Employing the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions for elastoplastic materials with incomplete damage affecting both the elastic tensor and the plastic yield surface, in a softening framework and in small strain assumptions.

}, issn = {1432-0835}, doi = {10.1007/s00526-015-0947-6}, url = {https://doi.org/10.1007/s00526-015-0947-6}, author = {Vito Crismale and Giuliano Lazzaroni} } @article {agrachev2016volume, title = {Volume geodesic distortion and Ricci curvature for Hamiltonian dynamics}, journal = {arXiv preprint arXiv:1602.08745}, year = {2016}, author = {Andrei A. Agrachev and Davide Barilari and Elisa Paoli} } @mastersthesis {2015, title = {Variational aspects of Liouville equations and systems}, year = {2015}, note = {The PHD thesis is composed of 112 pages and is recorded in PDF format}, school = {SISSA}, keywords = {Toda system}, author = {Aleks Jevnikar} } @mastersthesis {2015, title = {Variational aspects of singular Liouville systems}, year = {2015}, school = {SISSA}, abstract = {I studied singular Liouville systems on compact surfaces from a variational point of view. I gave sufficient and necessary conditions for the existence of globally minimizing solutions, then I found min-max solutions for some particular systems. Finally, I also gave some non-existence results.}, keywords = {Variational methods, Liouville systems, Moser-Trudinger inequalities, min-max methods}, author = {Luca Battaglia} } @mastersthesis {2015, title = {Volume variation and heat kernel for affine control problems}, year = {2015}, school = {SISSA}, abstract = {In this thesis we study two main problems. The first one is the small-time heat kernel expansion on the diagonal for second order hypoelliptic opeartors. We consider operators that can depend on a drift field and that satisfy only the weak H{\"o}rmander condition. In a first work we use perturbation techniques to determine the exact order of decay of the heat kernel, that depends on the Lie algebra generated by the fields involved in the hypoelliptic operator. We generalize in particular some results already obtained in the sub-Riemannian setting. In a second work we consider a model class of hypoelliptic operators and we characterize geometrically all the coefficients in the on-the diagonal asymptotics at the equilibrium points of the drift field. The class of operators that we consider contains the linear hypoelliptic operators with constant second order part on the Euclidean space. We describe the coefficients in terms only of the divergence of the drift field and of curvature-like invariants, related to the minimal cost of geodesics of the associated optimal control problem. In the second part of the thesis we consider the variation of a smooth volume along a geodesic. The structure of the manifold is induced by a quadratic Hamiltonian and the geodesic in described as the projection of the Hamiltonian flow. We find an expansion similar to the classical Riemannian one. It depends on the curvature operator associated to the Hamiltonian, on the symbol of the geodesic and on a new metric-measure invariant determined by the symbol of the geodesic and by the given volume.}, keywords = {Heat kernel asymptotics}, author = {Elisa Paoli} } @mastersthesis {2014, title = {A variational approach to statics and dynamics of elasto-plastic systems}, year = {2014}, school = {SISSA}, abstract = {We prove some existence results for dynamic evolutions in elasto-plasticity and delamination. We study the limit as the data vary very slowly and prove convergence results to quasistatic evolutions. We model dislocations by mean of currents, we introduce the space of deformations in the presence of dislocations and study the graphs of these maps. We prove existence results for minimum problems. We study the properties of minimizers.}, keywords = {delamination}, url = {http://urania.sissa.it/xmlui/handle/1963/7471}, author = {Riccardo Scala} } @article {10829, title = {A variational model for the quasi-static growth of fractional dimensional brittle fractures}, number = {Interfaces and Free Boundaries}, year = {2014}, publisher = {European Mathematical Society}, abstract = {We propose a variational model for the irreversible quasi-static evolution of brittle fractures having fractional Hausdorff dimension in the setting of two-dimensional antiplane and plane elasticity. The evolution along such irregular crack paths can be obtained as $\Gamma$-limit of evolutions along one-dimensional cracks when the fracture toughness tends to zero.

