ENG 
ESP Session S18 - Recent progress in non-linear PDEs and their applications
Talks
Monday, July 12, 12:00 ~ 12:50 UTC-3
On certain local and nonlocal $(p,q)$ systems in $\mathbb R^N$\\with critical and Hardy terms
Patrizia Pucci
Università degli Studi di Perugia, Itay - This email address is being protected from spambots. You need JavaScript enabled to view it.
Motivated by important applications in nonlinear elasticity, recently great attention has been devoted to the study of local and nonlocal nonlinear problems with $(p, q)$ growth conditions.
We present existence results for a class of parametric $(p,q)$ systems with critical and Hardy terms in $\mathbb R^N$, provided that the parameter is sufficiently large. The interest is twofold: on one hand, the simultaneous presence of critical terms, Hardy terms and the fact that the systems are studied in the whole $\mathbb R^N$ cause, roughly speaking, a {\it triple loss of compactness} which dramatically affects the applicability of standard variational methods. On the other hand, since we treat both the local and the nonlocal version of the system, the comparison of the results obtained for fractional Laplacian operators with their local counterpart is noteworthy.
Joint work with Letizia Temperini (Università degli Studi di Firenze).
Monday, July 12, 13:00 ~ 13:50 UTC-3
Blow-up analysis of a curvature prescription problem in the disk
María Medina
Universidad Autónoma de Madrid, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
We will establish necessary conditions on the blow-up points of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures, where a non local restriction will appear. Conversely, given a point satisfying these conditions, we will construct an explicit family of approximating solutions that explode at such a point. These results are contained in several works in collaboration with A. Jevnikar, R. López-Soriano and D. Ruiz, and with L. Battaglia and A. Pistoia.
Monday, July 12, 14:00 ~ 14:50 UTC-3
Lipschitz continuity of nonnegative minimizers of functional of Bernoulli type with nonstandard growth
Noemi Wolanski
IMAS-UBA-CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will report on research done in collaboration with Claudia Lederman from the University of Buenos Aires on the Lipschitz continuity of nonnegative minimizers to functionals \[J(u)=\int_\Omega F(x,u(x),\nabla u(x))+\lambda(x)\chi_{\{u>0\}}\,dx.\]
Here $F(x,s,\eta)$ is a function of $p(x)-$type growth with $p$ Lipschitz continuous, and $0<\lambda_1\le \lambda(x)\le\lambda_2<\infty$.
Some examples are $F(x,s,\eta)=a(x,s)|\eta|^{p(x)}+b(x) |s|^{p^*(x)}$, or $F(x,s,\eta)=G(|\eta|^{p(x)})+b(x) |s|^{p^*(x)}$ with $G$ strictly convex, under suitable assumptions.
Of independent interest are the results on existence, $L^\infty-$estimates, comparison principles and maximum principles for the associated equation \[\mbox{div}\big(A(x,u(x),\nabla u(x))\big)=B(x,u(x),\nabla u(x)),\] where $A=\nabla _\eta F$ and $B=F_s$.
Joint work with Claudia Lederman (Universidad de Buenos Aires, Argentina).
Monday, July 12, 15:00 ~ 15:50 UTC-3
a nonlocal version of the inverse problem of Donsker and Varadhan
Erwin Topp
Universidad de Santiago de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.
In their seminal paper of 1976, M.D. Donsker and S.R.S. Varadhan addressed the following ``inverse problem": let consider two linear, second-order, uniformly elliptic operators $L_1, L_2$ with the form \[ L_i \phi = Div(A_i(x) D \phi) + b_i(x) \cdot D\phi, \quad i =1,2. \]
If for every domain $\Omega$ and every smooth potential $V$, the operators $L_1 + V$ and $L_2 + V$ have the same principal eigenvalue in $\Omega$, then the diffusions are equal ($A_1 = A_2$), and either $L_1 \phi = L_2 (u \phi)/u$ for some $L_2$-harmonic function $u$, or $L_1 \phi = L_2^* (u \phi)/u$ for some $L_2^*$-harmonic function $u$.
