Research

My research interests lie in the areas of Calculus of Variations, Geometric Measure Theory, PDEs and Continuum Mechanics. Below you'll find the list of my papers, you can also find the same list with in addition a brief resume of the contents of some of them in my CV. To download a paper please visit either my personal webpage on the CVGMT website or, for the most recent ones, arXiv. The downloadable versions are not in the final version appeared on journals.

PAPERS

M. Focardi, Semicontinuity of vectorial functionals in Orlicz-Sobolev spaces, Rend. Ist. Mat. Univ. Trieste, vol. XXIX (1997), 141--161;

R. Alicandro, M. Focardi, M.S. Gelli, Finite-difference approximation of energies in fracture mechanics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), 671--709;

M. Focardi, E. Mascolo, Lower semicontinuity of quasiconvex functionals with nonstandard growth, Journal of Convex Analysis 8 (2001), 1--22;

M. Focardi, On the variational approximation of free-discontinuity problems in the vectorial case , Math. Models Methods Appl. Sci. 11 (2001), 663--684;

R. Alicandro, M. Focardi, Variational approximation of free-discontinuity energies with linear growth , Comm. Cont. Math., Vol. 4, No. 4 (2002), 685-723;

B. Bianconi, M. Focardi, E. Mascolo, Existence results for quasiconvex functionals with non-standard growth conditions, Ann. Mat. Pura Appl. (4) 180 (2002), 493--510;

M. Focardi, Variational Approximation of Free-Discontinuity Problems: the Discrete and Continuous Case, Tesi di Perfezionamento, Supervisor Prof. L. Ambrosio, Scuola Normale Superiore di Pisa, 2002;

M. Focardi, M.S. Gelli, Approximation results by difference schemes of fracture energies: the vectorial case, NoDEA Nonlinear Differential Equations Appl., Vol. 10, No. 4 (2003), 469--495;

M. Focardi, M.S. Gelli, Asymptotic analysis of the Mumford-Shah functional in periodically perforated domains , Interfaces and Free Boundaries 9 (2007), 107--132;

M. Focardi, A. Garroni, A 1-D macroscopic phase field model for dislocations and a second order $\Gamma$-limit, SIAM Multiscale Model. Simul. 6 (2007), No. 4, 1098--1124;

M. Focardi, M.S. Gelli, Relaxation of free-discontinuity energies with obstacles, Esaim COCV, 14 (2008), No. 4, 879-896;

M. Focardi, P.M. Mariano, Convergence of asynchronous variational integrators in linear elastodynamics, Internat. J. Numer. Methods in Engrg. 75 (2008), 755--769;

M. Focardi, P.M. Mariano, Discrete dynamics of complex bodies with substructural dissipation: variational integrators and convergence, Discr. Contin. Dyn. Syst. B, 11 1 (2009), 109--130;

M. Focardi, M.S. Gelli, M. Ponsiglione, Fracture mechanics in perforated domains: a variational model for brittle porous media, Math. Models Methods Appl. Sci., 19 11 (2009), 2065-2100;

M. Focardi, Homogenization of fractional obstacle problems via $\Gamma$-convergence, Comm. Partial Differential Equations 34 (2009),1607--1631;

M. Focardi, M.S. Gelli, G. Pisante, On a $1$-capacitary type problem in the plane, Commun. Pure Appl. Anal. 9 (2010), no. 5, 1319--1333;

M. Focardi, Aperiodic fractional obstacle problems, Adv. Math. 225 (2010), 3502--3544;

C. De Lellis, M. Focardi, E.N. Spadaro, Lower semicontinuous functionals for Almgren's multiple valued functions, Annales Academiae Scientiarum Fennicae Mathematica 36 (2011), 1--18;

M. Barchiesi, M. Focardi, Homogenization of the Neumann problem in perforated domains: an alternative approach, Calc. Var. Partial Differential Equations 42 (2011), no. 1-2, 257--288

M. Focardi, Vector-valued obstacle problems for non-local energies, Discr. Contin. Dyn. Syst. B 17 (2012), 487--507;

M. Focardi, E.N. Spadaro, An intrinsic approach to manifold constrained variational problems, Ann. Mat. Pura Appl. (4) 192 (2013), no. 1, 145--163.

C. De Lellis, M. Focardi, Density lower bound estimates for local minimizers of the Mumford-Shah energy in $2d$, Manuscripta Mathematica 132 (2013), 215--232

C. De Lellis, M. Focardi, Higher integrability of the gradient for minimizers of the $2d$ Mumford-Shah energy, J. Math. Pures Appl. 100 (2013), 391--409

M. Focardi, $\Gamma$-convergence: a tool to investigate physical phenomena across scales, Math. Methods Appl. Sci. 35 (2012), no. 14, 1613-1658.

M. Focardi, P. M. Mariano, E. N. Spadaro, Multi-value microstructural descriptors for finer scale material complexity: analysis of ground states, Arch. Rational Mech. Anal. 217 (2015), 899--933

M. Focardi, N. Fusco, C. Leone, P. Marcellini, E. Mascolo, A. Verde, Weak lower semicontinuity for polyconvex integrals in the limit case, Calc. Var. Partial Differential Equations 51 (2014), no. 1-2, 171--193

M. Focardi, F. Iurlano, Asymptotic analysis of Ambrosio-Tortorelli energies in linearized elasticity, SIAM J. Math. Anal. 46 (2014), no. 4, 2936--2955

M. Focardi, M.S. Gelli, E.N. Spadaro, Monotonicity formulas for obstacle problems with Lipschitz coeffcients, Calc. Var. Partial Differential Equations 54 (2015), 1547--1573