}, keywords = {Variational models}, doi = {10.4171/IFB/328}, url = {http://hdl.handle.net/1963/6983}, author = {Simone Racca and Rodica Toader} } @article {2014, title = {Vortex Partition Functions, Wall Crossing and Equivariant Gromov{\textendash}Witten Invariants}, number = {Communications in mathematical physics;volume 333; issue 2; pages 717-760;}, year = {2014}, publisher = {Springer}, abstract = {In this paper we identify the problem of equivariant vortex counting in a (2,2) supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov{\textendash}Witten invariants of the GIT quotient target space determined by the quiver. We provide new contour integral formulae for the I and J-functions encoding the equivariant quantum cohomology of the target space. Its chamber structure is shown to be encoded in the analytical properties of the integrand. This is explained both via general arguments and by checking several key cases. We show how several results in equivariant Gromov{\textendash}Witten theory follow just by deforming the integration contour. In particular, we apply our formalism to compute Gromov{\textendash}Witten invariants of the C3/Zn orbifold, of the Uhlembeck (partial) compactification of the moduli space of instantons on C2, and of An and Dn singularities both in the orbifold and resolved phases. Moreover, we analyse dualities of quantum cohomology rings of holomorphic vector bundles over Grassmannians, which are relevant to BPS Wilson loop algebrae.}, doi = {10.1007/s00220-014-2193-8}, url = {http://urania.sissa.it/xmlui/handle/1963/34652}, author = {Giulio Bonelli and Antonio Sciarappa and Alessandro Tanzini and Petr Vasko} } @article {2013, title = {A variational Analysis of the Toda System on Compact Surfaces}, journal = {Communications on Pure and Applied Mathematics, Volume 66, Issue 3, March 2013, Pages 332-371}, number = {arXiv:1105.3701;}, year = {2013}, note = {pre-peer version, to appear in Comm. Pure Applied Math}, publisher = {Wiley}, abstract = {In this paper we consider the Toda system of equations on a compact surface. We will give existence results by using variational methods in a non coercive case. A key tool in our analysis is a new Moser-Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u_1, u_2.}, doi = {10.1002/cpa.21433}, url = {http://hdl.handle.net/1963/6558}, author = {Andrea Malchiodi and David Ruiz} } @article {2012, title = {Variational implementation of immersed finite element methods}, journal = {Computer Methods in Applied Mechanics and Engineering. Volume 229-232, 1 July 2012, Pages 110-127}, number = {arXiv:1110.2063;}, year = {2012}, note = {42 pages, 5 figures, Revision 1}, publisher = {Elsevier}, abstract = {Dirac-delta distributions are often crucial components of the solid-fluid coupling operators in immersed solution methods for fluid-structure interaction (FSI) problems. This is certainly so for methods like the Immersed Boundary Method (IBM) or the Immersed Finite Element Method (IFEM), where Dirac-delta distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-delta distributions, either formally or practically. This has been shown in the Finite Element Immersed Boundary Method (FEIBM), where the variational structure of the problem is exploited to avoid Dirac-delta distributions at both the continuous and the discrete level. In this paper, we generalize the FEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic of differential type or purely elastic, and (iii) the solid to be and either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer to the discrete level automatically due to the non- matching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semi-discrete stability estimates and strong consistency of our spatial discretization.

