Prof. GIUSEPPE MINGIONE
Born 28 August 1972 in Caserta, Italy, is an Italian mathematician who is active in the fields of partial differential equations and calculus of variations.
Mingione received his Ph.D. in mathematics from the University of Naples Federico II in 1998 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 has found a positive answer for general integral functionals, thanks to the work of Kristensen and Mingione, who have also given explicit estimates for the dimension of the singular sets of minimisers. Subsequently, Mingione has 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 equationsand relate to and upgrade previous work of Kilpeläinen, Malý, Trudinger, Wang.
- Commendatore dell’Ordine al Merito della Repubblica Italiana per iniziativa del Presidente della Repubblica
- Amerio prize, 2016
- Invited speaker at the 7th European Congress of Mathematics, berlin 2016
- Nachdiplom lectures (list), ETH, 2015
- Highly Cited Researcher, Thomson Reuters-Clarivate Analytics (2014, 15, 16)
- Invited Speaker at the Italian Mathematical Society conference, 2011
- Caccioppoli prize 2010
- Invited speaker at the german mathematical society conference, 2008
- European research council award, 2007
- Stampacchia medal 2006
- Bartolozzi prize 2005
- Von Staudt chair, university of Erlangen-Nuremberg, 2004
The project has exploited basic properties of singular sets of vectorial problems, both in the interior and in the boundary case. Moreover, nonlinear potential techniques have been introduced in order to study the fine pointwise behavior of solutions to nonlinear elliptic and parabolic equations and systems. New gradient estimates for solutions in terms of linear and nonlinear potentials have been discovered. These allow to control regularity and singularities of solutions. Their use allows to determine sharp regularity criteria, leading to the solution of longstanding open issues in regularity theory.