Edgard A. Pimentel
Math Department @ PUC-Rio

Short Bio

I obtained my Ph.D. in Mathematics from the Instituto Superior Técnico at Universidade de Lisboa (Portugal, 2013) under the direction of Prof. Diogo A. Gomes. After the completion of my thesis, I held post-doctoral positions at CAMGSD-IST-UL (Lisbon), IMPA(Rio de Janeiro) and UFC/ICMC-USP (São Carlos). In November 2016, I completed my Habilitation (Livre-Docência) at USP (ICMC). As of January 2017, I joined faculty of the Department of Mathematics at the Pontifical Catholic University of Rio de Janeiro (PUC-Rio).

My research interests are in Analysis of Partial Differential Equations. In particular, I have studied the well-posedness of elliptic and parabolic mean-field games systems in the class of smooth solutions. In addition, I have worked on regularity theory for fully nonlinear equations of elliptic and parabolic type. As regards financial support, my work has been funded by the International Mathematical Union (IMU), São Paulo Research Foundation (FAPESP), The National Council for Research and Technology (CNPq-Brazil), Rio de Janeiro Research Foundation (FAPERJ) and PUC-Rio baseline funds. I am a member of the Brazilian Mathematical Society, the Israel Mathematical Union, the American Mathematical Society and the Society for Industrial and Applied Mathematics.

I am a Junior Associate Fellow of the International Center for Theoretical Physics (ICTP-Trieste) for the period 2018-2023. Since 2018, I am a Jovem Cientista do Estado do Rio de Janeiro, funded by the Fundação Carlos Chagas de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ). In 2018, I was selected as a Level-2 Researcher by the Brazilian National Council for Scientific and Technological Development (CNPq-Brazil). In April 2019 I was selected as a Grantee of Instituto Serrapilheira.

To download my complete CV, click here.

Publications

Papers submitted to scientific journals
1

Fully nonlinear mean-field games
w/ P. Andrade

2

Improved regularity for the porous medium equation along zero level-sets
w/ M. Santos

3

Existence and improved regularity for a nonlinear system with collapsing ellipticity
w/ J.-M. Urbano

4

Regularity of solutions to a class of variable-exponent fully nonlinear elliptic equations
w/ A. Bronzi, G. Rampasso and E. Teixeira

5

Improved regularity for the p-Poisson equation through approximation methods
w/ G. Rampasso and M. Santos

6

Singular mean-field games
w/ M. Cirant, D. Gomes and H. Sánchez-Morgado

Papers published in scientific journals
1

Geometric regularity for elliptic equations in double-divergence form
To appear in Analysis and PDE

2

Regularity theory for the Isaacs equation through approximation methods
Ann. Inst. H. Poincaré Anal. Non Linéaire (2019) 36

3

Interior Sobolev regularity for fully nonlinear parabolic equations
Calc. Var. PDE. (2017) 56
w/ R. Castillo

4

Regularity theory for second-order stationary mean-field games
Indiana Univ. Math. J. (2017) 66
w/ V. Voskanyan

5

Sharp Hessian integrability for nonlinear elliptic equations: an asymptotic approach
J. Math. Pures Appl. (2016) 106
w/ E. Teixeira

6

Time dependent mean-field games in the superquadratic case
ESAIM Control Optim. Calc. Var. (2016) 22
w/ D. Gomes and H. Sánchez-Morgado

7

Time dependent mean-field games with logarithmic nonlinearities
SIAM J. Math. Anal. (2015) 47
w/ D. Gomes

8

Time dependent mean-field games in the subquadratic case
Comm. Partial Differential Equations (2015) 40
w/ D. Gomes and H. Sánchez-Morgado

Book chapters
1

Asymptotic methods in regularity theory for nonlinear elliptic equations: a survey
P. Cardaliaguet et al. (Eds.) PDE Models for Multi-Agent Phenomenon, Springer-Verlag (2018)
w/ M. Santos

2

Regularity for mean-field games systems with initial-initial boundary conditions: the subquadratic case
J.-P. Bourguignon et al. (Eds.) Dynamics, Games and Sciences III, Springer-Verlag (2015)
w/ D. Gomes

Books published
1

Regularity theory for mean-field game systems.
SpringerBriefs in Mathematics, Springer-Verlag 2016
w/ D. Gomes and V. Voskanyan

2

Economic models and mean-field games theory.
IMPA Mathematical Publications - 30th Braz. Math. Colloquium 2015
w/ D. Gomes and L. Nurbekyan

Research

A brief research statement

My research interests are in Regularity Theory for (nonlinear) Partial Differential Equations (PDEs). In brief, the reasoning in this field can be stated as follows: once a function solves a given equation in some weaker sense, the goal is to derive further regularity, implied by the structure of the problem.