}, keywords = {Turbulent flow}, doi = {10.1016/j.cma.2012.04.001}, url = {http://hdl.handle.net/1963/6462}, author = {Luca Heltai and Francesco Costanzo} } @article {2012, title = {Vertices, vortices \& interacting surface operators}, journal = {JHEP 06(2012)178}, number = {SISSA;03/2011/FM-EP}, year = {2012}, note = {22 pages, 4 figures}, publisher = {SISSA}, abstract = {We show that the vortex moduli space in non-abelian supersymmetric N=(2,2) gauge theories on the two dimensional plane with adjoint and anti-fundamental matter can be described as an holomorphic submanifold of the instanton moduli space in four dimensions. The vortex partition functions for these theories are computed via equivariant localization. We show that these coincide with the field theory limit of the topological vertex on the strip with boundary conditions corresponding to column diagrams. Moreover, we resum the field theory limit of the vertex partition functions in terms of generalized hypergeometric functions formulating their AGT dual description as interacting surface operators of simple type. Analogously we resum the topological open string amplitudes in terms of q-deformed generalized hypergeometric functions proving that they satisfy appropriate finite difference equations.}, doi = {10.1007/JHEP06(2012)178}, url = {http://hdl.handle.net/1963/4134}, author = {Giulio Bonelli and Alessandro Tanzini and Zhao Jian} } @article {2012, title = {A Viscosity-driven crack evolution}, journal = {Advances in Calculus of Variations 5 (2012) 433-483}, number = {SISSA;63/2011/M}, year = {2012}, publisher = {SISSA}, abstract = {We present a model of crack growth in brittle materials which couples dissipative effects on the crack tip and viscous effects. We consider the 2 -dimensional antiplane case with pre-assigned crack path, and firstly prove an existence result for a rate-dependent evolution problem by means of time-discretization. The next goal is to describe the rate-independent evolution as limit of the rate-dependent ones when the dissipative and viscous effects vanish. The rate-independent evolution satisfies a Griffith{\textquoteright}s criterion for the crack growth, but, in general, it does not fulfil a global minimality condition; its fracture set may exhibit jump discontinuities with respect to time. Under suitable regularity assumptions, the quasi-static crack growth is described by solving a finite-dimensional problem.

}, doi = {10.1515/acv-2011-0012}, url = {http://hdl.handle.net/1963/5130}, author = {Simone Racca} } @article {2009, title = {A variational model for quasistatic crack growth in nonlinear elasticity: some qualitative properties of the solutions}, journal = {Boll. Unione Mat. Ital. (9) 2 (2009) 371-390}, number = {SISSA;41/2008/M}, year = {2009}, url = {http://hdl.handle.net/1963/2675}, author = {Gianni Dal Maso and Alessandro Giacomini and Marcello Ponsiglione} } @article {2009, title = {On viscosity solutions of Hamilton-Jacobi equations}, journal = {Trans. Amer. Math. Soc. 361 (2009) 41-59}, year = {2009}, publisher = {American Mathematical Society}, abstract = {We consider the Dirichlet problem for Hamilton-Jacobi equations and prove existence, uniqueness and continuous dependence on boundary data of Lipschitz continuous maximal viscosity solutions.}, doi = {10.1090/S0002-9947-08-04557-1}, url = {http://hdl.handle.net/1963/3420}, author = {Sandro Zagatti} } @article {2008, title = {A vanishing viscosity approach to quasistatic evolution in plasticity with softening}, journal = {Arch. Ration. Mech. Anal. 189 (2008) 469-544}, number = {SISSA;38/2006/M}, year = {2008}, abstract = {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.