Giuseppe Mingione

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• PhD
• Professor (Full) at University of Parma

Solutions to nonlinear integro-differential systems are regular outside a negligible closed subset whose Hausdorff dimension can be explicitly bounded from above. This subset can be characterized using quantitative, universal energy thresholds for nonlocal excess functionals. The analysis is carried out via the use of nonlinear potentials and allow...

Nonuniform ellipticity is a classical topic in the theory of partial differential equations. While several results in regularity theory have been adding up over decades, many basic issues, as for instance the validity of Schauder theory and sharp dependence of regularity upon data, remained opened for a while. In these notes we give an overview of...

Lecture notes from the CIME course 2022 "Geometric and analytic aspects of functional variational principles".

Schauder theory for nonuniformly elliptic problems is finally established under the best possible assumptions on the growth rate of the ellipticity ratio. Such optimal assumptions were discovered more than twenty years earlier by means of counterexamples.

Slides of the talk given at "Nonlinear PDEs in Salzburg" on November 6, 2023.

Local Schauder estimates hold in the nonuniformly elliptic setting. Specifically, first derivatives of solutions to nonuniformly elliptic variational problems and elliptic equations are locally Hölder continuous, provided coefficients are locally Hölder continuous.

Minima of functionals of the type $$ w\mapsto \int_{\Omega}\left[\snr{Dw}\log(1+\snr{Dw})+a(x)\snr{Dw}^{q}\right] \dx\,, \quad 0\leq a(\cdot) \in C^{0, \alpha}\,,$$ with $\Omega \subset \er^n$, have locally H\"older continuous gradient provided $1 < q < 1+\alpha/n$.

Minima of functionals of the type w↦∫Ω|Dw|log(1+|Dw|)+a(x)|Dw|qdx,0≤a(·)∈C0,α,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w\mapsto \int _{\varOmega...

Barcelona Analysis Seminar

Minimizers of functionals of the type $$\begin{aligned} w\mapsto \int _{\Omega }[|Dw|^{p}-fw]\,\textrm{d}x+\int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\frac{|w(x)-w(y)|^{\gamma }}{|x-y|^{n+s\gamma }}\,\textrm{d}x\,\textrm{d}y\end{aligned}$$ w ↦ ∫ Ω [ | D w | p - f w ] d x + ∫ R n ∫ R n | w ( x ) - w ( y ) | γ | x - y | n + s γ d x d y with $$...

Presentation given at the conference Nouniformly elliptic problems, IMPAN, Warsaw, September 2022.

Slides of the talk given at the conference "Trends in Calculus of Variations and PDEs", 18-20 May 2022, Jointly organised by University of Sussex, UK and Ghent Analysis & PDE Centre, UGent, Belgium.

Minimizers of functionals of mixed local and nonlocal type are locally $C^{1,\beta}$-regular.

Talk given at the International Prague Seminar on Function Spaces (Charles University and Czech Academy of Sciences).

Talk given at the 2022 Winter meeting on PDEs, Chinese Academy of Sciences. The video is available at (cut-and-paste) https://www.youtube.com/watch?v=KE-jli11cC0

Talk given at the Oxford PDE seminar.

A talk given at the conference One Day in Double Phase Problems

Local Schauder estimates hold in the nonuniformly elliptic setting. Specifically, first derivatives of solutions to nonuniformly elliptic variational problems and elliptic equations are locally H\"older continuous, provided coefficients are locally H\"older continuous.

We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with...

Recent regularity results for nonlinear elliptic problems.

For nonautonomous, nonuniformly elliptic integrals with so-called (p, q)-growth conditions, we show a general interpolation property allowing to get basic higher integrability results for Hölder continuous minimizers under improved bounds for the gap q/p. For this we introduce a new method, based on approximating the original, local functional, wit...

Function Spaces/Nonlinear Analysis and PDE’S Online Seminar, May 27, 2021

We provide an overview of recent results concerning elliptic variational problems with nonstandard growth conditions and related to different kinds of nonuniformly elliptic operators. Regularity theory is at the center of this paper.

For nonautonomous, nonuniformly elliptic integrals with so-called $(p,q)$-growth conditions, we show a general interpolation property allowing to get basic higher integrability results for H\"older continuous minimizers under improved bounds for the gap $q/p$. For this we introduce a new method, based on approximating the original, local functional...

We review some recent Lipischitz regularity results for solutions to nonlinear elliptic equations and systems. In particular, we deal with minima of integral functionals. Emphasis is put on the nonuniformly elliptic case.

