researchseminars.org

Camillo De Lellis (IAS, Princeton)
2020-04-15 20:00 - 21:00 UTC | https://researchseminars.org/talk/RJWAPDE/1/
Title: Flows of vector fields: classical and modern (https://researchseminars.org/talk/RJWAPDE/1/)
by Camillo De Lellis (IAS, Princeton) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
Consider a (possibly time-dependent) vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindel\"of) Theorem states that, if the vector field $v$ is Lipschitz in space, for every initial datum $x$ there is a unique trajectory $\gamma$ starting at $x$ at time $0$ and solving the ODE $\dot{\gamma} (t) = v (t, \gamma (t))$. The theorem looses its validity as soon as $v$ is slightly less regular. However, if we bundle all trajectories into a global map allowing $x$ to vary, a celebrated theory put forward by DiPerna and Lions in the 80es show that there is a unique such flow under very reasonable conditions and for much less regular vector fields. A long-standing open question is whether this theory is the byproduct of a stronger classical result which ensures the uniqueness of trajectories for {\em almost every} initial datum. I will give a complete answer to the latter question and draw connections with partial differential equations, harmonic analysis, probability theory and Gromov's $h$-principle.

Diogo A. Gomes (KAUST)
2020-04-22 12:00 - 13:00 | https://researchseminars.org/talk/RJWAPDE/2/
Title: A mean-field price model (https://researchseminars.org/talk/RJWAPDE/2/)
by Diogo A. Gomes (KAUST) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
We propose a mean-field game model for the price formation of a commodity whose production is subjected to deterministic or random fluctuations. Agents seek to minimize their average cost by choosing their trading rates with a price that is characterized by a balance between supply and demand. In the deterministic case, we establish the existence of a solution under general conditions and give a full characterization. In the stochastic case, we show that, for linear dynamics and quadratic costs, the optimal trading rates are determined in feedback form. Hence, the price arises as the solution to a stochastic differential equation, whose coefficients depend on the solution of a system of ordinary differential equations.

Cristiana De Filippis (Oxford)
2020-04-29 13:00 - 14:00 | https://researchseminars.org/talk/RJWAPDE/3/
Title: Vectorial problems: sharp Lipschitz bounds and borderline regularity (https://researchseminars.org/talk/RJWAPDE/3/)
by Cristiana De Filippis (Oxford) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
For the abstract of Cristiana's talk, please use the following link:

https://drive.google.com/open?id=1eU6lWvs808FKxdqswC2x3JPJ2jEmMgFz

To join the webinar, use the following information:

Link:

https://puc-rio.zoom.us/j/94917916635?pwd=[REDACTED]

Meeting ID: 949 1791 6635
Password: [REDACTED]

Tatiana Toro (U. Washington)
2020-05-14 18:00 - 19:00 | https://researchseminars.org/talk/RJWAPDE/4/
Title: Elliptic measure and the geometry of domains in Euclidean space (https://researchseminars.org/talk/RJWAPDE/4/)
by Tatiana Toro (U. Washington) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
In this talk we will discuss the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. In particular we will examine the relationship between the behavior of the elliptic measure of a certain class divergence form uniformly elliptic operators on a domain and the structure of its boundary. This work mixes elements of geometric measure theory, free boundary regularity problems and harmonic analysis.

To join Prof. Tatiana's talk, please use the following link and information:

https://puc-rio.zoom.us/j/402375411?pwd=[REDACTED]

Meeting ID: 402 375 411
Password: [REDACTED]

Giuseppe Mingione (University of Parma)
2020-05-07 14:00 - 15:00 | https://researchseminars.org/talk/RJWAPDE/5/
Title: Gradient estimates (https://researchseminars.org/talk/RJWAPDE/5/)
by Giuseppe Mingione (University of Parma) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
I am going to give a review of gradient estimates for solutions to nonlinear elliptic and parabolic equations, with special emphasis on Nonlinear Potential Theory. Non-uniformly elliptic problems will also be treated.