An example is the Schauder's theory for the Poisson equation in the presence of Hölder continuous right-hand side. It is known that the Hessian of the solutions is Hölder continuous. Surprisingly enough, this result also includes second-order derivatives which does not even appear in the equation.

The nonlinear setting has been studied extensively, for the last forty years, by several authors. As a consequence, important advances have been produced. For example, we mention the Krylov-Safonov and the Evans-Krylov theories and the seminal work of Caffarelli (Ann. Math., 1989 (130), 189-213). An important contribution to this theory is the so-called counter-examples of Nadirashvili and Vladut. Simply put, they set the Krylov-Safonov regularity as the best possible without further assumptions but ellipticity. My research is inspired by this tradition and aims at advancing this program.

With this respect, I have studied methods of regularity transmission by approximation methods. Here, the idea is to transport information among distinct - though close enough - classes of PDEs. In this realm, the key element is the limiting profile from which to import information, back to the original equation. The appropriate design of this target is paramount.

For example, when considering the Isaacs equation, a natural limiting profile would be the Bellman equation. Provided certain proximity regimes are met by the data of the problem, we produced a regularity theory in Sobolev and Hölder, spaces for the solutions of the Isaacs equation through this technique. Those developments are in my recent work entitled Regularity theory for the Isaacs equation through approximation methods (Ann. Institut Henri Poincaré - A.N., 2018).

A class of problems that I have started to examine recently regards degenerate equations. Here, there are two distinct directions to pursue. First, we can investigate how solutions approach their degeneracy sets. That is to ask for the behaviour of the solutions in the regions where the PDE fails to hold. Surprisingly enough, our techniques have unveiled improved regularity along the degeneracy set of the solutions, in a large class of equations. An example is the double-divergence equation. Although solutions are merely Hölder continuous in the entire domain, they reach their degeneracy set as Lipschitz functions. The regularity theory for the double-divergence equation is the subject of the research paper Geometric regularity for elliptic equations in double-divergence form, submitted for publication.

A similar behavior appears in the context of the porous medium equation (PME). In fact, if the exponent of the diffusion is close to 1, information can be imported from the heat equation. In this case, the solutions to the PME are shown to be asymptotically Lipschitz as they approach their degeneracy set. Important consequences on the pressure function can also be derived through the same set of techniques. Advances pertaining to this class of equations are in the paper entitled Geometric regularity for the porous medium equation, submitted for publication. This work is co-authored by my doctoral student Makson Santos.

In the last year, I started considering diffusions degenerating as a power of the gradient. In fact, I have examined the case where the degeneracy degree depends explicitly on the space-variable. In case such dependence is log-Hölder continuous, the theory is well-established. We have studied the boundedly-measurable setting. We called this case roughly degenerate diffusions. Here, we tackled both the variational and the non-variational setting. Our findings with this respect are in two papers; the first is in collaboration with Anne Bronzi (UNICAMP, Brazil), Giane Rampasso, and Eduardo Teixeira (UCF, Florida). The second onde is in collaboration with Giane Rampasso and Makson Santos.

Finally, I would mention the regularity transmission approach to the thermistor problem, the fully nonlinear mean-field games and my growing interest in free boundary problems. My research has been funded by several agencies, in Brazil and abroad. As part of my research practices, I highlight the supervision and direction of graduate students and post-doctoral interns.

Click here for my profile on GoogleScholar

Click here for my profile on Mathematical Reviews

Click here to watch a lecture of mine at the Hausdorff Research Institute (Bonn)

Click here to watch a lecture of mine at BIRS-Banff

Note: if you are a graduate student or a young post-doc interested in working in my group, please do not hesitate in dropping me a line.

Agenda

My scientific visits

  • Jan

    2020

    MIS MPG, Leipzig

    Scientific visit to prof. Benjamin Gess

  • Dec

    2019

    Guanajuato, Mexico

    Invited Speaker - XII Americas Conference on Differential Equations and Nonlinear Analysis

  • Aug

    2019

    Stockholm, KTH

    Invited Speaker - Nonlinear PDEs Conference

  • Aug

    2019

    Rio de Janeiro

    Visit of Prof. Yannick Sire (agenda blocked)

  • Jul-Aug

    2019

    Varese, Italy

    Invited Speaker - XI Brazilian-Italian Workshop

  • Jul

    2019

    Rio de Janeiro

    32nd Brazilian Math Colloquium

  • Jul

    2019

    Rio de Janeiro

    Invited Speaker - I Joint Meeting Brazil-France in Mathematics

  • Jun

    2019

    Belém

    Invited Speaker - II IMDE: Conference Amazon-Andalusia on PDEs

  • Jan-Feb

    2019

    Bonn, Germany

    HIM Trimestre Program - Evolution of Interfaces

  • Nov

    2018

    Goiânia, Brazil

    Invited Speaker - V Colóquio de Matemática do Centro Oeste

  • Nov

    2018

    Coimbra, Portugal

    Scientific visit to Prof. José Miguel Urbano

  • Aug

    2018

    Niterói, Brazil

    Invited Speaker - The Mathematics of Games in the Applied Science - Satellite event to the ICM 2018