In this talk we report a nonlocal a version of this problem, where both the diffusion and transport terms defining the involved operators have a fractional nature. We prove a similar conjugacy phenomena among operators having the same principal eigenvalues, by means of a minmax characterization for them, and developing new ideas to overcome the difficulties posed by the non locality.
Joint work with Gonzalo Dávila (UTFSM, Chile).
Monday, July 12, 17:00 ~ 17:50 UTC-3
Unbounded attractors for dynamical systems and applications in PDEs
Juliana Fernandes
Universidade Federal do Rio de Janeiro, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We present a general theory for unbounded attractors for semigroups in complete metric spaces, presenting sufficient conditions to ensure their existence. For semigroups, when a Lyapunov function exists, we also present the characterization of the maximal attractor as the unstable set of the critical elements (not necessarily fixed points). Concrete equations will be presented to illustrate the theory.
Joint work with Matheus Bortolan (Universidade Federal de Santa Catarina).
Monday, July 12, 18:00 ~ 18:50 UTC-3
Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds
Mariel Saez Trumper
Pontificia Universidad Católica, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk I will discuss bounds for the first eigenvalue of the Paneitz operator $P$ and its associated third-order boundary operator $B^3$ on four-manifolds. We restrict to orientable, simply connected, locally confomally flat manifolds that have at most two umbilic boundary components. The proof is based on showing that under the hypotheses of the main theorems, the considered manifolds are confomally equivalent to canonical models. The fact that $P$ and $B^3$ are conformal in four dimensions is key in the proof.
Joint work with Maria del Mar Gonzalez (Universidad Autonoma, Madrid, España).
Monday, July 12, 19:00 ~ 19:50 UTC-3
Efficiency functionals for the Lévy flight foraging hypothesis
Enrico Valdinoci
University of Western Australia, Australia - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider a forager diffusing via a fractional heat equation and we introduce several efficiency functionals whose optimality is discussed in relation to the Lévy exponent of the evolution equation. Several biological scenarios, such as a target close to the forager, a sparse environment, a target located away from the forager and two targets are specifically taken into account. The optimal strategies of each of these configurations are here analyzed explicitly also with the aid of some special functions of classical flavor and the results are confronted with the existing paradigms of the Lévy foraging hypothesis. Interestingly, one discovers bifurcation phenomena in which a sudden switch occurs between an optimal (but somehow unreliable) Lévy foraging pattern of inverse square law type and a less ideal (but somehow more secure) classical Brownian motion strategy. Additionally, optimal foraging strategies can be detected in the vicinity of the Brownian one even in cases in which the Brownian one is pessimizing an efficiency functional.
Joint work with Serena Dipierro and Giovanni Giacomin.
Tuesday, July 20, 16:00 ~ 16:50 UTC-3
Positive solutions to a nonlinear Choquard equation with symmetry
Liliane Maia
University of Brasília (UnB), Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We will present some recent results on existence of positive solutions for a class of Choquard equations with potential decaying at a positive limit at infinity. The solutions found are invariant under the action of a closed subgroup of linear isometries of $\mathbb{R}^N$, provided the potential is invariant under the group action, and satisfies suitable decay assumptions. We investigate superlinear, linear and sublinear nonlinearities, and we take into account an arbitrary number of bumps. Our results in particular include the physical case.
Joint work with Benedetta Pellacci (Dipartimento di Matematica e Fisica,, Universit\`a della Campania ``Luigi Vanvitelli'', Italy), and, Delia Schiera (Dipartimento di Matematica e Fisica, and Universit\`a della Campania ``Luigi Vanvitelli'', Italy).
Tuesday, July 20, 17:00 ~ 17:50 UTC-3
Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems
Julian Fernandez Bonder
University of Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this paper we analyze the asymptotic behavior of several fractional eigenvalue problems by means of Gamma-convergence methods. This method allows us to treat different eigenvalue problems under a unified framework. We are able to recover some known results for the behavior of the eigenvalues of the $p-$fractional laplacian when the fractional parameter $s$ goes to 1, and to extend some known results for the behavior of the same eigenvalue problem when $p$ goes to $\infty$. Finally we analyze other eigenvalue problems not previously covered in the literature. This is a joint work with A. Silva and J. Spedaletti from UNSL-Argentina.
