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.

1 aDal Maso, Gianni1 aOrlando, Gianluca1 aToader, Rodica uhttps://www.math.sissa.it/publication/lower-semicontinuity-class-integral-functionals-space-functions-bounded-deformation01104nas a2200145 4500008004100000245009100041210006900132300001200201490000800213520063400221100001800855700002100873700001900894856004500913 2017 en d00aA lower semicontinuity result for a free discontinuity functional with a boundary term0 alower semicontinuity result for a free discontinuity functional a952-9900 v1083 aWe 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.

1 aAlmi, Stefano1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/20.500.11767/1597901118nas a2200145 4500008004100000245013100041210006900172260001000241520052100251653010200772100002100874700002200895700001900917856003600936 2014 en d00aLaplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length0 aLaplace equation in a domain with a rectilinear crack higher ord bSISSA3 aWe 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.

10acracked domains, energy release rate, higher order derivatives, asymptotic expansion of solutions1 aDal Maso, Gianni1 aOrlando, Gianluca1 aToader, Rodica uhttp://hdl.handle.net/1963/727100709nas a2200169 4500008004100000245011000041210006900151260003000220300001200250490000700262520014000269653002500409100002600434700002100460700002200481856003600503 2012 en d00aLinear elasticity obtained from finite elasticity by Gamma-convergence under weak coerciveness conditions0 aLinear elasticity obtained from finite elasticity by Gammaconver bGauthier-Villars;Elsevier a715-7350 v293 aThe energy functional of linear elasticity is obtained as G-limit of suitable rescalings of the energies of finite elasticity...

10aNonlinear elasticity1 aAgostiniani, Virginia1 aDal Maso, Gianni1 aDeSimone, Antonio uhttp://hdl.handle.net/1963/426700398nas a2200121 4500008004300000245006200043210006100105260001300166100002100179700001800200700002200218856003600240 2002 en_Ud 00aLinearized elasticity as gamma-limit of finite elasticity0 aLinearized elasticity as gammalimit of finite elasticity bSpringer1 aDal Maso, Gianni1 aNegri, Matteo1 aPercivale, Danilo uhttp://hdl.handle.net/1963/305200462nas a2200121 4500008004100000245010500041210006900146260001800215100002100233700002500254700002500279856003600304 2000 en d00aLocal calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets0 aLocal calibrations for minimizers of the MumfordShah functional bSISSA Library1 aDal Maso, Gianni1 aMora, Maria Giovanna1 aMorini, Massimiliano uhttp://hdl.handle.net/1963/126100817nas a2200133 4500008004100000245009500041210006900136260001000205520036300215100002100578700002700599700002100626856003600647 1999 en d00aA Lipschitz selection from the set of minimizers of a nonconvex functional of the gradient0 aLipschitz selection from the set of minimizers of a nonconvex fu bSISSA3 aA 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.1 aDal Maso, Gianni1 aGoncharov, Vladimir V.1 aOrnelas, Antonio uhttp://hdl.handle.net/1963/643900455nas a2200133 4500008004100000245007400041210006900115260001000184100002100194700002300215700002300238700002400261856003600285 1998 en d00aLimits of variational problems for Dirichlet forms in varying domains0 aLimits of variational problems for Dirichlet forms in varying do bSISSA1 aDal Maso, Gianni1 aDe Cicco, Virginia1 aNotarantonio, Lino1 aTchou, Nicoletta A. uhttp://hdl.handle.net/1963/644000448nas a2200109 4500008004300000245007400043210006900117260007600186100002100262700001900283856003600302 1994 en_Ud 00aLimits of Dirichlet problems in perforated domains: a new formulation0 aLimits of Dirichlet problems in perforated domains a new formula bUniversitÃ degli Studi di Trieste, Dipartimento di Scienze Matematiche1 aDal Maso, Gianni1 aToader, Rodica uhttp://hdl.handle.net/1963/364900424nas a2200133 4500008004100000245005700041210005600098260001800154100002100172700002200193700001900215700002100234856003500255 1989 en d00aLimits of obstacle problems for the area functional.0 aLimits of obstacle problems for the area functional bSISSA Library1 aDal Maso, Gianni1 aCarriero, Michele1 aLeaci, Antonio1 aPascali, Eduardo uhttp://hdl.handle.net/1963/57700693nas a2200121 4500008004100000245006300041210006200104260001800166520030300184100002100487700002800508856003500536 1988 en d00aLimits of nonlinear Dirichlet problems in varying domains.0 aLimits of nonlinear Dirichlet problems in varying domains bSISSA Library3 aWe 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.1 aDal Maso, Gianni1 aDefranceschi, Anneliese uhttp://hdl.handle.net/1963/53600395nas a2200109 4500008004100000245007300041210006900114260001800183100002100201700002800222856003500250 1987 en d00aLimits of nonlinear Dirichlet problems in varying domains. (Italian)0 aLimits of nonlinear Dirichlet problems in varying domains Italia bSISSA Library1 aDal Maso, Gianni1 aDefranceschi, Anneliese uhttp://hdl.handle.net/1963/486