrete approximation of the Mumford-Shah functional in dimension two JF - M2AN 33 (1999) 651-672 Y1 - 1999 A1 - Antonin Chambolle A1 - Gianni Dal Maso PB - EDP Sciences UR - http://hdl.handle.net/1963/3588 U1 - 712 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Evans-Vasilesco theorem in Dirichlet spaces JF - Rendiconti di Matematica e delle sue Applicazioni. vol. 19, Issue 7, (1999), pages : 1-15 Y1 - 1999 A1 - Gianni Dal Maso A1 - Virginia De Cicco PB - SISSA UR - http://hdl.handle.net/1963/6436 U1 - 6376 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - A Lipschitz selection from the set of minimizers of a nonconvex functional of the gradient JF - Nonlinear Analysis, Theory, Methods and Applications. Volume 37, Issue 6, September 1999, Pages 707-717 Y1 - 1999 A1 - Gianni Dal Maso A1 - Vladimir V. Goncharov A1 - Antonio Ornelas AB - A constructive and improved version of the proof that there exist a continuous map that solves the convexified problem is presented. A Lipschitz continuous map is analyzed such that a map vector minimizes the functional at each vector satisfying Cellina\\\'s condition of existence of minimum. This map is explicitly given by a direct constructive algorithm. PB - SISSA UR - http://hdl.handle.net/1963/6439 U1 - 6379 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Renormalized solutions of elliptic equations with general measure data JF - Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 4, 741-808 Y1 - 1999 A1 - Gianni Dal Maso A1 - Francois Murat A1 - Luigi Orsina A1 - Alain Prignet PB - Scuola Normale Superiore di Pisa UR - http://hdl.handle.net/1963/1236 U1 - 2707 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Some properties of the solutions of obstacle problems with measure data JF - Ricerche Matematiche., Supplemento dedicato a Ennio De Giorgi, vol. 48 (1999), page : 99-116 Y1 - 1999 A1 - Paolo Dall'Aglio A1 - Gianni Dal Maso UR - http://hdl.handle.net/1963/6432 U1 - 6372 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Variational formulation of softening phenomena in fracture mechanics. The one-dimensional case JF - Arch. Ration. Mech. Anal. 146 (1999), no. 1, 23--58 Y1 - 1999 A1 - Andrea Braides A1 - Gianni Dal Maso A1 - Adriana Garroni AB - Starting from experimental evidence, the authors justify a variational model for softening phenomena in fracture of one-dimensional bars where the energy is given by the contribution and interaction of two terms: a typical bulk energy term depending on elastic strain and a discrete part that depends upon the jump discontinuities that occur in fracture. A more formal, rigorous derivation of the model is presented by examining the $\\\\Gamma$-convergence of discrete energy functionals associated to an array of masses and springs. Close attention is paid to the softening and fracture regimes. \\nOnce the continuous model is derived, it is fully analyzed without losing sight of its discrete counterpart. In particular, the associated boundary value problem is studied and a detailed analysis of the stationary points under the presence of a dead load is performed. A final, interesting section on the scale effect on the model is included. PB - Springer UR - http://hdl.handle.net/1963/3371 U1 - 959 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Asymptotic behavior of nonlinear Dirichlet problems in perforated domains JF - Ann. Mat. Pura Appl. (4) 174 (1998), 13--72 Y1 - 1998 A1 - Gianni Dal Maso A1 - Igor V. Skrypnik PB - SISSA Library UR - http://hdl.handle.net/1963/1064 U1 - 2738 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Limits of variational problems for Dirichlet forms in varying domains JF - Journal des Mathematiques Pures et Appliquees. Volume 77, Issue 1, January 1998, Pages 89-116 Y1 - 1998 A1 - Gianni Dal Maso A1 - Virginia De Cicco A1 - Lino Notarantonio A1 - Nicoletta A. Tchou PB - SISSA UR - http://hdl.handle.net/1963/6440 U1 - 6377 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Capacity theory for monotone operators JF - Potential Anal. 7 (1997), no. 4, 765-803 Y1 - 1997 A1 - Gianni Dal Maso A1 - Igor V. Skrypnik AB - If $Au=-div(a(x,Du))$ is a monotone operator defined on the Sobolev space $W^{1,p}(R^n)$, $1< p <+\\\\infty$, with $a(x,0)=0$ for a.e. $x\\\\in R^n$, the capacity $C_A(E,F)$ relative to $A$ can be defined for every pair $(E,F)$ of bounded sets in $R^n$ with $E\\\\subset F$. We prove that $C_A(E,F)$ is increasing and countably subadditive with respect to $E$ and decreasing with respect to $F$. Moreover we investigate the continuity properties of $C_A(E,F)$ with respect to $E$ and $F$. PB - Springer UR - http://hdl.handle.net/1963/911 U1 - 2880 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Comportamento asintotico delle soluzioni di problemi di Dirichlet JF - Bollettino della Unione Matematica Italiana A. Volume 7, Issue SUPPL. 