To access the seminar, please use the following information:

https://zoom.us/j/96230784442?pwd=[REDACTED]

Meeting ID: 962 3078 4442
Password: [REDACTED]

Susanna Terracini (U. Torino)
2020-05-21 15:00 - 16:00 | https://researchseminars.org/talk/RJWAPDE/6/
Title: Segregation, interaction of species and related free boundary problems. (https://researchseminars.org/talk/RJWAPDE/6/)
by Susanna Terracini (U. Torino) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
Partial differential equations are often used to describe physical laws governing a complicated system in a confined space with relatively simpler behaviour on the boundary. Thus, reaction-diffusion systems with strong interaction terms appear in many multi-species physical problems as well as in population dynamics, chemistry and material science. The qualitative properties of the solutions and their limiting profiles in different regimes will be considered, as well as the interfaces.

To join Prof. Terracini's lecture, please use the following information:

https://puc-rio.zoom.us/j/99546616886?pwd=[REDACTED]

Meeting ID: 995 4661 6886
Password: [REDACTED]

Filomena Pacella (U. Roma, Sapienza)
2020-05-28 13:00 - 14:00 | https://researchseminars.org/talk/RJWAPDE/7/
Title: Critical exponents for a class of fully nonlinear equations (https://researchseminars.org/talk/RJWAPDE/7/)
by Filomena Pacella (U. Roma, Sapienza) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
We consider a class of radial fully nonlinear equations involving the extremal Pucci's operators and power nonlinearities.
For such equations some critical exponents were introduced by P.Felmer and A.Quaas in 2003, motivated by the study of positive entire radial solutions.
In the first part of the talk I will discuss the role and properties of such exponents and its relation with concentration phenomena and energy invariance.
In the second part I will present an alternative proof of the existence and uniqueness of these critical exponents, entirely based on the study of an associated quadratic dynamical system. This approach also allows to get in a unified and simple way new existence and classification results for singular solutions as well as to prove that the same critical exponents give a threshold for the existence or nonexistence of positive radial solutions for the Dirichlet problem in exterior domains. This last result was recently proved by G.Galise, A.Iacopetti and F. Leoni (2019) with a different proof.
The results presented are contained in several joint papers with I.Birindelli, G.Galise, F.Leoni, L.Maia, G.Saller Nornberg.

To join Prof. Pacella's talk, please use the following information.

Link to join the talk:
https://puc-rio.zoom.us/j/91799849377?pwd=[REDACTED]

Meeting ID: 917 9984 9377
Password: [REDACTED]

Mariana Smit Vega Garcia (Western Washington University)
2020-06-12 17:00 - 18:00 | https://researchseminars.org/talk/RJWAPDE/8/
Title: Regularity of almost minimizers with free boundary (https://researchseminars.org/talk/RJWAPDE/8/)
by Mariana Smit Vega Garcia (Western Washington University) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
We study almost minimizer for functionals which yield a free boundary, as in the work of Alt-Caffarelli and Alt-Caffarelli-Friedman. The almost minimizing property can be understood as the defining characteristic of a minimizer in a problem which explicitly takes noise into account. In this talk, we discuss the regularity of almost minimizers to energy functionals with variable coefficients. This is joint work with Guy David, Max Engelstein & Tatiana Toro.

