Research

Research Areas

Calculus of variations and partial differential equations
Homogenisation (deterministic and stochastic)
Harmonic analysis

I am interested in several applications in material science:

plasticity and dislocations
non-local aggregation problems
nonlinear elasticity and rod and plate theories
fracture and damage theories

Industrial Partners

Tata Steel

PhD students

Xavier Pellet (2015-2019)

Publications

Preprints available here

L. Scardia: The nonlinear bending-torsion theory for curved rods as Gamma-limit of three-dimensional elasticity. Asymptot. Anal., 47 (2006) 317-343.
L. Scardia: Damage as Gamma-limit of microfractures in anti-plane linearized elasticity. Math. Models Methods Appl. Sci., 18 (2008) 1703-1740.
L. Scardia: Asymptotic models for curved rods derived from nonlinear elasticity by Gamma-convergence. Proc. Roy. Edinb. Soc. A, 139 (2009), 1037-1070.
T. Roubicek, L. Scardia and C. Zanini: Quasistatic delamination problem. Contin. Mech. Thermodyn., 21 (2009), 223-235.
L. Scardia: Damage as the Gamma-limit of microfractures in linearized elasticity under the non-interpenetration constraint. Adv. Calc. Var., 3 (2010), 423-458.
F. Cagnetti and L. Scardia: An extension theorem in SBV and an application to the homogenization of the Mumford-Shah functional in perforated domains. J. Math. Pures Appl., 95 (2011), 349-381.
L. Scardia, A. Schloemerkemper and C. Zanini: Boundary layer energies for nonconvex discrete energies. Math. Models Methods Appl. Sci., 21 (2011), 777-817.
M.G. Mora and L. Scardia: Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density. J. Differential Equations, 252 (2012), 35-55.
L. Scardia, A. Schloemerkemper and C. Zanini: Towards uniformly Gamma-equivalent theories for non-convex discrete systems. Discrete Contin. Dyn. Syst. Ser. B, 17/2 (2012), 661-686.
L. Scardia and C.I. Zeppieri: Line Tension for Plasticity as the Gamma-limit of a nonlinear dislocation energy. SIAM J. Math. Anal., 44/4 (2012), 2372-2400.
M.G.D. Geers, R.H.J. Peerlings, M.A. Peletier and L. Scardia: Asymptotic behaviour of a pile-up of infinite walls of edge dislocations. Arch. Rational Mech. Anal., 209/2 (2013), 495-539.
L. Scardia, R.H.J. Peerlings, M.A. Peletier and M.G.D. Geers: Mechanics of dislocation pile-up: A unification of scaling regimes. J. Mech. Phys. Solids, 70 (2014), 42-61.
S. Mueller, L. Scardia and C.I. Zeppieri: Geometric rigidity for incompatible fields and an application to stain-gradient plasticity. Indiana Univ. Math. J., 63 (2014), 1365-1396.
S. Mueller, L. Scardia and C.I. Zeppieri: Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations. In: Conti, S. and Hackl, K., eds. Analysis and Computation of Microstructure in Finite Plasticity. Switzerland: Springer (2015), 175-204.
A. Garroni, P. van Meurs, M.A. Peletier and L. Scardia: Boundary-layer analysis of a pile-up of walls of edge dislocations at a lock. Math. Models Methods Appl. Sci., 21 (2016).
L. Scardia: Continuum limits of discrete models via Gamma-convergence. In: Muntean et al eds. Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity. Lect. Notes Appl. Math. Mech., 3, Springer (2016), 145-186.
M.G. Mora, M.A. Peletier and L. Scardia: Convergence of interaction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane confinement. SIAM J. Math. Anal., 49/5 (2017), 4149-4205.
M.G. Mora, L. Rondi and L. Scardia: The equilibrium measure for a nonlocal dislocation energy. Comm. Pure Appl. Math., 72(1)(2019), 136-158.
J.A. Carrillo, J. Mateu, M.G. Mora, L. Rondi L. Scardia and J. Verdera: The ellipse law: Kirchhoff meets dislocations. Commun. Math. Phys., 373(2) (2020), 507-524.
F. Cagnetti, G. Dal Maso, L. Scardia and C.I. Zeppieri: Gamma-convergence of free-discontinuity problems. Ann. Inst. H. Poincaré Anal. Non Linéaire, 36(4) (2019), 1035-1079.
F. Cagnetti, G. Dal Maso, L. Scardia and C.I. Zeppieri: Stochastic homogenisation of free-discontinuity problems. Arch. Rational Mech. Anal., 233(2) (2019), 935-974.
X. Pellet, L. Scardia and C.I. Zeppieri: Homogenization of high-contrast Mumford-Shah energies. SIAM J. Math. Anal., 51(3) (2019), 1696-1729.
A. Garroni, P. van Meurs, M.A. Peletier and L. Scardia: Convergence and non-convergence of many-particle evolutions with multiple signs. Arch. Rational Mech. Anal., 235(1) (2020), 3-49.
J. Mateu, M.G. Mora, L. Rondi L. Scardia and J. Verdera: A maximum-principle approach to the minimisation of a nonlocal dislocation energy. Mathematics in Engineering, 2 (2) (2020), 253-263.
J. Mateu, M.G. Mora, L. Rondi L. Scardia and J. Verdera: Explicit minimisers of some nonlocal anisotropic energies: A short proof. Izv. Math., 85:3 (2021).
J.A. Carrillo, J. Mateu, M.G. Mora, L. Rondi L. Scardia and J. Verdera: The equilibrium measure for an anisotropic nonlocal energy. Calc. Var. Partial Differ. Equ., 60:109 (2021).
X. Pellet, L. Scardia and C.I. Zeppieri: Stochastic homogenisation of free-discontinuity functionals in random perforated domains. Published online in Adv. Calc. Var.
F. Cagnetti, A. Chambolle, M. Perugini and L. Scardia: An extension result for generalised special functions of bounded deformation. J. Convex Anal., 28(2) (2021).
F. Cagnetti, A. Chambolle and L. Scardia: Korn and Poincaré-Korn inequalities for functions with a small jump set. Math. Ann. , 383 (3-4) (2022), 1179-1216.
F. Cagnetti, G. Dal Maso, L. Scardia and C.I. Zeppieri: A global method for deterministic and stochastic homogenisation in BV. Annals of PDE, 8(8) (2022).
J. Mateu, M.G. Mora, L. Rondi L. Scardia and J. Verdera: Stability of ellipsoids as the energy minimisers of perturbed Coulomb energies. Published online in SIAM J. Math. Anal.
J. Mateu, M.G. Mora, L. Rondi L. Scardia and J. Verdera: Energy minimisers of perturbed dislocation energies. Nonlinear Analysis, 231, (2023).
J. Mateu, M.G. Mora, L. Rondi L. Scardia and J. Verdera: Explicit minimisers for anisotropic Coulomb energies in 3D. Published online in Advances in Mathematics.
L. Scardia, K. Zemas and C.I. Zeppieri: Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations. Submitted. CVGMT preprint .