We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with...

Talk given at the conference "Singular Problems Associated to Quasilinear Equations",

Slides of a talk given on May 12, 2020, at the One world PDE seminar hosted by the University of Bath.

Slides of a talk presented at the Rio de Janeiro webinar on analysis and partial differential equations.

Slides of a talk given at Shanghaitech university via Zoom

Slides of the Lectures given at Shanghaitech University, via Zoom

Nonlinear Potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered. In recent years there has been a substantial development of this subject, mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials. Here we give...

Nonlinear Potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered. In recent years there has been a substantial development of this subject, mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials. Here we give...

Slides of a talk given at Augsburg on September 9, 2019

A talk given in honour of Juan Manfredi, the p-Laplacean godfather

We report on new techniques and results in the regularity theory of general non-uniformly elliptic variational integrals. By means of a new potential theoretic approach we reproduce, in the non-uniformly elliptic setting, the optimal criteria for Lipschitz continuity known in the uniformly elliptic one and provide a unified approach between non-uni...

Slides of a talk given at Nihon University on June 14, 2019

Slides of a talk given at rims on June 11, 2019

Slides of the Colloquium talk given at the Tokyo University of Science on June 7th, 2019

Slides of the talk given at the PDE real analysis seminar, The University of Tokyo, on June 4th, 2019

Talk given at the conference "Harmonic Analysis and PDE", Holon Institute of Technology, Israel, in honor of Vladimir Maz'ya

We consider regularity issues for minima of non-autonomous functionals in the Calculus of Variations exhibiting non-uniform ellipticity features. We provide a few sharp regularity results for local minimizers that also cover the case of functionals with nearly linear growth. The analysis is carried out provided certain necessary approximation-in-en...

We consider regularity issues for minima of non-autonomous functionals in the Calculus of Variations exhibiting non-uniform ellipticity features. We provide a few sharp regularity results for local minimizers that also cover the case of functionals with nearly linear growth. The analysis is carried out provided certain necessary approximation-in-en...

We consider the problem of minimizing variational integrals defined on nonlinear Sobolev spaces of competitors taking values into the sphere. The main novelty is that the underlying energy features a non-uniformly elliptic integrand exhibiting different polynomial growth conditions and no homogeneity. We develop a few intrinsic methods aimed at pro...

We consider the problem of minimizing variational integrals defined on \cc{nonlinear} Sobolev spaces of competitors taking values into the sphere. The main novelty is that the underlying energy features a non-uniformly elliptic integrand exhibiting different polynomial growth conditions and no homogeneity. We develop a few intrinsic methods aimed a...

We show, in a borderline case which was not covered before, the validity of nonlinear Calderón-Zygmund estimates for a class of non-uniformly elliptic problems driven by double phase energies.

We show, in a borderline case which was not covered before, the validity of nonlinear Calder\'on-Zygmund estimates for a class of non-uniformly elliptic problems driven by double phase energies.

Slides of a talk given at the conference "Workshop on Nonlinear PDE", Columbia University,. on December 1, 2018

Tank given at Courant Institute, New York University, November 2018

We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial...

A talk given at Urbino

Slides of a talk given at the Oxford Lunchtime Seminar in PDE

We report on new techniques and results in the regularity theory of general non-uniformly elliptic variational integrals. By means of a new potential theoretic approach we reproduce , in the non-uniformly elliptic setting, the optimal criteria for Lipschitz continuity known in the uniformly elliptic one and provide a unified approach between non-un...

Slides of the talk given in Rome on March 19, 2018, at the Stampacchia memorial conference

Slides of a talk given at a conference at Accademia dei Lincei to celebrate the 80th birthday of the great mathematician Vladimir Maz'ya

The article summarizes the scientific achievements of Professor Carlo Sbordone

We prove sharp regularity results for a general class of functionals of the type w → F (x, w, Dw) dx , featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral w → b(x, w)(|Dw| p + a(x)|Dw| q) dx , 1 < p < q , a(x) ≥ 0 , with 0 < ν ≤ b(·) ≤ L. This changes its ellipticity...

We settle the longstanding problem of establishing pointwise potential estimates for vectorial solutions u\colon \Omega \to \mathbb R^{N} to the non-homogeneous p -Laplacean system -\mathrm{div} (|Du|^{p-2}Du)=\mu \qquad \text{in}\ \Omega \subset \mathbb R^{n}\,, where \mu is a \mathbb R^{N} -valued measure with finite total mass. In particular, fo...

