Abstract : Nonconvex variational problems regularized by higher order terms have been used to describe many physical systems, including, for example, martensitic phase transformation, micromagnetics, and the Ginzburg–Landau model of nucleation. These problems exhibit microstructure formation, as the coefficient of the higher order term tends to zero.  They can be naturally embedded in a whole family of problems of the form: minimize E(u)= S(u)+N(u) over an admissible class of functions u taking only two values, say -1 and 1, with a nonlocal interaction N favoring small-scale phase oscillations, while the interfacial energy S penalizes them. In this talk I will report on joint work with Tobias Ried, in which we establish scaling laws for the global and local energies of minimizers of an energy functional that naturally arises when analyzing the behavior of uniaxial ferromagnets using the Landau-Lifschitz model. These scaling laws strongly suggest that minimizers have a self-similar behavior.

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/96642349167?pwd=MkRVbWxzOFBUUXlCTWFicW0reWZ6dz09

Abstract : In an attempt to find a more intuitive proof of the Prime Number Theorem, Lord Cherwell derived, through heuristic arguments, the equation:
f'(x) = -(f(x) f(\sqrt{x})/(2x),
where f(x) represents the “density of primes at x”. Through a simple change of variables, the differential equation can be rewritten as the following delay differential equation:
h'(u) = -(ln 2)(h(u) + 1)h(u – 1),
which marks the first appearance of this type of equation in number theory.
In this talk, we present other families of differential equations, both with delay and advance, related to various problems in number theory. Regarding these equations, we will explore some known results and emphasize the importance of studying the asymptotic behavior of their solutions. With this in mind, we will provide global bounds for the solutions using the theory of regularly varying functions.

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/96642349167?pwd=MkRVbWxzOFBUUXlCTWFicW0reWZ6dz09

Abstract : Let $(M^n, [\hat{g}])$ be the conformal infinity of an asymptotically hyperbolic Einstein (AHE) manifold $(X^{n+1},g^+).$ We will take the scattering operator associated to the AHE filling in as the fractional conformal Laplacian. Equipped with fractional conformal Laplacians defined via the AHE manifold, we can define a fractional Yamabe problem, looking for a conformal metric of $(M^n,[\hat{g}])$ which has constant fractional scalar curvature. We will present some new developments on the fractional Yamabe problem assuming an AHE filling in.

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/96642349167?pwd=MkRVbWxzOFBUUXlCTWFicW0reWZ6dz09

Abstract : We obtain the existence and uniqueness of a classical solution to a Cauchy problem for a quasilinear parabolic partial integro-differential equation (PIDE) which arises due to its association with a forward-backward stochastic differential equation (FBSDEs) with jumps. More specifically, we embed the associated PIDE into a suitable class of non-local quasilinear parabolic PDEs which allows us to extend the methodology of Ladyzhenskaya et al (O. Ladyzenskaja, V. Solonnikov, N.N. Uralceva. Linear and Quasi-Linear Equations of Parabolic Type, 1968) to non-local PDEs of this class. The maximum principle, gradient estimate, and Hölder norm estimates are obtained in order to show the existence of a solution to an initial-boundary value problem by means of the Leray-Schauder theorem. The existence of a classical solution to the Cauchy problem is then obtained by employing the diagonalization argument.

Venue: DIM seminar room, Beauchef 851, 4th floor.

Zoom: https://uchile.zoom.us/j/96642349167?pwd=MkRVbWxzOFBUUXlCTWFicW0reWZ6dz09

Abstract : In this talk, we examine a concentration phenomenon for solutions to the constant Q-curvature equation, a critical fourth-order equation on a closed Riemannian manifold. The challenge of finding constant Q-curvature metrics is closely linked to the Yamabe problem and arises from the goal of identifying optimal metrics for a given compact, boundaryless manifold.
In this work, we address the problem on a product Riemannian manifold. We will start by briefly introducing the concept of Q-curvature and then outline the main ideas for finding solutions that concentrate around specific points on the manifold, which are of particular geometric significance.

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/96642349167?pwd=MkRVbWxzOFBUUXlCTWFicW0reWZ6dz09

Abstract : In this talk I will first  present a 10 years old result about the asymptotic stability of the kink in the classical φ^4 model under the assumption of oddness of the initial perturbations. I will explain how the problem can be decomposed into radiation and internal modes and how the components  can be controlled through virial estimates. This result depends on some numerical approximations and its proof can be viewed as computer assisted. Recently, we were able to generalize the asymptotic stability result to one dimensional scalar field models with one internal mode. I will show how using the Darboux factorization of the linearized operator around the kink one can avoid  numerical approximations.

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/96642349167?pwd=MkRVbWxzOFBUUXlCTWFicW0reWZ6dz09

Abstract : In this talk, we consider a compact Riemannian surface (M,g) with nonempty boundary and negative Euler characteristic. Given two smooth non-constant functions f in M and h in the boundary of M with max f = max h = 0, under a suitable condition on the maximum points of f and h, we prove that for sufficiently small positive constants λ and  μ, there exist at least two distinct conformal metrics g_{λ,μ}=e^{2u_{μ,λ}}g and g^{λ,μ}=e^{2u^{μ,λ}}g with prescribed sign-changing Gaussian and geodesic curvature equal to f+μ and h+λ, respectively. Additionally, we employ the method Borer et al. (2015) used to study the blowing up behavior of the large solution u^{μ,λ} when μ↓0 and λ↓0. This is joint work with R. Caju (Universidad de Chile) and T. Cruz (UFAL).

Venue: DIM seminar room, Beauchef 851, 5th floor.

Zoom: https://uchile.zoom.us/j/96642349167?pwd=MkRVbWxzOFBUUXlCTWFicW0reWZ6dz09

Abstract : The Vlasov-Poisson system describes a macroscopic number of particles with their mutual gravitational attraction in a mean-field approximation. Its steady-state solutions are known as “polytropes” and a popular model for galaxies in the astronomy literature. In many cases, they are known to be neutrally stable, but the question of asymptotic stability is widely open. The goal of this talk is to present some results on the linearized equation around a steady state.
This is based on joint work with Matías Moreno and Paola Rioseco.

Venue: DIM seminar room, Beauchef 851, 5th floor.

Abstract : We prove Ambrosetti-Prodi type results for periodic solutions of some one-dimensional nonlinear problems that can have drift term whose principal operator is the fractional Laplacian of order s ∈ (0, 1). We establish conditions for the existence and nonexistence of solutions of those problems. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. We also obtain a priori bounds in order to get multiplicity results. We also prove that the solutions are C1,α under some regularity assumptions in the nonlinearities, that is, the solutions of the mentioned equations are classical.

Venue: DIM seminar room, Beauchef 851, 4th floor.

Abstract : In this talk, we will discuss  different versions of the classical Hopf’s boundary lemma in the setting of the fractional $p-$Laplacian, for $p \geq 2$. We will start with a  Hopf’s lemma   based on comparison principles and for  constant-sign potentials. Afterwards, we will present a Hopf’s result for sign-changing potentials describing the behavior of the fractional normal derivative of solutions around boundary points. As we wiil see, the main contribution here is that we do not need to impose a global condition on the sign of the solution. Applications of the main results to boundary point lemmas and a discussion of  non-local non-linear overdetermined problems will also be discussed.

This is a joint work with Dr. Ariel Salort (UBA).

Venue: DIM seminar room, Beauchef 851, 4th floor.