}, doi = {10.1007/s00205-008-0117-5}, url = {http://hdl.handle.net/1963/1844}, author = {Gianni Dal Maso and Antonio DeSimone and Maria Giovanna Mora and Massimiliano Morini} } @article {Berti2008391, title = {Variational methods for Hamiltonian PDEs}, journal = {NATO Science for Peace and Security Series B: Physics and Biophysics}, year = {2008}, note = {cited By (since 1996)0}, pages = {391-420}, abstract = {We present recent existence results of periodic solutions for completely resonant nonlinear wave equations in which both "small divisor" difficulties and infinite dimensional bifurcation phenomena occur. These results can be seen as generalizations of the classical finite-dimensional resonant center theorems of Weinstein-Moser and Fadell-Rabinowitz. The proofs are based on variational bifurcation theory: after a Lyapunov-Schmidt reduction, the small divisor problem in the range equation is overcome with a Nash-Moser implicit function theorem for a Cantor set of non-resonant parameters. Next, the infinite dimensional bifurcation equation, variational in nature, possesses minimax mountain-pass critical points. The big difficulty is to ensure that they are not in the "Cantor gaps". This is proved under weak non-degeneracy conditions. Finally, we also discuss the existence of forced vibrations with rational frequency. This problem requires variational methods of a completely different nature, such as constrained minimization and a priori estimates derivable from variational inequalities. {\textcopyright} 2008 Springer Science + Business Media B.V.}, isbn = {9781402069628}, issn = {18746500}, doi = {10.1007/978-1-4020-6964-2-16}, author = {Massimiliano Berti} } @article {2007, title = {Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients}, journal = {J. Hyperbolic Differ. Equ. 4 (2007) 771-795}, number = {SISSA;23/2003/M}, year = {2007}, publisher = {World Scientific}, abstract = {We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define a viscosity solution by treating the discontinuities in the coefficients analogously to \\\"internal boundaries\\\". By defining an appropriate penalization function, we prove that viscosity solutions are unique. The existence of viscosity solutions is established by showing that a sequence of front tracking approximations is compact in $L^\\\\infty$, and that the limits are viscosity solutions.}, doi = {10.1142/S0219891607001355}, url = {http://hdl.handle.net/1963/2907}, author = {Giuseppe Maria Coclite and Nils Henrik Risebro} } @article {2006, title = {On variational approach to differential invariants of rank two distributions}, journal = {Differential Geom. Appl. 24 (2006) 235-259}, number = {arXiv.org;math/0402171v2}, year = {2006}, abstract = {n the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n=5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his \\\"reduction-prolongation\\\" procedure. After Cartan\\\'s work the following questions remained open: first the geometric reason for existence of Cartan\\\'s tensor was not clear; secondly it was not clear how to generalize this tensor to other classes of distributions; finally there were no explicit formulas for computation of Cartan\\\'s tensor. Our paper is the first in the series of papers, where we develop an alternative approach, which gives the answers to the questions mentioned above. It is based on the investigation of dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the general theory of unparametrized curves in the Lagrange Grassmannian, developed in our previous works with A. Agrachev . In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n greater than 4.\\nFor n=5 we give an explicit method for computation of these invariants and demonstrate it on several examples. In our next paper we show that in the case n=5 our fundamental form coincides with Cartan\\\'s tensor.}, doi = {10.1016/j.difgeo.2005.09.004}, url = {http://hdl.handle.net/1963/2188}, author = {Igor Zelenko} } @article {2006, title = {Variational problems in fracture mechanics}, number = {SISSA;10/2006/M}, year = {2006}, abstract = {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.}, url = {http://hdl.handle.net/1963/1816}, author = {Gianni Dal Maso} } @article {2005, title = {Vanishing viscosity solutions of nonlinear hyperbolic systems}, journal = {Ann. of Math. 161 (2005) 223-342}, number = {SISSA;86/2001/M}, year = {2005}, publisher = {Annals of Mathematics}, abstract = {We consider the Cauchy problem for a strictly hyperbolic, $n\\\\times n$ system in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has small total variation.