To join Mariana's webinar, please use the following information:

Link to joni Mariana's talk:
https://puc-rio.zoom.us/j/94772587702?pwd=[REDACTED]

Meeting ID: 947 7258 7702
Password: [REDACTED]

Hugo Tavares (Instituto Superior Técnico, IST-Lisbon)
2020-06-18 15:00 - 16:00 | https://researchseminars.org/talk/RJWAPDE/9/
Title: Behavior near criticality of sign-changing solutions to the Lane-Emden equation with Dirichlet and Neumann boundary conditions (https://researchseminars.org/talk/RJWAPDE/9/)
by Hugo Tavares (Instituto Superior Técnico, IST-Lisbon) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
In this talk we will be concerned with several recent results on the classical Lane-Emden equation in a bounded domain, with either Dirichlet or Neumann boundary conditions.
In the first part of the talk, for radial solutions and in the case of the ball in dimension 3 or higher, we provide sharp rates and constants describing the asymptotic behavior (as we approach the Sobolev critical exponent) of all local minima and maxima of the solutions, as well as its derivative at roots. As corollaries, we complement a known asymptotic approximation of the Dirichlet nodal solution in terms of a tower of bubbles and present a similar formula for the Neumann problem.
In the second part or the talk we analyse the nonradial case with Neumann boundary conditions, namely the existence (and symmetry) of least energy solutions and their dependence on the exponent of the nonlinearity up to the Sobolev critical exponent, discussing also the slightly supercritical case.
The talk is based on joint works with Alberto Saldaña, Angela Pistoia and Massimo Grossi.

To join Hugo's talk, please use the following information:

Link to join Hugo's talk:
https://puc-rio.zoom.us/j/96415145570?pwd=[REDACTED]

Meeting ID: 964 1514 5570
Password: [REDACTED]

Yehuda Pinchover (Technion)
2020-08-13 13:00 - 14:00 | https://researchseminars.org/talk/RJWAPDE/10/
Title: How large can Hardy-weight be? (https://researchseminars.org/talk/RJWAPDE/10/)
by Yehuda Pinchover (Technion) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
In the first part of the talk we will discuss the existence of optimal Hardy-type inequalities with 'as large as possible' Hardy-weight for a general second-order elliptic operator defined on noncompact Riemannian manifolds and discrete graphs, while the second part of the talk will be devoted to a sharp answer to the question: "How large can Hardy-weight be?"

Noemi Wolanski (U. Buenos Aires)
2020-08-27 14:00 - 15:00 | https://researchseminars.org/talk/RJWAPDE/11/
Title: Fractional heat equations with memory in $\R^N$. Behavior in $L^p(\R^N)$ (https://researchseminars.org/talk/RJWAPDE/11/)
by Noemi Wolanski (U. Buenos Aires) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
To access the abstract of Prof. Noemi Wolanski's talk, please, use the following link:

https://www.dropbox.com/s/zb8tan4j0fk1clm/abstract_wolanski.pdf?dl=0

Maria J. Esteban (CEREMADE)
2020-09-17 13:00 - 14:00 | https://researchseminars.org/talk/RJWAPDE/12/
Title: Magnetic interpolation inequalities (https://researchseminars.org/talk/RJWAPDE/12/)
by Maria J. Esteban (CEREMADE) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
In this talk I will present various results concerning interpolation inequalities, best constants and information about the extremal functions involving Schrödinger magnetic operators in dimensions 1, 2 and 3. The particular, and physical interesting, cases of constant and of Aharonov-Bohm magnetic fields will be discussed in detail.

These works have been made in collaboration with D. Bonheure, J. Dolbeault, A. Laptev and M. Loss.

Mark Allen (Brigham Young U.)
2020-09-24 16:00 - 17:00 | https://researchseminars.org/talk/RJWAPDE/13/
Title: A boundary Harnack principle for equations with nonzero right hand side (https://researchseminars.org/talk/RJWAPDE/13/)
by Mark Allen (Brigham Young U.) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
The boundary Harnack principle allows one to compare the behavior of two harmonic solutions when they both vanish on the boundary. This principle has proven useful in PDE, harmonic analysis, and free boundary problems. In this talk I will review the principle and some of its applications. I will then introduce a new boundary Harnack principle for functions whose Laplacian has nonzero right hand sides. This new principle has applications to Hele-Shaw flow as well as free boundary problems. I will also show how we have recently proven this result for fully nonlinear equations as well as the p-Laplacian. This is joint work with Dennis Kriventsov and Henrik Shahgholian.

