My research focuses on problems in the Calculus of Variations and Partial Differential Equations.
My research topics include:
Fracture dynamics and damage in continuum mechanics
Mathematical models of elasticity, plasticity, and dislocations
Γ-convergence and asymptotic analysis of local and nonlocal energies
Variational methods for elliptic PDEs involving fractional and nonlocal operator
The following preprints can be found here
M. Caponi: A Survey on the Div-Curl Lemma and Some Extensions to Fractional Sobolev Spaces, Preprint (2025), arXiv: 2509.11734
S. Almi, M. Caponi, M. Friedrich, F. Solombrino: Riesz fractional gradient functionals defined on partitions: nonlocal-to-local variational limits, Preprint (2025), arXiv: 2510.04881
A preprint version of the following papers can be found here
M. Caponi, A. Carbotti, A. Maione: H-compactness for nonlocal linear operators in fractional divergence form, Calc. Var. Partial Differential Equations 64 (2025), Paper No. 290, 36 pp.
S. Almi, M. Caponi, M. Friedrich, F. Solombrino: A fractional approach to strain-gradient plasticity: beyond core-radius of discrete dislocations, Math. Ann. 391 (2025), 4063-4115
M. Caponi, V. Crismale: A rate-independent model for geomaterials under compression coupling strain gradient plasticity and damage, ESAIM, Control Optim. Calc. Var. 31 (2025), Paper No. 14, 49 pp.
S. Almi, M. Caponi, M. Friedrich, F. Solombrino: Geometric rigidity on Sobolev spaces with variable exponent and applications, NoDEA Nonlinear Differential Equations Appl. 32, (2025), Paper No. 12, 50 pp.
M. Caponi, A. Carbotti, F. Sapio: The viscoelastic paradox in a nonlinear Kelvin-Voigt type model of dynamic fracture, J. Evol. Equ. 24 (2024), Paper No. 63, 38 pp.
N. Cangiotti, M. Caponi, A. Maione, E. Vitillaro: Schrödinger-Maxwell equations driven by mixed local-nonlocal operators, Fract. Calc. Appl. Anal. 27 (2024), 677-705
N. Cangiotti, M. Caponi, A. Maione, E. Vitillaro: Klein-Gordon-Maxwell equations driven by mixed local-nonlocal operators, Milan J. Math. 91 (2023), 375-403
M. Caponi, F. Sapio: An existence result for the fractional Kelvin-Voigt's model on time-dependent cracked domains, J. Evol. Equ. 21 (2021), 4095-4143
M. Caponi, F. Sapio: A dynamic model for viscoelastic materials with prescribed growing cracks, Ann. Mat. Pura Appl. 199 (2020), 1263-1292
M. Caponi: Existence of solutions to a phase-field model of dynamic fracture with a crack-dependent dissipation, NoDEA Nonlinear Differential Equations Appl. 27 (2020), Paper No. 14, 48 pp.
M. Caponi, I. Lucardesi, E. Tasso: Energy-dissipation balance of a smooth moving crack, J. Math. Anal. Appl. 483 (2020), 123656, 32 pp.
M. Caponi: Linear Hyperbolic Systems in Domains with Growing Cracks, Milan J. Math. 85 (2017), 149-185
M. Caponi, P. Pucci: Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl. 195 (2016), 2099-2129