  • Aug

    2018

    Rio de Janeiro, Brazil

    Member of the Organizing Committee - International Congress of Mathematicians - ICM2018

  • Jul

    2018

    São Paulo, Brazil

    Invited Speaker - 7th IST-IME, in honor of Prof. Paulo Cordaro - Satellite event to the ICM 2018

  • Jul

    2018

    Fortaleza, Brazil

    Invited Speaker - Satellite Conference on Nonlinear Partial Differential Equations - Satellite event to the ICM 2018

  • Jun-Jul

    2018

    Trieste, Italy

    Scientific visit to ICTP

  • May

    2018

    Haifa, Israel

    Scientific visit to Prof. Yechuda Pinchover - Technion

  • May

    2018

    Haifa, Israel

    Annual Meeting of the Israel Mathematical Union - Technion

  • Feb

    2018

    London, UK

    Scientific visit to Prof. José Carrillo - Imperial College

  • Jan

    2018

    Recife, Brazil

    Scientific visit to Prof. Roberto Capistrano - UFPE

  • Jan

    2018

    Fortaleza, Brazil

    Scientific visit to Prof. Raimundo Leitão - UFC

  • Jan

    2018

    Recife, Brazil

    Scientific visit to Prof. Roberto Capistrano - UFPE

  • Jan

    2018

    Fortaleza, Brazil

    Scientific visit to Prof. Raimundo Leitão - UFC

  • Jan

    2018

    Rio de Janeiro, Brazil

    Scientific visit of Prof. Anne C. Bronzi - PUC-Rio

  • Jan

    2018

    Rio de Janeiro, Brazil

    Scientific visit of Prof. Ricardo Ribeiro - PUC-Rio

  • Dec

    2017

    Baltimore, USA

    SIAM Conference on Partial Differential Equations

  • Dec

    2017

    Waco, USA

    Scientific visit to Prof. Jameson Graber

  • Nov

    2017

    Rio de Janeiro, Brazil

    Scientific visit of Prof. Diogo Gomes - PUC-Rio

  • Nov

    2017

    Rio de Janeiro, Brazil

    Scientific visit of Prof. Marcos Pimenta (UNESP) - PUC-Rio

  • Oct-Nov

    2017

    Rio de Janeiro, Brazil

    Scientific visit of Prof. Daniela Tonon (Paris-Dauphine and CEREMADE) - Funded by the Réseau Franco-Brésilien en Mathématiques

  • Nov

    2017

    Campinas, Brazil

    Scientific visit to Prof. Anne Bronzi - IMECC-Unicamp

  • Oct-Nov

    2017

    Rio de Janeiro, Brazil

    Scientific visit of Prof. Daniela Tonon (Paris-Dauphine and CEREMADE) - Funded by the Réseau Franco-Brésilien en Mathématiques

  • Sep

    2017

    Rio de Janeiro, Brazil

    Scientific visit of Prof. Julián Fernandez Bonder (UBA) - PUC-Rio

  • Sep

    2017

    Brasília, Brazil

    10th Workshop on Nonlinear Partial Differential Equations

  • Aug

    2017

    Thuwal, Saudi Arabia

    Scientific visit to Prof. Diogo A. Gomes - KAUST

  • Aug

    2017

    Edmonton, Canada

    XI Americas Conference on Differential Equations and Nonlinear Analysis

  • Jul

    2017

    Rio de Janeiro, Brazil

    31st Brazilian Mathematical Colloquium

  • Jul

    2017

    Montreal, Canada

    Mathematical Congress of the Americas - MCA2017

  • May-Jun

    2017

    Trieste, Italy

    Advanced School/Workshop on Nonlocal PDEs and Applications to Geometry, Physics and Probability

  • Apr

    2017

    Banff, Canada

    BIRS Workshop - Mostly Maximum Principle

  • Mar

    2017

    João Pessoa, Brazil

    Scientific visit to Prof. Damião J. Araújo

  • Feb

    2017

    Brasília, Brazil

    IX UnB Workshop de Verão em Matemática

  • Feb

    2017

    São Carlos, Brazil

    ICMC Summer Meeting on Differential Equations

Teaching

My students

Makson S. Santos

Doctoral student

October/2017 - present

Pêdra Daricléa S. Andrade

Doctoral student

January/2018 - present

Contact

Department of Mathematics - PUC-Rio

Rua Marquês de São Vicente 225

22453-900 Rio de Janeiro, RJ Brazil

Ed. Cardeal Leme, 8th Floor, Room 870

pimentel [at] puc-rio [dot] br