# Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions

### Date: Thursday, December 6, 2022 at 12 Santiago time

Abstract:  I will present some recent results obtained in collaboration with A. Pistoia and H. Tavares for a Lane-Emden system on a bounded regular domain with Neumann boundary conditions and critical nonlinearities. We show that, under suitable conditions on the exponents in the nonlinearities, least-energy (sign-changing) solutions exist. In the proof we exploit a dual variational formulation which allows to deal with the strong indefinite character of the problem, and we establish a compactness condition which is based on a new Cherrier type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. I will also briefly present existence of least-energy solutions for the particular case in which the system reduces to a biharmonic equation, and some symmetry results in the case the domain is an annulus.

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Gabrielle Nornberg

# Non-existence results for an eigenvalue problem involving Lipschitzian nonlinearities with non-positive primitive and applications

### Date: Thursday, December 1, 2022 at 14:15 Santiago time

Abstract: We discuss here existence and non existence results for nonlinear eigenvalues problems like

$$\Delta u = \lambda \sin u, \lambda\geq 0$$

in a bounded domain of $$\mathbb{R}^D$$ with homogeneous Dirichlet conditions.
We then infer some applications for the long time behavior of solutions to sine-Gordon equations. This work is a joint work with B. Ricceri (Catane, Italy).

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Claudio Muñoz

# Nonlocal Aggregation-Difusion Equations: entropies, gradient flows, phase transitions and applications

### Date: Tuesday, November 29, 2022 at 12 Santiago time

Abstract: This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics and nancial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear difusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential. The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear di usion, one can work in the L2 framework, nonlinear difusion needs the stronger Linfty topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient ow structure of these problems. The theoretical analysis of the asymptotic stability of the diferent branches of solutions is a challenging open problem.

This overview talk is based on several works in collaboration with R. Bailo, A. Barbaro, J. A. Canizo, X. Chen, P. Degond, R. Gvalani, J. Hu, G. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, and L. Zhang.

This research has been funded by EPSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Claudio Muñoz

# On Space-Time Formulations and Boundary Integral Equations for the Wave Equation

### Date: Tuesday, November 24, 2022 at 12 Santiago time

Abstract: Space-time discretization methods are becoming increasingly popular, since they allow adaptivity in space and time simultaneously, and can use parallel iterative solution strategies for time-dependent problems. However, in order to exploit these advantages, one needs to have a complete numerical analysis of the corresponding Galerkin methods.

In this talk, we consider the wave equation with on a Lipschitz bounded domain, with either Dirichlet or Neumann boundary conditions, and with zero initial conditions. The first step to build the required numerical analysis was to show new existence and uniqueness results for the weak formulations of these initial boundary value problems. With this, we are able to propose a new approach to boundary integral equations for the wave equation that allows us to prove that the associated boundary integral operators are continuous and satisfy inf-sup conditions in trace spaces of the same regularity, which are closely related to standard energy spaces with the expected regularity in space and time. This feature is crucial from a numerical perspective, as it provides the foundations to derive sharper error estimates and paves the way to devise stale and efficient adaptive space-time boundary element methods.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Claudio Muñoz

# From sign-changing solutions of the Yamabe equation to critical competitive systems

### Date: Tuesday, November 17, 2022 at 14:15 Santiago time

Abstract: In this talk we will analyze the existence and the structure of different sign-changing solutions to the Yamabe equation in the whole space and we will use them to find positive solutions to critical competitive systems in dimension 4.

Venue: Online via Zoom / Sala de seminarios DIM, 5th floor, Beauchef 851
Chair: Claudio Muñoz

# Existence and stability of steady states of Flat Vlasov Poisson with a central mass density

### Date: Tuesday, November 15, 2022 at 12 Santiago time

Abstract: El Sistema de Vlasov-Poisson Plano es usado para modelar objetos astronómicos extremadamente planos, a través de la evolución de una distribución de partículas en el plano de fase, las cuales autointeractúan a través del campo gravitacional inducido por ellas mismas.

En este trabajo se estudia la construcción de estados estacionarios del Sistema de Vlasov-Poisson Plano en presencia de un potencial gravitacional externo, inducido por una densidad de masa fija. La construcción de dichos estados se realiza a través de la minimización de los funcionales de Energía-Casimir, los cuales permiten probar resultados de estabilidad no lineal para dichos estados estacionarios.

Venue: Online via Zoom / Sala multimedia CMM, 6th floor, Beauchef 851
Chair: Paola Rioseco

# Simulations of black hole binaries and their gravitational waves

### Date: Tuesday, November 22, 2022 at 12 Santiago time

Abstract: Our recently acquired ability to detect gravitational waves has expanded our senses and our possibilities of inquiring about the Universe. As a new era of gravitational wave detections rapidly unfolds, the importance of having accurate models for their signals becomes increasingly important. In this context, we will discuss numerical simulations of black hole binaries. In particular, we will focus on how they are made, what they help us achieve, and what are the current challenges in this research area.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Paola Rioseco

# Recent developments in Fourier interpolation theory

### Date: Tuesday, November 8, 2022 at 12 Santiago time

Abstract: In this talk we will discuss some problems related to the notion of Fourier interpolation:  formulas where one can recover functions from its values over a certain discrete set, and the values of its Fourier transform over a dual discrete set. These formulas arrive naturally in many situations, and we will mention a few related to certain kinds of uncertainty principles, and the theory of sphere packing.

This talk is meant for a broad audience with basic knowledge in analysis.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851

Chair: Rayssa Caju

# Qualitative Properties for a class of fully nonlinear elliptic equations

### Date: Tuesday, October 27, 2022 at 2 p.m. Santiago time

Abstract: In joint work with F. Pacella, we study the existence and asymptotic behavior of radial positive solutions of some fully nonlinear equations involving Pucci’s extremal operators in dimension two and higher. In particular we prove the existence of a positive solution of a fully nonlinear version of the Liouville equation in the plane. Moreover, for the (negative) Pucci P^- operator, we show the existence of a critical exponent and give bounds for it. The same technique is then applied in higher dimensions to improve the previously known bounds.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Gabrielle Nornberg

# Existence and multiplicity of positive solutions to the scalar-field equation on large annuli in the three-dimensional sphere

### Date: Tuesday, October 25, 2022 at 12 Santiago time

Abstract: In this talk, we consider the Dirichlet problem for the scalar-field equation in a large annulus in the three-dimensional sphere. We obtain the existence, uniqueness, and multiplicity results of the positive solutions depending only on the latitude. This is joint work with Noaki Shioji (Yokohama National University) and Kohtaro Watanabe (National Defense Academy).

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Hanne Van Den Bosch