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.

}, doi = {10.1515/acv-2015-0036}, author = {Gianni Dal Maso and Gianluca Orlando and Rodica Toader} } @article {2015, title = {A lower semicontinuity result for a free discontinuity functional with a boundary term}, journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es}, volume = {108}, year = {2017}, pages = {952-990}, chapter = {952}, abstract = {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.

}, doi = {10.1016/j.matpur.2017.05.018}, url = {http://hdl.handle.net/20.500.11767/15979}, author = {Stefano Almi and Gianni Dal Maso and Rodica Toader} } @article {2014, title = {Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length}, number = {Nonlinear Differential Equations and Applications}, year = {2014}, publisher = {SISSA}, abstract = {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.

}, keywords = {cracked domains, energy release rate, higher order derivatives, asymptotic expansion of solutions}, doi = {10.1007/s00030-014-0291-0}, url = {http://hdl.handle.net/1963/7271}, author = {Gianni Dal Maso and Gianluca Orlando and Rodica Toader} } @article {2012, title = {Linear elasticity obtained from finite elasticity by Gamma-convergence under weak coerciveness conditions}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire}, volume = {29}, number = {SISSA;30/2011/M}, year = {2012}, pages = {715-735}, publisher = {Gauthier-Villars;Elsevier}, abstract = {The energy functional of linear elasticity is obtained as G-limit of suitable rescalings of the energies of finite elasticity...

}, keywords = {Nonlinear elasticity}, doi = {10.1016/j.anihpc.2012.04.001}, url = {http://hdl.handle.net/1963/4267}, author = {Virginia Agostiniani and Gianni Dal Maso and Antonio DeSimone} } @article {2002, title = {Linearized elasticity as gamma-limit of finite elasticity}, journal = {Set-Valued Anal. 10 (2002), p.165-183}, number = {SISSA;69/2001/M}, year = {2002}, publisher = {Springer}, doi = {10.1023/A:1016577431636}, url = {http://hdl.handle.net/1963/3052}, author = {Gianni Dal Maso and Matteo Negri and Danilo Percivale} } @article {2000, title = {Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets}, journal = {J. Math. Pures Appl. 79, 2 (2000) 141-162}, number = {SISSA;47/99/M}, year = {2000}, publisher = {SISSA Library}, doi = {10.1016/S0021-7824(99)00140-3}, url = {http://hdl.handle.net/1963/1261}, author = {Gianni Dal Maso and Maria Giovanna Mora and Massimiliano Morini} } @article {1999, title = {A Lipschitz selection from the set of minimizers of a nonconvex functional of the gradient}, journal = {Nonlinear Analysis, Theory, Methods and Applications. Volume 37, Issue 6, September 1999, Pages 707-717}, year = {1999}, publisher = {SISSA}, abstract = {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.}, doi = {10.1016/S0362-546X(98)00067-4}, url = {http://hdl.handle.net/1963/6439}, author = {Gianni Dal Maso and Vladimir V. Goncharov and Antonio Ornelas} } @article {1998, title = {Limits of variational problems for Dirichlet forms in varying domains}, journal = {Journal des Mathematiques Pures et Appliquees. Volume 77, Issue 1, January 1998, Pages 89-116}, year = {1998}, publisher = {SISSA}, doi = {10.1016/S0362-546X(98)00067-4}, url = {http://hdl.handle.net/1963/6440}, author = {Gianni Dal Maso and Virginia De Cicco and Lino Notarantonio and Nicoletta A. Tchou} } @article {1994, title = {Limits of Dirichlet problems in perforated domains: a new formulation}, journal = {Rend. Istit. Mat. Univ. Trieste 26 (1994) 339-360}, number = {SISSA;167/1994/M}, year = {1994}, publisher = {Universit{\`a} degli Studi di Trieste, Dipartimento di Scienze Matematiche}, url = {http://hdl.handle.net/1963/3649}, author = {Gianni Dal Maso and Rodica Toader} } @article {1989, title = {Limits of obstacle problems for the area functional.}, journal = {Partial differential equations and the calculus of variations : essays in honor of Ennio De Giorgi. - Boston : Birkhauser, 1989. - p. 285-309}, number = {SISSA;96/87/M}, year = {1989}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/577}, author = {Gianni Dal Maso and Michele Carriero and Antonio Leaci and Eduardo Pascali} } @article {1988, title = {Limits of nonlinear Dirichlet problems in varying domains.}, journal = {Manuscripta Math. 61 (1988), no. 3, 251-278.}, number = {SISSA;54/87/A}, year = {1988}, publisher = {SISSA Library}, abstract = {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.}, url = {http://hdl.handle.net/1963/536}, author = {Gianni Dal Maso and Anneliese Defranceschi} } @article {1987, title = {Limits of nonlinear Dirichlet problems in varying domains. (Italian)}, journal = {Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 81, 1987, no. 2, 111-118}, number = {SISSA;4/87/M}, year = {1987}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/486}, author = {Gianni Dal Maso and Anneliese Defranceschi} }