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Giuseppe Mingione
Giuseppe Mingione (born 28 August 1972) is an Italian mathematician who is active in the fields of partial differential equations and calculus of variations.
==Scientific activity==
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.〔(【引用サイトリンク】url=http://dml.cz/bitstream/handle/10338.dmlcz/134645/AplMat_51-2006-4_4.pdf )〕 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.〔(【引用サイトリンク】url=http://link.springer.com/article/10.1007/s00205-005-0402-5 )〕〔(【引用サイトリンク】url=http://link.springer.com/article/10.1007/s00205-006-0036-2 )〕 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 equations〔(【引用サイトリンク】url=http://umi.dm.unibo.it/caccioppoli/vincitori-en.html#2010 )〕〔(【引用サイトリンク】url=http://link.springer.com/article/10.1365%2Fs13291-010-0004-5 )〕〔(【引用サイトリンク】url=http://www.ems-ph.org/journals/show_abstract.php?issn=1435-9855&vol=13&iss=2&rank=7&srch=searchterm%7CMingione )〕 and relate to and upgrade previous work of Kilpeläinen, Malý, Trudinger, Wang.
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