Calculus of Variations, Geometric Measure Theory and Nonlinear Partial Differential Equations
My research lies at the interface between the Calculus of Variations, Geometric Measure Theory and nonlinear Partial Differential Equations, with connections to continuum mechanics and materials science.
A central theme of my research is the analysis of variational models and evolution problems involving discontinuities, interfaces, singularities, multiple scales and non-standard growth. My work combines variational methods, Γ-convergence, geometric measure theory, BV and BD techniques, nonlinear PDE methods, and variational time-discretization schemes.
I study variational models at different scales, with particular emphasis on the passage from discrete or microscopic descriptions to continuum theories.
My research includes:
atomistic and discrete variational models;
motion and nucleation of discrete interfaces;
discrete-to-continuum limits;
homogenization and Γ-convergence;
finite-difference approximations of variational energies;
non-local approximations of continuum functionals.
A major part of my research concerns variational problems involving discontinuities, interfaces and surface energies.
Topics include:
Griffith-type fracture energies and their discrete and non-local approximations;
free-discontinuity functionals;
integral representation and Γ-convergence;
lower semicontinuity of surface integrals;
free-discontinuity problems with non-standard and spatially dependent growth;
regularity of minimizers and almost-minimizers;
one-phase and two-phase free-boundary problems.
I investigate nonlinear variational problems and elliptic and parabolic systems with general or non-standard growth.
Topics include:
nonlinear elasticity and relaxation;
nematic elastomers;
Orlicz-Sobolev and generalized Orlicz settings;
variational integrals with non-standard growth;
double-phase problems and double-phase fluids;
partial regularity for elliptic and parabolic systems;
boundary regularity and regularity of minimizers.
I study dissipative evolution problems through variational approximation and time-discrete methods.
Topics include:
minimizing movements and geometric minimizing movements;
gradient flows;
multiscale analysis of variational dynamics;
delayed loss of stability;
quasi-static limits;
vanishing-viscosity and vanishing-inertia limits;
Balanced Viscosity solutions;
Inertial Balanced Viscosity and Inertial Virtual Viscosity solutions for rate-independent systems.
A further research direction concerns pairing theories and integration-by-parts formulas in nonsmooth settings.
Topics include:
Anzellotti-type pairings and their generalizations;
pairings involving measure fields with divergence measure;
divergence-measure vector and tensor fields;
Gauss–Green formulas on rough domains and sets of finite perimeter;
div-curl type results;
stress-strain duality and slicing methods.
Calculus of Variations · Geometric Measure Theory · Γ-convergence · BV · BD · GSBD · Free-Discontinuity Problems · Fracture Mechanics · Discrete-to-Continuum Analysis · Homogenization · Nonlinear Elasticity · Nematic Elastomers · Orlicz Growth · Double-Phase Problems · Partial Regularity · Gradient Flows · Rate-Independent Systems · Divergence-Measure Fields · Pairings · Gauss–Green Formulas
My profiles on Google Scholar and Researchgate provide additional information on my research activity.
[1] G. Scilla, Variational problems with percolation: rigid spin systems. Adv. Math. Sci. Appl. 23 (2013), 187-207. (preprint)
[2] A. Braides and G. Scilla, Motion of discrete interfaces in periodic media. Interfaces Free Bound. 15 (2013), 451-476. (preprint)
[3] G. Scilla, Motion of discrete interfaces in low contrast periodic media. Netw. Heterog. Media 9 (2014), 169-189. (preprint)
[4] A. Braides and G. Scilla, Nucleation and backward motion of discrete interfaces. C. R. Acad. Sci. Paris (2013), Vol. 351, Issues 21-22, 803-806. (preprint)
[5] A. Barbagallo and G. Scilla, Stochastic weighted variational inequalities in non-pivot Hilbert spaces with applications to a transportation model, J. Math. Anal. Appl. 457 (2) (2018), 1118-1134. 10.1016/j.jmaa.2017.07.067
[6] G. Scilla and V. Vallocchia, Chirality transitions in frustrated ferromagnetic spin chains: a link with the gradient theory of phase transitions, J. Elasticity 132(2) (2018), 271-293 (preprint)
[7] G. Scilla and F. Solombrino. Delayed loss of stability in singularly perturbed finite-dimensional gradient flows, Asymptot. Anal. 110 (1-2) (2018), 1-19 (preprint)
[8] G. Scilla and F. Solombrino, Multiscale analysis of singularly perturbed finite dimensional gradient flows: the minimizing movement approach, Nonlinearity 31(11) (2018), 5036-5074. (preprint)
[9] G. Scilla and F. Solombrino, A variational approach to the quasistatic limit of viscous dynamic evolutions in finite dimension, J. Differential Equations 267 (2019), 6216-6264. (preprint)
[10] G. Scilla and B. Stroffolini, Relaxation of nonlinear elastic energies related to Orlicz-Sobolev nematic elastomers, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 31(2020), 349-389. (preprint)
[11] G. Scilla, Motion of discrete interfaces on the triangular lattice. Milan J. Math. 88(2) (2020), 315-346 (preprint)
[12] V. Crismale, G. Scilla and F. Solombrino, A derivation of Griffith functionals from discrete finite-difference models, Calc. Var. Partial Differential Equations 59:193 (2020). (preprint)
[13] G. Scilla and F. Solombrino, Non-local approximation of the Griffith functional, NoDea Nonlinear Differential Equations Appl. 28:17 (2021). (preprint)
[14] A. Braides, G. Scilla and A. Tribuzio, Nucleation and growth of lattice crystals, J. Nonlinear Sci. 31:97 (2021). (preprint)
[15] C. Goodrich, G. Scilla and B. Stroffolini, Partial regularity for minimizers of discontinuous quasiconvex integrals with general growth, Proc. Roy. Soc. Edinburgh Sect. A (2021), doi: 10.1017/prm.2021.53 (preprint)
[16] F. Farroni, G. Scilla and F. Solombrino, On some non-local approximation of nonisotropic Griffith-type functionals, Math. Eng. 4(4)(2022), 1-22 (preprint)
[17] G. Scilla and B. Stroffolini, Invertibility of Orlicz-Sobolev maps, In: Español, M.I., Lewicka, M., Scardia, L., Schlömerkemper, A. (eds) Research in Mathematics of Materials Science. Association for Women in Mathematics Series, vol 31. Springer, Cham. (2022) https://doi.org/10.1007/978-3-031-04496-0_13 , (preprint).
