Mingione received his Ph.D. in mathematics from the
University of Naples Federico II in 1999 having
Nicola Fusco as advisor; he is professor of mathematics at the
University of Parma. He has mainly worked on regularity aspects of the Calculus of Variations, solving a few longstanding questions about the
Hausdorff dimension of the singular sets of minimisers of vectorial integral functionals and the boundary singularities of solutions to
nonlinear elliptic systems. This connects to the work of authors as
Almgren,
De Giorgi,
Morrey,
Giusti, who proved theorems asserting regularity of solutions outside a singular set (i.e. a closed subset of
null measure) both in
geometric measure theory and for variational systems of partial differential equations. These are indeed called partial regularity results and one of the main issues is to establish whether the dimension of the singular set is strictly less than the ambient dimension. This question found a positive answer for general integral functionals, thanks to the work of
Kristensen and Mingione, who also gave explicit estimates for the dimension of the singular sets of minimisers. Mingione also worked on nonlinear
potential theory obtaining potential estimates for solutions to nonlinear elliptic and
parabolic equations. Such estimates allow to give a unified approach to the regularity theory of quasilinear, degenerate equations and relate to and upgrade previous work of Kilpeläinen, Malý,
Trudinger,
Wang. Together with
Cristiana De Filippis, Mingione proved a Schauder type theory for nonuniformly elliptic equations and functionals. ==Recognition==