C. De Lellis, M. Focardi, B. Ruffini, A note on the Hausdorff dimension of the singular set for minimizers of the Mumford-Shah energy, Adv. Calc. Var. 7 (2014), no. 4, 539--545

G. De Philippis, S. Di Marino, M. Focardi, Lower semicontinuity for non-coercive polyconvex integrals in the limit case, Proc. Roy. Soc. Edinburgh Sect. A 145A (2015), 1--23

M. Focardi, P. Marcellini, On the lower semicontinuity of quasiconvex integrals, Bull. Inst. Math. Acad. Sin. (N.S.) 9 (2014), no. 2, 245--266

S. Conti, M. Focardi, F. Iurlano, Phase field approximation of cohesive fracture models, Ann. I. H. Poincar&eacute - AN 33 (2016), issue 4, 1033--1067

M. Focardi, A. Marchese, E.N. Spadaro, Improved estimate of the singular set of Dir-minimizing $Q$-valued functions via an abstract regularity result, J. Funct. Anal. 268 (2015), 3290--3325

S. Conti, M. Focardi, F. Iurlano, Which special functions of bounded deformation have bounded variation?, Proc. Roy. Soc. Edinburgh Sect. A 148A (2017), 33--50

M. Focardi, E.N. Spadaro, An Epiperimetric Inequality for the Thin Obstacle problem, Adv. Differential Equations 21 (2016), no. 1-2, 153--200

S. Conti, M. Focardi, F. Iurlano, Integral representation for functionals defined on $SBD^p$ in dimension two, Arch. Ration. Mech. Anal. 223 (2017), no. 3, 1337--1374

S. Conti, M. Focardi, F. Iurlano, Some recent results on the convergence of damage to fracture, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016), no. 1, 51--60

S. Conti, M. Focardi, F. Iurlano, Existence of minimizers for the 2d stationary Griffith fracture model, C. R. Math. Acad. Sci. Paris 354 (2016), no. 11, 1055--1059

M. Focardi, Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability, of the gradient and Mumford-Shah conjecture, Free Discontinuity Problems, 1--68, CRM Series, 19, Ed. Norm., Pisa, 2016

M. Focardi, F. Geraci, E. Spadaro, The classical obstacle problem for nonlinear variational energies, Nonlinear Anal. 154 (2017), 71--87

S. Conti, M. Focardi, F. Iurlano, Existence of strong minimizers for the Griffith static fracture model, in dimension two}, Ann. I. H. Poincar\'e - AN 36 (2019), Issue 2, 455--474

M. Focardi, E. Spadaro, On the measure and the structure of the free boundary in the lower dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 (2018), no. 1, 125--184

G. Cupini, M. Focardi, F. Leonetti, E. Mascolo, Local boundedness of vectorial minimizers of non-convex functionals, Bruno Pini Mathematical Analysis Seminar 2018, 20--40, Bruno Pini Math. Anal. Semin., 9, Univ. Bologna, Alma Mater Stud., Bologna, 2018

S. Conti, M. Focardi, F. Iurlano, Approximation of fracture energies with $p$-growth via piecewise affine finite elements, ESAIM Control Optim. Calc. Var. 25 (2019), Art. 34, 12 pp.

M. Focardi, F. Geraci, E. Spadaro, Quasi-monotonicity formulas for classical obstacle problems with Sobolev coefficients and applications, J. Optim. Theory Appl. 184 (2020), no. 1, 125--138

S. Conti, M. Focardi, F. Iurlano, A note on the Hausdorff dimension of the singular set of solutions to elasticity type systems, Comm. Cont. Math. (2019), 21 (2019), no. 6, 1950026, 58 pp.

G. Cupini, M. Focardi, F. Leonetti, E. Mascolo, On the H\"older continuity for a class of vector-valued problems, Adv. Nonlinear Anal. 9 (2020), no. 1, 1008--1025

M. Focardi, E. Spadaro, How a minimal surface leaves a thin obstacle, Ann. I. H. Poincar\'e - AN 37 (2020), 1017--1046

M. Caroccia, M. Focardi, N. Van Goethem, On the integral representation of variational functionals on $BD$, SIAM J. Math. Anal. 52 (2020), no. 4, 4022--4067

M. Focardi, E. Spadaro, The local structure of the free boundary in the fractional obstacle problem, to appear on Adv. Calc. Var., preprint arXiv:1903.05909

C. De Lellis, M. Focardi, S. Ghinassi, Endpoint regularity for $2$d Mumford-Shah minimizers: on a theorem by Andersson and Mikayelyan, preprint arXiv:2010.04888

Proceedings

M. Focardi, P.M. Mariano, Convergence of asynchronous variational integrators in linear elastodynamics. Proceedings AIMETA 2007

M. Focardi, Asymptotic analysis and relaxation of Mumford-Shah type energies with obstacles. Proceedings in Applied Mathematics and Mechanics 2007. PAMM, Proc. Appl. Math. Mech. 7, 1150807-1150808 (2007)/DOI 10.1002/pamm.200700448

M. Focardi, P.M. Mariano, Convergence of AVI's for the linear elastodynamics of simple bodies. Proceedings IACM/ECCOMAS 2008

M. Focardi, Regularity issues for local minimizers of the Mumford-Shah energy in 2d. Bruno Pini Mathematical Analysis Seminar (2012), 14-32.

Some links:

Web page of our research group in Florence (Gruppo di Calcolo delle Variazioni ed Equazioni alle Derivate Parziali di Firenze);

Web page of the research group in Pisa (Gruppo di Calcolo delle Variazioni e Teoria Geometrica della Misura di Pisa), and the direct link to my web page there.

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