[18] F. Riva, G. Scilla and F. Solombrino, The notions of Inertial Balanced Viscosity and Inertial Virtual Viscosity solution for rate-independent systems, Adv. Calc. Var. (2022) https://doi.org/10.1515/acv-2021-0073 , (preprint)
[19] J. Ok, G. Scilla and B. Stroffolini, Boundary partial regularity for minimizers of discontinuous quasiconvex integrals with general growth, Comm. Pure Appl. Anal. 21(12)(2022), 4173-4214, http://dx.doi.org/10.3934/cpaa.2022140 (preprint)
[20] V. De Cicco and G. Scilla, Lower semicontinuity in GSBD for nonautonomous surface integrals, To appear on ESAIM:COCV 29:13 (2023), https://doi.org/10.1051/cocv/2023001 (preprint)
[21] G. Scilla and B. Stroffolini, Partial regularity for steady double phase fluids, Math. Eng. 5(5) (2023), 1-47, https://doi.org/10.3934/mine.2023088 (preprint)
[22] G. Scilla, F. Solombrino and B. Stroffolini, Integral representation and Gamma-convergence for free-discontinuity problems with p(x)-growth, Calculus of Variations and Partial Differential Equations 62 (2023), art. 213 (preprint)
[23] C. Leone, G. Scilla, F. Solombrino and A. Verde, Regularity of minimizers for free-discontinuity problems with p(x)-growth, ESAIM: Control, Optimisation and Calculus of Variations 29 (2023), art n. 78 (preprint)
[24] J. Ok, G. Scilla and B. Stroffolini, Regularity theory for parabolic systems with Uhlenbeck structure, Journal de Mathématiques Pures et Appliquées 182 (2024), 116-163 (preprint)
[25] J. Ok, G. Scilla and B. Stroffolini, Partial regularity for degenerate parabolic systems with general growth via caloric approximations, Calculus of Variations and Partial Differential Equations 63 (2024), art. n. 105 (preprint)
[26] F. Riva, G. Scilla and F. Solombrino, Inertial Balanced Viscosity (IBV) solutions to infinite-dimensional rate-independent systems, Journal of Functional Analysis 288(7) (2025), (preprint)
[27] C. Leone, G. Scilla, F. Solombrino and A. Verde, Strong existence for free-discontinuity problems with non-standard growth, SIAM: Journal on Mathematical Analysis 57(3) (2025), (preprint)
[28] J. Ok, G. Scilla and B. Stroffolini, Partial regularity for degenerate systems of double phase type, Journal of Differential Equations 432 (2025), 113207 (preprint)
[29] G. E. Comi, V. De Cicco and G. Scilla, Beyond BV: new pairings and Gauss-Green formulas for measure fields with divergence measure, Communications in Contemporary Mathematics (2026) DOI: https://doi.org/10.1142/S0219199726500410, (preprint)
[30] C. Leone, G. Scilla, F. Solombrino and A. Verde, Lipschitz regularity of almost-minimizers in two-phase free boundary problems with generalized Orlicz growth, Nonlinear Analysis 271 (2026), 114155 (preprint)
[31] G. Scilla, Variational motion of discrete interfaces (PhD Thesis, 2014). (PDF)
[32] C. Leone, G. Scilla, F. Solombrino and A. Verde, Lipschitz regularity of almost-minimizers in one-phase problems with generalized Orlicz growth, Preprint (2025), (preprint)
[33] J. Ok, G. Scilla and B. Stroffolini, Partial regularity for parabolic systems of double phase type, Preprint (2025), (preprint)
[34] V. De Cicco and G. Scilla, Gauss-Green formulas for divergence measure tensor fields on rough domains, Preprint (2026), (preprint)
[35] G. Scilla and B. Stroffolini, Orlicz meets double phase: steady fluids, Preprint (2026), (preprint)
[36] V. De Cicco and G. Scilla, A slicing approach to stress-strain duality, Preprint (2026), (preprint)
[37] G. E. Comi, V. De Cicco and G. Scilla, Pairing as duality product and div-curl lemma, In progress
[38] S. Almi, C. Leone, G. Manzo and G. Scilla, Regularity results for orthotropic obstacle problems, In progress
[39] C. Leone, G. Scilla, F. Solombrino and A. Verde, Regularity of the free boundary of almost-minimizers in one-phase problems with generalized Orlicz growth, In progress
[40] G. Di Fratta, F. Dipasquale, G. Scilla and B. Stroffolini, In progress