11 part 2, June 1997, Pages 253-277 Y1 - 1997 A1 - Gianni Dal Maso UR - http://hdl.handle.net/1963/6438 N1 - This is the title of the conference which was delivered by prof. Dal Maso in Padova during the XV Congress of the \\\"Unione Mathematica Italiana\\\" (Padova, 11-16 settembre 1995) U1 - 6378 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Some properties of reachable solutions of nonlinear elliptic equations with measure data JF - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. Sér. 4, 25 no. 1-2 (1997), p. 375-396 Y1 - 1997 A1 - Gianni Dal Maso A1 - Annalisa Malusa PB - SISSA UR - http://hdl.handle.net/1963/6434 U1 - 6375 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - A capacity method for the study of Dirichlet problems for elliptic systems in varying domains JF - Rend. Sem. Mat. Univ. Padova 96 (1996), 257--277 Y1 - 1996 A1 - Gianni Dal Maso A1 - Rodica Toader PB - SISSA Library UR - http://hdl.handle.net/1963/989 U1 - 2867 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Capacity and Dirichlet problems in varying domains Y1 - 1995 A1 - Gianni Dal Maso PB - SISSA Library UR - http://hdl.handle.net/1963/950 U1 - 3504 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Special functions of bounded deformation Y1 - 1995 A1 - Giovanni Bellettini A1 - Alessandra Coscia A1 - Gianni Dal Maso PB - SISSA Library UR - http://hdl.handle.net/1963/978 U1 - 3476 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Limits of Dirichlet problems in perforated domains: a new formulation JF - Rend. Istit. Mat. Univ. Trieste 26 (1994) 339-360 Y1 - 1994 A1 - Gianni Dal Maso A1 - Rodica Toader PB - Università degli Studi di Trieste, Dipartimento di Scienze Matematiche UR - http://hdl.handle.net/1963/3649 U1 - 656 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A variational method in image segmentation: existence and approximation result JF - Acta Math. 168 (1992), no.1-2, p. 89-151 Y1 - 1992 A1 - Gianni Dal Maso A1 - Jean-Michel Morel A1 - Sergio Solimini PB - SISSA Library UR - http://hdl.handle.net/1963/808 U1 - 2983 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Shape optimization for Dirichlet problems: relaxed formulations and optimally conditions JF - Appl.Math.Optim. 23 (1991), no.1, p. 17-49. Y1 - 1991 A1 - Giuseppe Buttazzo A1 - Gianni Dal Maso PB - SISSA Library UR - http://hdl.handle.net/1963/880 U1 - 2911 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On systems of ordinary differential equations with measures as controls JF - Differential Integral Equations 4 (1991), no.4, p.739-765. Y1 - 1991 A1 - Gianni Dal Maso A1 - Franco Rampazzo PB - SISSA Library UR - http://hdl.handle.net/1963/840 U1 - 2951 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Correctors for the homogeneization of monotone operators JF - Differential Integral Equations 3 (1990), no.6, p.1151-1166. Y1 - 1990 A1 - Gianni Dal Maso A1 - Anneliese Defranceschi PB - SISSA Library UR - http://hdl.handle.net/1963/812 U1 - 2979 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - G-convergence of monotone operators JF - Ann. Inst. H. Poincare\\\' Anal. Non Linére 7 (1990), no. 3, 123-160 Y1 - 1990 A1 - Valeria Chiadò Piat A1 - Gianni Dal Maso A1 - Anneliese Defranceschi PB - SISSA Library UR - http://hdl.handle.net/1963/680 U1 - 3246 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A general chain rule for distributional derivatives JF - Proc. Amer. Math. Soc. 108 (1990), no. 3, 691-702 Y1 - 1990 A1 - Luigi Ambrosio A1 - Gianni Dal Maso PB - SISSA Library UR - http://hdl.handle.net/1963/650 U1 - 3276 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Shape optimization for Dirichlet problems: relaxed solutions and optimality conditions JF - Bull. Amer. Math. Soc. (N.S.) , 23 (1990), no.2, 531-535. Y1 - 1990 A1 - Giuseppe Buttazzo A1 - Gianni Dal Maso PB - SISSA Library UR - http://hdl.handle.net/1963/809 U1 - 2982 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - An approach to the thin obstacle problem for variational functionals depending on vector JF - Comm. Partial Differential Equations, 14 (1989), no.12, 1717-1743. Y1 - 1989 A1 - Gianni Dal Maso A1 - Roberta Musina PB - SISSA Library UR - http://hdl.handle.net/1963/802 U1 - 2989 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Convergence of