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commentaria [features a rapid description of some of my work and its impact]
ten selected papers [you can click on the green words]
c. de filippis, g. mingione: the sharp growth rate in nonuniformly elliptic schauder theory -- duke mathematical journal 175 (2025) 1775-1848 [schauder type estimates hold for solutions to nonuniformly elliptic problems under optimal assumptions on the growth of the ellipticity ratio]
c. de filippis, g. mingione: nonuniformly elliptic schauder theory -- inventiones mathematicae 234 (2023) 1109-1196 [solves two longstanding open problems. first, schauder type estimates for solutions to nonuniformly elliptic equations are proved; since the 60s dependence on coefficients was never allowed to be hölder continuous to get gradient hölder continuity. second, hölder gradient continuity of minima of non-differentiable, nonuniformly elliptic variational integrals is proved; in this case only the uniformly elliptic result was available, being a classical result of giaquinta & giusti, and of manfredi for the degenerate case, dating back to the 80s. based on a novel approach that allows for the first time to get gradient bounds directly, bypassing the standard perturbation arguments of the classical uniformly elliptic theory, linear and/or nonlinear. the results of the paper are the main ones appearing in cristiana de filippis' 2024 ems prize citation] [a review]
l. beck, g. mingione: lipschitz bounds and nonuniform ellipticity -- communications on pure and applied mathematics 73 (2020) 944-1034 [sharp lipschitz continuity criteria for nonuniformly elliptic problems; blueprint for several developments in the theory] [>100 citations on scopus]
t. kuusi, g. mingione: vectorial nonlinear potential theory -- journal of the european mathematical society 20 (2018) 929-1004 [solves the longstanding open problem of proving pointwise potential estimates for solutions, and for their gradients, to nonlinear degenerate elliptic systems] [>100 citations on scopus] [a review]
m. colombo, g. mingione: regularity for double phase variational problems -- archive for rational mechanics and analysis 215 (2015) 443-496 [optimal regularity theory of a class of variational integrals, that we called double phase functionals and that were first considered by zhikov in the context of homogenization and lavrentiev phenomenon; first regularity results were given with esposito and leonetti, jde 2004 where the sharp bound on the ellipticity ratio was discovered. more general cases were considered with baroni in calc var 2018. this and the other arma 2015 paper opened a wide lane of research and developments. the results of this paper are part of the maria colombo's 2024 stampacchia medal citation] [>450 citations on scopus]
g. mingione: gradient potential estimates -- journal of the european mathematical society 13 (2011) 459-486 [solves in the non-degenerate case the longstanding open problem, dating back to the work of kilpeläinen & malý, of proving gradient potential estimates for solutions to nonlinear equations in divergence form; the degenerate case is covered in later papers with duzaar, jfa 2010, ajm 2011 and kuusi, arma 2013, jfa 2012. parabolic cases were later treated with kuusi, jems 2014, ann sns pisa 2013, arma 2014. these papers opened a wide lane of research and developments] [>150 citations on scopus] [a review] [a review] [a review] [a review] [a review] [a review]
g. mingione: the calderón-zygmund theory for elliptic problems with measure data -- annali della scuola normale superiore di pisa, classe di scienze (ser. v) 6 (2007) 195-261 [first and sharp gradient fractional differentiability results for solutions to measure data problems; the previously existing literature, spanning over 40 years, only reported gradient integrability results] [>250 citations on scopus] [a review]
e. acerbi, g. mingione: gradient estimates for a class of parabolic systems -- duke mathematical journal 136 (2007) 285-320 [solves the longstanding open problem of proving calderón-zygmund estimates for solutions to nonlinear, potentially degenerate parabolic equations and systems; the elliptic analog was previously achieved between '83 and '94 by t. iwaniec, dibenedetto & manfredi and caffarelli & peral and the techniques of these papers do not extend to parabolic problems. the harmonic analysis free techniques introduced in this paper eventually became of standard use in the literature. when applied to the basic, linear elliptic case, the methods introduced here allow for a total elementary proof of calderòn-zygmund estimates which is only based on mean value formulas for harmonic functions and vitali covering] [>350 citations on scopus] [a review]
f. duzaar, j. kristensen, g. mingione: the existence of regular boundary points for non-linear elliptic systems -- journal für die reine und angewandte mathematik (crelles journal) 602 (2007) 17-58 [solves the longstanding open problem, dating back to the beginning of the 80s, of proving partial boundary regularity of solutions to general elliptic systems, i.e., that a.e. boundary point is regular when boundary data are smooth. see page 247 in giaquinta, ann. math. studies 105, princeton univ. press, 1983. prior to this paper, even the existence of one boundary regular point was unknown. the case of functionals is treated in arma 2010]
j. kristensen, g. mingione: the singular set of minima of integral functionals -- archive for rational mechanics & analysis 180 (2006) 331-398 [solves the longstanding open problem, dating back to the beginning of the 80s with the work of giaquinta & giusti, evans and others, of proving that the hausdorff dimension of the singular set of minima of vectorial functionals of the calculus of variations is strictly less than the ambient dimension. see open problem (a), page 117 in giaquinta, eth lectures in math., birkhäuser 1993. later on, we dealt with the quasiconvex case, arma 2007. the case of systems was previously dealt with in arma 2003 and calc var 2003. the same cirle of ideas allows to deal with dini continuous coefficients comm pde 2004 and parabolic problems ann. ihp 2005, mems 2011] [>150 citations on scopus] [a review]
survey papers and lecture notes [a bird-eye view on some of my results]
c. de filippis, g. mingione: nonuniform ellipticity in variational problems and regularity -- notices of the american mathematical society to appear [a streamlined introduction to the regularity theory of nonuniformly elliptic problems, with special emphasis on the methods introduced by the authors in order to prove nonuniformly elliptic schauder estimates]
g. mingione: nonlinear potential theoretic methods in nonuniformly elliptic problems -- lecture notes in mathematics 2348 springer (2024) 65-149 [outgrow of a series of cime lectures i delivered in 2022. based on work with beck, colombo and de filippis, summarises progresses on nonuniformly elliptic regularity made via extensive use of nonlinear potential theoretic methods]
g. mingione, v. radulescu: recent developments in problems with nonstandard growth and nonuniform ellipticity -- journal of mathematical analysis and applications 501 (2021) 125197 [content as of the title] [>150 citations on scopus]
g. mingione: recent progress in nonlinear potential theory -- proceedings of the 7th european congress of mathematics, ems press (2018) 501-524 [lecture i gave at the 7th ecm in berlin, 2016]
g. mingione: short tales from nonlinear calderón-zygmund theory -- lecture notes in mathematics 2186 springer (2017) 159-204 [outgrow of a series of cime lectures i delivered in 2016]
t. kuusi, g. mingione: guide to nonlinear potential estimates -- bulletin of mathematical sciences 4 (2024) 1-82 [this basically surveys the elliptic scalar results on nonlinear potential estimates obtained with kuusi, and adds new ones extending those in jfa 2012. previous results on nonlinear potential estimates in the literature are included as well] [>150 citations on scopus]
g. mingione: regularity of minima: an invitation to the dark side of the calculus of variations -- applications of mathematics 51 (2006) 355-426 [summarizes the situation with regularity in the calculus of variations as long as general integral functionals are concerned up to 2005, when this was written as an outgrow of the lectures i gave at the 9th Paseky school in fluid mechanics. It was also conceived as an update (with no proofs) to giaquinta’s orange book and giusti’s direct methods] [>300 citations on scopus]
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