Tuesday, July 20, 18:00 ~ 18:50 UTC-3
The Vázquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients
Boyan Sirakov
PUC-Rio, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We develop a new, unified approach to the following two classical questions on elliptic PDE: (i) the strong maximum principle for equations with non-Lipschitz nonlinearities, and (ii) the at most exponential decay of solutions in the whole space or exterior domains. Our results apply to divergence and nondivergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish.
Joint work with Philippe Souplet (Université Sorbonne Paris Nord, France).
Tuesday, July 20, 19:00 ~ 19:50 UTC-3
Regularity estimates for the Boltzmann equation without cutoff
Luis Silvestre
University of Chicago, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We study the regularization effect of the inhomogeneous Boltzmann equation without cutoff. We obtain a priori estimates for all derivatives of the solution depending only on bounds of its hydrodynamic quantities: mass density, energy density and entropy density. As a consequence, a classical solution to the equation may fail to exist after a certain time T only if at least one of these hydrodynamic quantities blows up. Our analysis applies to the case of moderately soft and hard potentials. We use methods that originated in the study of nonlocal elliptic and parabolic equations: a weak Harnack inequality in the style of De Giorgi, and a Schauder-type estimate, for integro-differential equations.
Joint work with Cyril Imbert (CNRS), Clement Mouhot (Cambridge) and Jamil Chaker (Bielefeld).
Posters
Uniquennes continuation for the Cauchy problem of nonlinear interactions of Schrondinger type
Isnaldo Isaac Barbosa
UFAL, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
This work concerns with the study of the Uniqueness Continuation problem of Cauchy's smooth solutions of a system of equations which arises in the context of nonlinear optics problems. The main results are prove that if $(u, v)$ are sufficiently smooth solutions to the Cauchy problem associated with a system of Schrodinger equations with quadratic nonlinear interactions such that $a,\ b\in \mathbb{R}$ with supp $u(t_j) \subset (a,\ \infty)$ e supp $v(t_j) \subset (b,\ \infty)$ for $j=1$ or $2$ ($t_1\neq t_2$) then $u\equiv v\equiv 0$.}
Basically, we will study the following mathematical model:
$$ \begin{cases} i\partial_{t} u(x,t)+p\partial^2_{x} u(x,t) -\theta u(x,t)+ \bar{u}(x,t)v(x,t)=0, \ \ x\in \mathbb{R},\; t\ge 0,\\ i\sigma\partial_{t} v(x,t)+q\partial^2_{x} v(x,t) -\alpha v(x,t)+\tfrac{a}{2}u^2(x,t)=0,\ \ \\ u(x,0)=u_{0}(x),\quad v(x,0)=v_{0}(x), \end{cases} $$ where $u$ and $v$ are functions that assume complex values, $\alpha$, $\theta$ and $a: =1/\sigma$ are real numbers that play a physical parametrical role of the system, for which $\sigma >0$ e $p, \ q\ =\pm1$.
The main result of this work is the following:
\textbf{Theorem}
Let $(u,v)\in C([0,T]: H^3\times H^3)\cap C^1([0,T]:L^2(\mathbb{R})\times L^2(\mathbb{R}))$ be a strong solution of above system. Suppose that there exist $a,b \in \mathbb{R}$ with supp $u(0)$, supp $u(T)\subset (a,\infty)$ and supp $v(0)$, supp $v(T)\subset (b,\infty)$, then $u\equiv v\equiv 0.$ in $[0,T]\times \mathbb{R}$.
\tectbf{Sketch of the proof:}
The proof follows the ideas of the works [1], [2], [3] and [4].
This result is part of a more complete work aimed at establishing uniqueness continuation results for systems of dispersive equations whose Schrodinger equation is one of the coupled equations.