Francesco Maggi (UT Austin)
2020-10-09 16:00 - 17:00 | https://researchseminars.org/talk/RJWAPDE/14/
Title: Symmetry of Plateau Surfaces and the Moving Planes Method with Singularities (https://researchseminars.org/talk/RJWAPDE/14/)
by Francesco Maggi (UT Austin) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
By employing the method of moving planes in a novel way we extend some classical symmetry and rigidity results for smooth minimal surfaces to surfaces that have singularities of the sort typically observed in soap films. This is a joint work with Jacob Bernstein at Johns Hopkins University.

Alessio Porretta (Roma Tor Vergata)
2020-10-16 13:00 - 14:00 | https://researchseminars.org/talk/RJWAPDE/16/
Title: Mean field games: a bridge between Hamilton-Jacobi and transport-diffusion equations (https://researchseminars.org/talk/RJWAPDE/16/)
by Alessio Porretta (Roma Tor Vergata) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
Mean field game theory was developed since 2006 by J.‑M. Lasry and P.‑L. Lions in order to adapt the concept of Nash equilibria to differential games with infinitely many players. In this context, the value function of any small player depends on the distribution law of the dynamical state of the system. This model leads to systems of PDEs coupling Hamilton–Jacobi with Fokker–Planck (or continuity) equations. In this talk I will describe some features of mean field game systems and their connection with optimal control and optimal transport, pointing out the role played by weak solutions, renormalized formulations, convex analysis and adjoint methods.

Héctor Chang-Lara (CIMAT)
2020-11-13 16:00 - 17:00 | https://researchseminars.org/talk/RJWAPDE/17/
Title: Eikonal vs. Brownian: Regularity for the solution of an equation with gradient constraint. (https://researchseminars.org/talk/RJWAPDE/17/)
by Héctor Chang-Lara (CIMAT) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
Two controllers are in charge of steering a spaceship in some domain Omega. The first controller wants to spend as much time as possible exploring Omega while the second wants to get out of it as quickly as possible. The first controller determines minute by minute whether the ship is moving by a Brownian motion or with constant speed, in which case it is the second controller who chooses the direction. Under these instructions, determining the optimal strategies for each player leads us to solve the equation $\min (-\Delta u, |Du|) = 1$ which has several interesting characteristics. Among them is the presence of a free boundary which separates the regions where a Poisson or an Eikonal equation is satisfied. In a recent collaboration with Edgard Pimentel (PUC-Rio) we showed that the solutions are Lipschitz continuous and that $|Du|$ is continuous, even though the gradient is discontinuous in numerous examples. This problem is a simplification of interesting models in financial mathematics related with the optimal strategy for the payment of dividends from multiple insurances.

Edriss S. Titi (U. Cambridge, Texas A&M U. and Weizmann Institute of Science)
2020-11-27 14:00 - 15:00 | https://researchseminars.org/talk/RJWAPDE/18/
Title: Recent Advances Concerning the Navier-Stokes and Euler Equations (https://researchseminars.org/talk/RJWAPDE/18/)
by Edriss S. Titi (U. Cambridge, Texas A&M U. and Weizmann Institute of Science) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
In this talk I will discuss some recent progress concerning the Navier-Stokes and Euler equations of incompressible fluid. In particular, issues concerning the lack of uniqueness using the convex integration machinery and their physical relevance. Moreover, I will show the universality of the critical $1/3$ H\"older exponent, conjectured by Onsager for the preservation of energy in Euler equations, by extending the Onsager conjecture for the preservation of generalized entropy in general conservation laws. In addition, I will present a blow-up criterion for the 3D Euler equations based on a class of inviscid regularization for these equations and the effect of physical boundaries on the potential formation of singularity.

