00694nas a2200121   4500008004100000245007500041210006900116260001000185520028900195100002100484700001900505856004800524       2018                        en  d00aA minimization approach to the wave equation on time-dependent domains0 aminimization approach to the wave equation on timedependent doma  bSISSA3 aWe 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.1 aDal Maso, Gianni1 aDe Luca, Lucia  uhttp://preprints.sissa.it/handle/1963/3531800690nas a2200109   4500008004100000245007500041210006900116520030100185100002100486700002200507856005100529       2016                        en  d00aA model for the quasistatic growth of cracks with fractional dimension0 amodel for the quasistatic growth of cracks with fractional dimen3 aWe 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.1 aDal Maso, Gianni1 aMorandotti, Marco  uhttp://urania.sissa.it/xmlui/handle/1963/3517501609nas a2200121   4500008004100000245009900041210006900140260001800209520118400227100002101411700001901432856003601451       2002                        en  d00aA model for the quasi-static growth of a brittle fracture: existence and approximation results0 amodel for the quasistatic growth of a brittle fracture existence  bSISSA Library3 aWe 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.1 aDal Maso, Gianni1 aToader, Rodica  uhttp://hdl.handle.net/1963/157101599nas a2200121   4500008004100000245008900041210006900130260001800199520118400217100002101401700001901422856003601441       2002                        en  d00aA model for the quasi-static growth of brittle fractures based on local minimization0 amodel for the quasistatic growth of brittle fractures based on l  bSISSA Library3 aWe 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.1 aDal Maso, Gianni1 aToader, Rodica  uhttp://hdl.handle.net/1963/162101237nas a2200121   4500008004300000245009800043210006900141260001300210520081600223100002101039700001901060856003601079       2002                        en_Ud  00aA Model for the Quasi-Static Growth of Brittle Fractures: Existence and Approximation Results0 aModel for the QuasiStatic Growth of Brittle Fractures Existence   bSpringer3 aWe 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.1 aDal Maso, Gianni1 aToader, Rodica  uhttp://hdl.handle.net/1963/305601054nas a2200121   4500008004100000245008200041210006900123260001800192520064300210100002100853700002200874856003600896       2001                        en  d00aA monotonicity approach to nonlinear Dirichlet problems in perforated domains0 amonotonicity approach to nonlinear Dirichlet problems in perfora  bSISSA Library3 aWe 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.1 aDal Maso, Gianni1 aSkrypnik, Igor V.  uhttp://hdl.handle.net/1963/1555