For instance: Benney System
$$ \left\{\begin{array}{ll}{i \partial_t u+\partial_x^2 u=u v+\beta|u|^{2} u,} & \quad t \in \mathbb{R}, \quad x \in \mathbb{R} \\ {v_{t}=\partial_x\left(|u|^{2}\right)} & {} \\ {u(x, 0)=u_{0}(x),} & {v(x, 0)=v_{0}(x)}\end{array}\right. , $$
and Schrodinger-Debye System $$\left\{\begin{array}{l}{i \partial_t u+\frac{1}{2} \partial_{x}^{2} u=u v, \quad t \in \mathbb{R}, \quad x \in \mathbb{R}} \\ {\sigma \partial_{t} v+v=\epsilon|u|^{2}} \\ {u(x, 0)=u_{0}(x), \quad v(x, 0)=v_0(x)}\end{array}\right. .$$
[1] {\sc Barbosa, I.I.} { The Cauchy problem for nonlinear quadratic interactions of the Schr{\"o}dinger type in one dimensional space}, {\it Journal of Mathematical Physics}, {\bf 59}, {7},{2018}
[2] {\sc Angulo, J. and Linares, F.} {Periodic pulses of coupled nonlinear Schr\"odinger equations in optics.}, {\it Indiana University Mathematics Journal}, 56(2):847?878. 2007
[3] {\sc Kenig, C.E., Ponce, G. and Vega, L.} - On unique continuation for nonlinear Schrodinger equations. {\it Comm. Pure Appl. Math.,}, {\bf 56}, 1247-1262, 2002.
[4] {\sc Urrea, J. J.} - On the support of solutions to the NLS-KDV system. {\it Differential and Integral Equations}, {\bf 25}, 611--618, 2012.
Fully non-linear singularly perturbed models with non-homogeneous degeneracy
Elzon Cezar Bezerra Junior
Universidade Federal do Ceará , Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.
This work is devoted to study non-variational, non-linear singularly perturbed elliptic models enjoying a double degeneracy character with prescribed boundary value in a domain. In its simplest form, for each $\varepsilon>0$ fixed, we seek a non-negative function $u^{\epsilon}$ satisfying $$ \left\{ \begin{array}{rclcl} \left[|\nabla u^{\varepsilon}|^p + \mathfrak{a}(x)|\nabla u^{\varepsilon}|^q \right] \Delta u^{\varepsilon} & = & \zeta_{\varepsilon}(x, u^{\varepsilon}) & \mbox{in} & \Omega,\\ u^{\varepsilon}(x) & = & g(x) & \mbox{on} & \partial \Omega, \end{array} \right. $$ in the viscosity sense for suitable data $p, q \in (0, \infty)$, $\mathfrak{a}$, $g$, where $\zeta_{\varepsilon}$ one behaves singularly of order $\mbox{O} \left(\epsilon^{-1} \right)$ near $\epsilon$-level surfaces. In such a context, we establish existence of certain solutions. We also prove that solutions are locally (uniformly) Lipschitz continuous, and they grow in a linear fashion. Moreover, solutions and their free boundaries possess a sort of measure-theoretic and weak geometric properties. Particularly, for a restricted class of non-linearities, we prove the finiteness of the $(N-1)$-dimensional Hausdorff measure of level sets. At the end, we address a complete and in-deep analysis concerning the asymptotic limit as $\varepsilon \to 0^{+}$, which is related to one-phase solutions of inhomogeneous non-linear free boundary problems in flame propagation and combustion theory.
Keywords: Singular perturbation methods, doubly degenerate fully non-linear operators, geometric regularity theory.
Joint work with João Vitor da Silva and Gleydson Chaves Ricarte.
Improved regularity for a time-dependent Isaacs equation
Giane Casari Rampasso
University of Campinas, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
The purpose of this work is to discuss a regularity theory for viscosity solutions to a parabolic problem driven by the Isaacs operator. Under distinct smallness regime imposed on the coefficients, our findings are three-fold; first, we produce estimates in Sobolev spaces. It includes operators with dependence on the gradient. Then we examine the regularity in Hölder spaces. Here we deal with the boderline case and, if we refine the smallness regime, estimates in $\mathcal{C}^{2+\gamma,\frac{2+\gamma}{2}}$ are produced. This is done through geometric and approximation techniques with preliminary compactness and localization arguments.