YanYan Li (Rutgers U.)
2020-12-03 15:00 - 16:00 | https://researchseminars.org/talk/RJWAPDE/19/
Title: Gradient estimates for the insulated conductivity problem (https://researchseminars.org/talk/RJWAPDE/19/)
by YanYan Li (Rutgers U.) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
In this talk, we discuss the insulated conductivity problem with multiple inclusions embedded in a bounded domain in $n$-dimensional Euclidean space. The gradient of a solution may blow up as two inclusions approach each other. The optimal blow up rate was known in dimension $n=2$. It was not known whether the established upper bound of the blow up rates in higher dimensions were optimal. We answer this question by improving the previously known upper bound of the blow up rates in dimension $n>2$. This is a joint work with Zhuolun Yang.

Irene Fonseca (Carnegie Mellon University)
2021-03-05 15:00 - 16:00 | https://researchseminars.org/talk/RJWAPDE/20/
Title: Phase Transitions in Heterogeneous Media: Equilibria and Geometric Flows (https://researchseminars.org/talk/RJWAPDE/20/)
by Irene Fonseca (Carnegie Mellon University) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
A variational model in the context of the gradient theory for fluid-fluid phase transitions with small scale heterogeneities is studied. In the case where the scale of the small homogeneities is of the same order of the scale governing the phase transition, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy.

The underlying gradient flow provides unconditional convergence results for an Allen-Cahn type bi-stable reaction diffusion equation in a periodic medium. The limiting dynamics are given by an analog for anisotropic mean curvature flow, of the formulation due to Ken Brakke. As an essential ingredient in the analysis, an explicit expression for the effective surface tension, which dictates the limiting anisotropic mean curvature, is obtained.

This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands), Adrian Hagerty, Cristina Popovici, Rustum Choksi (McGill), Jessica Lin (McGill), and Raghavendra Venkatraman (CMU).

Herbert Koch (Mathematisches Institut, U. Bonn)
2021-03-18 14:00 - 15:00 | https://researchseminars.org/talk/RJWAPDE/21/
Title: Sobolev regularity for the infinity Laplace equation (https://researchseminars.org/talk/RJWAPDE/21/)
by Herbert Koch (Mathematisches Institut, U. Bonn) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
The infinity Laplace equation is a very degenerate elliptic equation. I report on Sobolev regularity of $|Du|^a$, both for solutions to the homogeneous and the inhomogeneous equation in 2d. This is joint work with Yi Ru-Ya Zhang and Yuan Zhou.

Sylvia Serfaty (CIMS, NYU)
2021-04-09 14:00 - 15:00 | https://researchseminars.org/talk/RJWAPDE/22/
Title: Mean-Field limits of Coulomb-type dynamics (https://researchseminars.org/talk/RJWAPDE/22/)
by Sylvia Serfaty (CIMS, NYU) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
We consider a system of $N$ particles evolving according to the gradient flow of their Coulomb or Riesz interaction, or a similar conservative flow, and possible added random diffusion. By Riesz interaction, we mean inverse power $s$ of the distance. We present a convergence result as $N$ tends to infinity to the expected limiting mean field evolution equation. We also discuss the derivation of Vlasov-Poisson from newtonian dynamics in the monokinetic case, as well as related results for Ginzburg-Landau vortex dynamics.

Gabrielle S. Nornberg (Universidade de São Paulo (ICMC-USP))
2021-05-07 14:00 - 15:00 | https://researchseminars.org/talk/RJWAPDE/23/
Title: Qualitative properties of Lane-Emden type systems (https://researchseminars.org/talk/RJWAPDE/23/)
by Gabrielle S. Nornberg (Universidade de São Paulo (ICMC-USP)) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
In this talk we discuss some qualitative properties of Lane-Emden type systems and their applications to existence and nonexistence of solutions.