\\nWe show that the solutions of the viscous approximations $u_t+A(u)u_x=\\\\ve u_{xx}$ are defined globally in time and satisfy uniform BV estimates, independent of $\\\\ve$. Moreover, they depend continuously on the initial data in the $\\\\L^1$ distance, with a Lipschitz constant independent of $t,\\\\ve$. Letting $\\\\ve\\\\to 0$, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where $A=Df$ is the Jacobian of some flux function $f:\\\\R^n\\\\mapsto\\\\R^n$, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws $u_t+f(u)_x=0$.}, url = {http://hdl.handle.net/1963/3074}, author = {Stefano Bianchini and Alberto Bressan} } @article {2004, title = {Virasoro Symmetries of the Extended Toda Hierarchy}, journal = {Comm. Math.\\nPhys. 250 (2004) 161-193.}, number = {arXiv.org;math/0308152}, year = {2004}, abstract = {We prove that the extended Toda hierarchy of \\\\cite{CDZ} admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators $L_m$, $m\\\\geq -1$ of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the $CP^1$ Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.}, doi = {10.1007/s00220-004-1084-9}, url = {http://hdl.handle.net/1963/2544}, author = {Boris Dubrovin and Zhang Youjin} } @article {2000, title = {Value Functions for Bolza Problems with Discontinuous Lagrangians and Hamilton-Jacobi inequalities}, journal = {ESAIM Control Optim. Calc. Var., 5 (2000), n. 5, p. 369-393.}, number = {SISSA;56/00/M}, year = {2000}, publisher = {SISSA Library}, doi = {10.1051/cocv:2000114}, url = {http://hdl.handle.net/1963/1514}, author = {Gianni Dal Maso and Helene Frankowska} } @article {1999, title = {Vanishing viscosity solutions of hyperbolic systems on manifolds}, number = {SISSA;24/99/M}, year = {1999}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1238}, author = {Stefano Bianchini and Alberto Bressan} } @article {1999, title = {Variational formulation of softening phenomena in fracture mechanics. The one-dimensional case}, journal = {Arch. Ration. Mech. Anal. 146 (1999), no. 1, 23--58}, number = {SISSA;160/97/M}, year = {1999}, publisher = {Springer}, abstract = {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.}, doi = {10.1007/s002050050135}, url = {http://hdl.handle.net/1963/3371}, author = {Andrea Braides and Gianni Dal Maso and Adriana Garroni} } @article {1999, title = {The vector measures whose range is strictly convex}, journal = {J. Math. Anal. Appl. 232 (1999) 1-19}, number = {SISSA;108/96/M}, year = {1999}, publisher = {Elsevier}, doi = {10.1006/jmaa.1998.6215}, url = {http://hdl.handle.net/1963/3546}, author = {Stefano Bianchini and Carlo Mariconda} } @article {1997, title = {Viscosity solutions and uniquenessfor systems of inhomogeneous balance laws}, journal = {Discrete Contin. Dynam. Systems 3 (1997), no. 4, 477--5}, number = {SISSA;67/95/M}, year = {1997}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/969}, author = {Graziano Crasta and Benedetto Piccoli} } @article {1994, title = {A version of Olech\\\'s lemma in a problem of the calculus of variations}, journal = {SIAM J. Control Optim. 32 (1994) 1114-1127}, year = {1994}, publisher = {SIAM}, abstract = {This paper studies the solutions of the minimum problem for a functional of the gradient under linear boundary conditions. A necessary and sufficient condition, based on the facial structure of the epigraph of the integrand, is provided for the continuous dependence of the solutions on boundary data.}, doi = {10.1137/S0363012992234669}, url = {http://hdl.handle.net/1963/3514}, author = {Arrigo Cellina and Sandro Zagatti} } @article {1992, title = {A variational method in image segmentation: existence and approximation result}, journal = {Acta Math. 168 (1992), no.1-2, p. 89-151}, number = {SISSA;48/89/M}, year = {1992}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/808}, author = {Gianni Dal Maso and Jean-Michel Morel and Sergio Solimini} } @article {1988, title = {Variational inequalities for the biharmonic operator with variable obstacles.}, journal = {Ann. Mat. Pura Appl. (4) 153 (1988), 203-227 (1989)}, number = {SISSA;50/87/M}, year = {1988}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/531}, author = {Gianni Dal Maso and Gabriella Paderni} } @mastersthesis {1988, title = {Variational Problems with Obstructions}, year = {1988}, school = {SISSA}, url = {http://hdl.handle.net/1963/5832}, author = {Roberta Musina} }