- 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
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.,
- 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., in press.
- J.A. Carrillo, J. Mateu, M.G. Mora, L. Rondi L. Scardia and J. Verdera: The ellipse law: Kirchhoff meets dislocations. Submitted.
- F. Cagnetti, G. Dal Maso, L. Scardia and C.I. Zeppieri: Gamma-convergence of free-discontinuity problems. Submitted.
- F. Cagnetti, G. Dal Maso, L. Scardia and C.I. Zeppieri: Stochastic homogenisation of free-discontinuity problems. Submitted.
- X. Pellet, L. Scardia and C.I. Zeppieri: Homogenization of high-contrast Mumford-Shah energies. Submitted.
Recent conferences, seminars and visits
- The mathematical design of new materials, Newton Institute, 1-28 June 2019.
- Emergence of Structures in Particle Systems: Mechanics, Analysis and Computation, Oberwolfach, 28 October-3 November 2018.
- Meeting in Applied Mathematics and Calculus of Variations 2018, Rome, 3-6 September 2018.
- 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Taiwan, 5-9 July 2018.
- Calculus of Variations and Geometric Measure Theory at Sussex, University of Sussex, 2-4 July 2018.
- University of Bristol, 1 May 2018.
- Universitat Autònoma de Barcelona, 21-26 April 2018.
- Heriot-Watt University, 17 April 2018.
- Operators and Spectral theory, University of Reading, 4 April 2018.
- Geometry Day, Durham University, 2 March 2018.
- Universität Münster, 19-23 February 2018.
- University of Cardiff, 12 February 2018.
- Variational Methods for the Modelling of Inelastic Solids, Oberwolfach, 4-10 February 2018.
- Mini workshop on Analysis and Probability, University of Sussex, 22-25 January 2018.
- Variational Approaches to Problems in Solid Mechanics, University of Warwick, 18 December 2017.
- Universität Münster, 28 November - 2 December 2017.
- University of Pavia, 23-29 October 2017.
- Material Theories, Oberwolfach, 16-22 July 2017.
- Workshop on "Aggregation-Diffusion PDEs:Variational Principles, Nonlocality and Systems",
Anacapri, Italy, 10-14 July, 2017.
- BIRS Workshop "Analysis of Dislocation Models for Crystal Defects", Oaxaca (CMO, Mexico), 25-30 June 2017.
- Institut Henri Poincare, Paris, 15-16 June 2017.
- 7th meeting of the Reading-Bath-Cardiff network on generalised solutions for nonlinear PDE, University of Reading, 14 June 2017.
- Non-local equations and fractional diffusion, University of Warwick, 22-26 May 2017.
- University of Pavia, 1-5 May 2017.
- Miniworkshop on dislocations, plasticity, and fracture, SISSA, Trieste, Italy, 13-16 February 2017.
- University of Warwick, 3 February 2017.
- Universitat Autònoma de Barcelona, 26-28 January 2017.
Organisation of conferences
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).
Current seminar organisation
- EPSRC First Grant EP/N035631/1 on "Dislocation Patterns beyond equilibrium" (£ 95699), 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.