Andrzej Święch (Georgia Institute of Technology)
2021-04-22 14:00 - 15:00 | https://researchseminars.org/talk/RJWAPDE/24/
Title: Singular perturbations and optimal control of stochastic systems in infinite dimension (https://researchseminars.org/talk/RJWAPDE/24/)
by Andrzej Święch (Georgia Institute of Technology) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
We will discuss a stochastic optimal control problem for a two scale system driven by an infinite dimensional stochastic differential equation which consists of ''slow'' and ''fast'' components. We will consider a rather general case where the evolution is given by an abstract semilinear stochastic differential equation with nonlinear dependence on the controls. We will present a PDE approach to the problem based on the theory of viscosity solutions in Hilbert spaces. This approach allows to prove that as the speed of the fast component goes to infinity, the value functions of the optimal control problems converge to the viscosity solution of a reduced effective equation. Our results generalize to the infinite dimensional case the finite dimensional results of Alvarez and Bardi and complement recent results in Hilbert spaces obtained by Guatteri and Tessitore.

Isabella Ianni (Sapienza Università di Roma)
2021-05-21 14:00 - 15:00 | https://researchseminars.org/talk/RJWAPDE/25/
Title: Radial and non-radial solutions for the Lane-Emden problem in the disk (https://researchseminars.org/talk/RJWAPDE/25/)
by Isabella Ianni (Sapienza Università di Roma) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
We discuss sharp asymptotic analysis results on radial solutions of the Lane-Emden problem in the disk, which in particular lead to a Morse index formula.
As a consequence we can prove the existence of new unexpected solutions. The talk is based on results mainly contained in [ADI], [IS], [GI] and [DIP].

[ADI] A. Amadori, F. De Marchis, I. Ianni, Morse index computation for radial solutions of the Henon problem in the disk, preprint

[IS] I. Ianni, A. Saldana, Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems, preprint

[GI] I. Ianni, F. Gladiali, Quasi-radial solutions for the Lane-Emden problem in the ball, NoDEA 27 (2) (2020)

[DIP] F. De Marchis, I. Ianni, F. Pacella, Exact Morse index computation for nodal radial solutions of Lane-Emden problems, Mathematische Annalen 367 (2017)

Diego R. Moreira (Universidade Federal do Ceará)
2021-06-18 14:00 - 15:00 | https://researchseminars.org/talk/RJWAPDE/26/
Title: Up to the boundary gradient estimates for solutions to nonlinear free boundary problems with unbounded measurable ingredients (https://researchseminars.org/talk/RJWAPDE/26/)
by Diego R. Moreira (Universidade Federal do Ceará) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
In this talk, we discuss recent advances on up to the boundary gradient estimates for viscosity solutions of free boundary problems governed by fully nonlinear and quasilinear equations with unbounded coefficients. We present the new Inhomogeneous Pucci Barriers as new elements for the proof. If time permits, we discuss some of the main steps in the proof, namely, the trace estimate of the solution on the points of the fixed boundary that projects nontangentially over the free boundary. These methods are inspired by some ideas of Carlos Kenig in Harmonic Analysis.

Yannick Sire (Johns Hopkins U.)
2021-07-15 14:00 - 15:00 | https://researchseminars.org/talk/RJWAPDE/27/
Title: Blow-up solutions via parabolic gluing (https://researchseminars.org/talk/RJWAPDE/27/)
by Yannick Sire (Johns Hopkins U.) as part of Rio de Janeiro webinar on analysis and partial differential equations

Abstract
We will present some recent results on the construction of blow-up solutions for critical parabolic problems of geometric flavor. Initiated in the recent years, the inner/outer parabolic gluing is a very versatile parabolic version of the well-known Lyapunov-Schmidt reduction in elliptic PDE theory. The method allows to prove rigorously some formal matching asymptotics (if any available) for several PDEs arising in porous media, geometric flows, etc….I will give an overview of the strategy and will present several applications to (variations of) the harmonic map flow, Yamabe flow and Yang-Mills flow. I will also present some open questions.