Recent conferences, seminars and visits

Conference on Calculus of Variations in Lille - 4th edition -, Lille, 24-28 June 2024.
Measures amd Materials, University of Warwick, 25-28 March 2024.
Variational methods for evolution, Oberwolfach, 3-8 December 2023.
Paris-London Analysis Seminar, King's College, 20 October 20223.
Variational methods for nonlinear PDEs, University of Cardiff, 4-9 July 2023.
Spring School "Recent trends in the mathematics of complex materials", 20-24 March 2023, HIM, Bonn.
ESI Vienna, 20-24 February 2023.
University of Pavia, 14-18 November 2022.
Workshop on Discrete Variational Problems, TUM Munich, 7-11 November 2022.
University of Pavia, 26-30 September 2022.
Calculus of Variations, Oberwolfach, 14-20 August 2022.
Universitat Autònoma de Barcelona, 27-30 July 2022.
Theoretical and Applied Aspects for nonlocal Models, Banff, 17-22 July 2022, online.
Workhop on Analysis and PDEs, University of Sussex, 13-15 June 2022.
Frontiers in kinetic theory: connecting microscopic to macroscopic scales, Newton Institute, 1-31 May 2022.

Organisation of seminars and conferences

Optimal Transport and he Calculus of Variations, ICMS Edinburgh, 11-15 December 2023, co-orgnised with David Bourne and Alpár Mészáros.

Polycrystals: Microstructure ad Plasticity, ICMS, Edinburgh, 19-22 April 2022, co-organised with John Ball, David Bourne and Myrto Galanopoulou.

One World PDE seminar, co-organised with Juan Davila (Bath) and Manuel Del Pino (Bath) since April 2021.

Nonlocal interactions: Dislocations and beyond, University of Bath, 11-14 June 2018 (co-organised with Maria Giovanna Mora).

Workshop Young Applied Analysts in the UK, University of Bath, 26-27 May 2016 (co-organised with David Bourne).

Third Scottish PDE Colloquium, University of Strathclyde, 1-2 June 2015 (co-organised with G. Barrenechea).

Workshop Young Applied Analysts in the UK, University of Glasgow, 29-30 May 2014 (co-organised with David Bourne).

Grants

EPSRC Grant EP/V00204X/1 on "Mathematical Theory of polycrystalline materials" (£ 576,848), 2021-2024, with J. Ball (PI) and D.P. Bourne (coI).
EPSRC Network Grant EP/V008897/1 on "Generalised and Low-Regularity Solutions of Nonlinear Partial Differential Equations" (£ 35,960), 2021-2024.
EPSRC First Grant EP/N035631/1 on "Dislocation Patterns beyond equilibrium" (£ 95,699), 2016.
London Mathematical Society, as PI (£ 4000), 2015.
Edinburgh Mathematical Society, as PI (£ 1200), 2014.
Carnegie Trust - Small Research Grant, as PI (£ 980), 2014.
Glasgow Mathematical Journal Trust Fund, as PI (£ 2800), 2014.
Edinburgh Mathematical Society, as PI (£ 1200), 2013.