unilateral problems for monotone operators JF - J. Analyse Math. 53 (1989), 269--289 Y1 - 1989 A1 - Gianni Dal Maso A1 - Anneliese Defranceschi PB - SISSA Library UR - http://hdl.handle.net/1963/722 U1 - 3069 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Limits of obstacle problems for the area functional. JF - Partial differential equations and the calculus of variations : essays in honor of Ennio De Giorgi. - Boston : Birkhauser, 1989. - p. 285-309 Y1 - 1989 A1 - Gianni Dal Maso A1 - G. Carere A1 - Antonio Leaci A1 - Eduardo Pascali PB - SISSA Library UR - http://hdl.handle.net/1963/577 U1 - 3327 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A pointwise regularity theory for the two-obstacle problem JF - Acta Math. 163 (1989), no. 1-2, 57-107 Y1 - 1989 A1 - Gianni Dal Maso A1 - Umberto Mosco A1 - Maria Agostina Vivaldi PB - SISSA Library UR - http://hdl.handle.net/1963/643 U1 - 3810 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A Kellogg property for µ-capacities JF - Boll. Un. Mat. Ital. A (7) 2, 1988, no. 1, 127-135 Y1 - 1988 A1 - Gianni Dal Maso A1 - Anneliese Defranceschi PB - SISSA Library UR - http://hdl.handle.net/1963/492 U1 - 3412 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Limits of nonlinear Dirichlet problems in varying domains. JF - Manuscripta Math. 61 (1988), no. 3, 251-278. Y1 - 1988 A1 - Gianni Dal Maso A1 - Anneliese Defranceschi AB - We study the general form of the limit, in the sense of gamma-convergence, of a sequence of nonlinear variational problems in varying domains with Dirichlet boudary conditions. The asymptotic problem is characterized in terms of the limit of suitable nonlinear capacities associated to the domains. PB - SISSA Library UR - http://hdl.handle.net/1963/536 U1 - 3368 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Some properties of a class of nonlinear variational $m$-capacities JF - J.Funct.Anal. 79, 1988, no. 2, 476-492 Y1 - 1988 A1 - Gianni Dal Maso A1 - Anneliese Defranceschi PB - SISSA Library UR - http://hdl.handle.net/1963/485 U1 - 3419 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Variational inequalities for the biharmonic operator with variable obstacles. JF - Ann. Mat. Pura Appl. (4) 153 (1988), 203-227 (1989) Y1 - 1988 A1 - Gianni Dal Maso A1 - Gabriella Paderni PB - SISSA Library UR - http://hdl.handle.net/1963/531 U1 - 3373 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Integral representation of some convex local functionals. JF - Ricerche Mat. 36 (1987), no. 2, 197-214 Y1 - 1987 A1 - Gianni Dal Maso A1 - Gabriella Paderni PB - SISSA Library UR - http://hdl.handle.net/1963/497 U1 - 3407 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Limits of nonlinear Dirichlet problems in varying domains. (Italian) JF - Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 81, 1987, no. 2, 111-118 Y1 - 1987 A1 - Gianni Dal Maso A1 - Anneliese Defranceschi PB - SISSA Library UR - http://hdl.handle.net/1963/486 U1 - 3418 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Convergence of unilateral convex sets. Optimization and related fields (Erice, 1984) JF - Berlin : Springer-Verlag, 1986, Lecture notes in mathematics, v.1190, p. 181-190 Y1 - 1986 A1 - Gianni Dal Maso PB - SISSA Library UR - http://hdl.handle.net/1963/353 U1 - 3614 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Dirichlet problems for demicoercive functionals JF - Nonlinear anal. 10(1986), no.6, 603-613 Y1 - 1986 A1 - Gabriele Anzellotti A1 - Giuseppe Buttazzo A1 - Gianni Dal Maso PB - SISSA Library UR - http://hdl.handle.net/1963/390 U1 - 3577 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Some necessary and sufficient conditions for the convergence of sequences of unilateral convex sets JF - J. Funct. Anal. 62 (1985), no. 2, 119--159 Y1 - 1985 A1 - Gianni Dal Maso PB - SISSA Library UR - http://hdl.handle.net/1963/318 U1 - 3649 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Some singular perturbation problems in the calculus of variations. JF - Ennio De Giorgi Colloquium, p. 41-49, Research Notes in Mathematics, v.125, London : Pitman, 1985 Y1 - 1985 A1 - Gianni Dal Maso PB - SISSA Library UR - http://hdl.handle.net/1963/297 U1 - 3670 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Weak convergence of measures on spaces of semicontinuous functions. JF - Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 79 (1985), no. 5, 98-106 Y1 - 1985 A1 - Gianni Dal Maso A1 - Ennio De Giorgi A1 - Luciano Modica PB - SISSA Library UR - http://hdl.handle.net/1963/463 U1 - 3440 U2 - Mathematics U3 - Mathematical Physics ER -