Joint work with Pêdra D. S. Andrade (Centro de Investigación en Matemáticas, Mexico) and Makson S. Santos (Centro de Investigación en Matemáticas, Mexico).
Regularization of a chemoattraction model with consumption
André Luiz Corrêa Vianna Filho
Federal University of Paraná (UFPR), Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
Chemotaxis is the movement of cells in response to the concentration gradient of a chemical substance. It plays an important role in wound healing, migration of immune cells, migration of bacteria and other relevant situations which make of chemotaxis a matter of very practical interest. Cells can be attracted (chemoattraction) or repeled (chemorepulsion) by the by the greater concentration of this chemical substance. In this work we investigate a chemoattraction model, which we call the original model, with consumption, in non-smooth bounded domains of $R^N$ ($N = 1, 2, 3$). The non-linear term related to the chemotaxis phenomenon introduces many difficulties to the analysis of the original model, mainly for its numerical simulation. In order to overcome these difficulties, we propose and analyze a regularized model, establishing results on existence, uniqueness, regularity’s properties and convergence of its solutions towards the original model.
Joint work with Francisco Guillén-González (Universidad de Sevilla, Spain) and Pedro Danizete Damázio (Federal University of Paraná, Brazil).
Nonlocal PDEs in Ecology
Erin Ellefsen
University of Colorado Boulder, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Nonlocal models can be very useful to describe some phenomena in Ecology. However, they also pose both analytical and computational challenges. We investigate territory development of meerkats by studying a system of nonlocal continuum equations, and we take advantage of the gradient-flow structure of the system in order to find equilibrium solutions. We first find explicit equilibrium solutions of the model. We then perform a long-wave approximation of this system to investigate a local approximation. We compare the equilibrium solutions of the local and nonlocal models to determine if the local model is an appropriate approximation for future work. Finally, we explore finding equilibrium solutions of the system using spectral methods.
Joint work with Nancy Rodriguez (University of Colorado Boulder).
A GAME THEORETICAL APPROACH FOR AN ELLIPTIC SYSTEM WITH TWO DIFFERENT OPERATORS (THE LAPLACIAN AND THE INFINITY LAPLACIAN)
Alfredo Miranda
Universidad de Bs. As., Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
Abstract. In this paper we analyze the behaviour of value functions of some Tug-of-War / random walks games as a parameter that controls the length of the steps goes to zero. We show that these value functions converge uniformly to a viscosity solution of an elliptic system governed by two different operators (the Laplacian and the infinity Laplacian).
Joint work with Julio D. Rossi (Universidad de Bs. As.).
Improved Regularity for a Class of Nonlocal $\mathcal{L}_0$-Elliptic Equations with an Asymptotic Property
Aelson Sobral
Federal University of Paraíba(UFPB), Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We establish improved $\mathcal{C}^{1,(\sigma-1)^-}$ regularity estimates to viscosity solutions of a class of nonlocal $\mathcal{L}_0(\sigma)$-elliptic equations with an asymptotic property. We introduce the notion of recession operator to the nonlocal context, discuss the main features and compare with its classical version. The role of this class of equations is detailed on some examples and an improved regularity result.
Joint work with Disson dos Prazeres(Federal University of Sergipe(UFS), Brazil).
A Mathematical Model of Wealth Distribution Through an Amenities-Based Theory
Lyndsey Wong
University of Colorado Boulder, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Gentrification refers to the influx of income into a community leading to the improvement of an area through renovation or the introduction of local amenities. This is usually accompanied by an increase in the cost of living, which displaces lower income populations. To better understand this problem, we will introduce a model for the dynamics of wealth based on amenities. In order to find when we have inhomogeneous solutions to this model, we present two approaches. The first is to perform a linear stability analysis in order to find when small perturbations of constant equilibrium solutions become unstable. The second is to prove the existence of a global bifurcation of these solutions from the constant equilibrium solution.