Lecture given at the Korea Institute for Advanced Studies in February 2018

We prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems. Specifically, we deal with the limiting case of the classical theory of Calder\'on \& Zygmund in the setting of nonlinear, possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of sol...

Slides of a talk given at the conference ”Sub-Riemannian Geometry Harmonic analysis, pde and applications”, Accademia delle Scienze dell’Istituto di Bologna, January 24, 2018

Slides of a talk given at the conference ”Non-standard growth phenomena 2017” at Turku University, in august 2017

We prove sharp regularity results for a general class of functionals of the type $$ w \mapsto \int F(x, w, Dw) \, dx\;, $$ featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral $$ w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx\;,\quad 1 <p < q\,, \quad a(x)\geq 0\;, $$...

Slides of the lectures given at the 4th Summer School in Analysis, University of Chicago -- June 2017

Closing plenary address at the 23th Rolf Nevanlinna Colloquium - ETH

Plenary talk at the German Mathematical Society Conference 2008

Slides of a talk given at ICTP on May 31, 2017

Nonlinear Calderón-Zygmund Theory aims at reproducing, in the nonlinear setting, the classical linear theory originally developed by Calderón and Zygmund. This topic has large intersections with Nonlinear Potential Theory. We survey here the main results of this theory.

Presenting a selection of topics in the area of nonlocal and nonlinear diffusions, this book places a particular emphasis on new emerging subjects such as nonlocal operators in stationary and evolutionary problems and their applications, swarming models and applications to biology and mathematical physics, and nonlocal variational problems. The aut...

We connect classical partial regularity theory for elliptic systems to Nonlinear Potential Theory of possibly degenerate equations. More precisely, we find a potential theoretic version of the classical $\varepsilon $-regularity criteria leading to regularity of solutions of elliptic systems. For non-homogenous systems of the type $-\mathrm{div}\,...

This is the foreword to the special issue of Nonlinear Analysis dedicated to Professor Nicola Fusco on his 60th birthday

Nonlinear Potential Theory aims at reproducing, in the nonlinear setting, the classical results of potential theory concerning the fine and regularity properties of solutions to linear elliptic and parabolic equations. Potential estimates, integrability, differentiability and continuity properties of solutions are at the heart of the matter. Here w...

Slides of the course give at CIME summer school in July 2016

Slides of the talk given at the Telc Conference in Regularity, Charles University Prague, in April 2016

We give a summary of recent results from Nonlinear Potential Theory, focusing on linear and nonlinear potential estimates of solutions to non-homogeneous equations and systems. We start with the cases of quasilinear, possibly degenerate equations of -Laplacian type, and, passing through fully nonlinear elliptic equations, finally move to systems. I...

Slides of a talk delivered at the PDE Seminar of the Universitat Politècnica de Catalunya

We consider a class of non-uniformly nonlinear elliptic equations whose model is given by −div (|Du| p−2 Du + a(x)|Du| q−2 Du) = −div (|F | p−2 F + a(x)|F | q−2 F) where p < q and a(x) ≥ 0, and establish the related nonlinear Calderón-Zygmund theory. In particular, we provide sharp conditions under which the natural, and optimal, Calderón-Zygmund t...

Bounded minimisers of the functional $$ w \mapsto \int (|Dw|^p+a(x)|Dw|^q) \, dx\;, $$ where $0\leq a(\cdot) \in C^{0, \alpha}$ and $1< p < q$, are $C^{1, \beta}$-regular provided the sharp bound $ q \leq p+\alpha $ holds.

Solutions to nonlocal equations with measurable coefficients are higher differentiable. Specifically, we consider nonlocal integrodifferential equations with measurable coefficients whose model is given by where the kernel K (·) is a measurable function and satisfies the bounds with 0 < α < 1, Λ > 1, while f ∈ Lqloc (ℝn) for some q > 2n/(n + 2α). T...

We prove a Harnack inequality for minimisers of a class of non-autonomous functionals with non-standard growth conditions. They are characterised by the fact that their energy density switches between two types of different degenerate phases.

We consider a class of non-autonomous functionals characterised by the fact that the energy density changes its ellipticity and growth properties according to the point, and prove some regularity results for related minimis-ers. These results are the borderline counterpart of analogous ones previously derived for non-autonomous functionals with (p,...

One of the basic achievements in nonlinear potential theory is that the typical linear pointwise estimates via fundamental solutions find a precise analog in the case of nonlinear equations. We give a comprehensive account of this fact and prove new unifying families of potential estimates. We also describe new fine properties of solutions to measu...