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 postdoctoral positions at CAMGSDISTUL (Lisbon), IMPA(Rio de Janeiro) and UFC/ICMCUSP (São Carlos). In November 2016, I completed my Habilitation (LivreDocência) at USP (ICMC). As of January 2017, I joined faculty of the Department of Mathematics at the Pontifical Catholic University of Rio de Janeiro (PUCRio).
My research interests are in Analysis of Partial Differential Equations. In particular, I have studied the wellposedness of elliptic and parabolic meanfield 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 (CNPqBrazil), Rio de Janeiro Research Foundation (FAPERJ) and PUCRio 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 (ICTPTrieste) for the period 20182023. 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 Level2 Researcher by the Brazilian National Council for Scientific and Technological Development (CNPqBrazil). In April 2019 I was selected as a Grantee of Instituto Serrapilheira.
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1 
On a Mean Field Optimal Control Problem


2 
Fully nonlinear meanfield games

3 
Improved regularity for the porous medium equation along zero levelsets

4 
Existence and improved regularity for a nonlinear system with collapsing ellipticity

5 
Regularity of solutions to a class of variableexponent fully nonlinear elliptic equations

6 
Improved regularity for the pPoisson equation through approximation methods

7 
Singular meanfield games

1 
Geometric regularity for elliptic equations in doubledivergence form


2 
Regularity theory for the Isaacs equation through approximation methods

3 
Interior Sobolev regularity for fully nonlinear parabolic equations

4 
Regularity theory for secondorder stationary meanfield games

5 
Sharp Hessian integrability for nonlinear elliptic equations: an asymptotic approach

6 
Time dependent meanfield games in the superquadratic case

7 
Time dependent meanfield games with logarithmic nonlinearities

8 
Time dependent meanfield games in the subquadratic case

1 
Asymptotic methods in regularity theory for nonlinear elliptic equations: a survey


2 
Regularity for meanfield games systems with initialinitial boundary conditions: the subquadratic case

1 
Regularity theory for meanfield game systems.


2 
Economic models and meanfield games theory.

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 righthand side. It is known that the Hessian of the solutions is Hölder continuous. Surprisingly enough, this result also includes secondorder 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 KrylovSafonov and the EvansKrylov theories and the seminal work of Caffarelli (Ann. Math., 1989 (130), 189213). An important contribution to this theory is the socalled counterexamples of Nadirashvili and Vladut. Simply put, they set the KrylovSafonov 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 doubledivergence equation. Although solutions are merely Hölder continuous in the entire domain, they reach their degeneracy set as Lipschitz functions. The regularity theory for the doubledivergence equation is the subject of the research paper Geometric regularity for elliptic equations in doubledivergence 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 coauthored 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 spacevariable. In case such dependence is logHölder continuous, the theory is wellestablished. We have studied the boundedlymeasurable setting. We called this case roughly degenerate diffusions. Here, we tackled both the variational and the nonvariational 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 meanfield 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 postdoctoral interns.
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Click here to watch a lecture of mine at the Hausdorff Research Institute (Bonn)
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Note: if you are a graduate student or a young postdoc interested in working in my group, please do not hesitate in dropping me a line.
Jan
2020
Scientific visit to prof. Benjamin Gess
Dec
2019
Invited Speaker  XII Americas Conference on Differential Equations and Nonlinear Analysis
Nov
2019
Visit of Prof. Hitoshi Ishii (agenda blocked)
Oct
2019
Visit of Prof. José Miguel Urbano (agenda blocked)
Oct
2019
Scientific visit to Prof. Henrik Shahgholian
Sep
2019
Visit of Prof. Disson dos Prazeres (agenda blocked)
Aug
2019
Invited Speaker  Nonlinear PDEs Conference
Aug
2019
Visit of Prof. Yannick Sire (agenda blocked)
JulAug
2019
Invited Speaker  XI BrazilianItalian Workshop
Jul
2019
32nd Brazilian Math Colloquium
Jul
2019
Invited Speaker  I Joint Meeting BrazilFrance in Mathematics
Jun
2019
Invited Speaker  II IMDE: Conference AmazonAndalusia on PDEs
JanFeb
2019
HIM Trimestre Program  Evolution of Interfaces
Nov
2018
Invited Speaker  V Colóquio de Matemática do Centro Oeste
Nov
2018
Scientific visit to Prof. José Miguel Urbano
Aug
2018
Invited Speaker  The Mathematics of Games in the Applied Science  Satellite event to the ICM 2018
Aug
2018
Member of the Organizing Committee  International Congress of Mathematicians  ICM2018
Jul
2018
Invited Speaker  7th ISTIME, in honor of Prof. Paulo Cordaro  Satellite event to the ICM 2018
Jul
2018
Invited Speaker  Satellite Conference on Nonlinear Partial Differential Equations  Satellite event to the ICM 2018
JunJul
2018
Scientific visit to ICTP
May
2018
Scientific visit to Prof. Yechuda Pinchover  Technion
May
2018
Annual Meeting of the Israel Mathematical Union  Technion
Feb
2018
Scientific visit to Prof. José Carrillo  Imperial College
Jan
2018
Scientific visit to Prof. Roberto Capistrano  UFPE
Jan
2018
Scientific visit to Prof. Raimundo Leitão  UFC
Jan
2018
Scientific visit to Prof. Roberto Capistrano  UFPE
Jan
2018
Scientific visit to Prof. Raimundo Leitão  UFC
Jan
2018
Scientific visit of Prof. Anne C. Bronzi  PUCRio
Jan
2018
Scientific visit of Prof. Ricardo Ribeiro  PUCRio
Dec
2017
SIAM Conference on Partial Differential Equations
Dec
2017
Scientific visit to Prof. Jameson Graber
Nov
2017
Scientific visit of Prof. Diogo Gomes  PUCRio
Nov
2017
Scientific visit of Prof. Marcos Pimenta (UNESP)  PUCRio
OctNov
2017
Scientific visit of Prof. Daniela Tonon (ParisDauphine and CEREMADE)  Funded by the Réseau FrancoBrésilien en Mathématiques
Nov
2017
Scientific visit to Prof. Anne Bronzi  IMECCUnicamp
OctNov
2017
Scientific visit of Prof. Daniela Tonon (ParisDauphine and CEREMADE)  Funded by the Réseau FrancoBrésilien en Mathématiques
Sep
2017
Scientific visit of Prof. Julián Fernandez Bonder (UBA)  PUCRio
Sep
2017
10th Workshop on Nonlinear Partial Differential Equations
Aug
2017
Scientific visit to Prof. Diogo A. Gomes  KAUST
Aug
2017
XI Americas Conference on Differential Equations and Nonlinear Analysis
Jul
2017
31st Brazilian Mathematical Colloquium
Jul
2017
Mathematical Congress of the Americas  MCA2017
MayJun
2017
Advanced School/Workshop on Nonlocal PDEs and Applications to Geometry, Physics and Probability
Apr
2017
BIRS Workshop  Mostly Maximum Principle
Mar
2017
Scientific visit to Prof. Damião J. Araújo
Feb
2017
IX UnB Workshop de Verão em Matemática
Feb
2017
ICMC Summer Meeting on Differential Equations
Doctoral student
October/2017  present
Doctoral student
January/2018  present
Rua Marquês de São Vicente 225
22453900 Rio de Janeiro, RJ Brazil
Ed. Cardeal Leme, 8th Floor, Room 870
pimentel [at] pucrio [dot] br