# Modeling chemotaxis with a nonlinear Schrödinger equation: solitary waves

### Date: May 31, 2022 at 12 Santiago time

Abstract: In this talk I will show how chemotaxis can be modeled by using a nonlinear Schrödinger equation with  well-known quantum dissipative mechanisms. This relation will allow us to find explicit new solitary wave solutions.

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

# Fractional Sobolev regularity for fully nonlinear elliptic equations

### Date: May 24, 2022 at 12 Santiago time

Abstract: We study high-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations, in the presence of unbounded source terms. Our techniques are based on touching the solution with C1,α cone-like functions to produce a decay rate of the measure of certain sets.

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

# Una nueva visión del Laplaciano fraccionario vía redes neuronales profundas

### Date:  May 10, 2022 at 12 Santiago time

Venue: Online via Zoom / Sala de seminarios John Von Neumann, Beauchef 851,  piso 7
Chair: Jessica Trespalacios

Abstract: El Laplaciano fraccionario ha sido fuertemente estudiado durante las últimas décadas. En esta charla presentamos un nuevo enfoque al problema de Dirichlet asociado, usando técnicas recientes de aprendizaje profundo. En efecto, últimamente se ha demostrado que las soluciones aciertas ecuaciones en derivadas parciales se pueden representar de manera estocástica, y aproximar dicha representación mediante redes neuronales profundas, superando la llamada maldición de la dimensionalidad. Entre estas ecuaciones se encuentran las de tipo parabólicas sobre el espacio R^d, y las de tipo elípticas sobre un dominio acotado.

# Nonlocal truncated Laplacians: representation formulas and Liouville results

### Date:  May 3, 2022 at 12 Santiago time

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Erwin Topp

Abstract: We consider some nonlinear extremal integral operators that approximate the, so called, degenerate truncated Laplacians. For these operators we obtain representation formulas that lead to the construction of “fundamental solutions” and  to Liouville type results. Differences with respect to both the local case and the uniformly elliptic framework will be emphasized.

# Optimal design problems for a degenerate operator in Orlicz-Sobolev spaces

### Date:  April 26, 2022 at 12 Santiago time

Abstract: An optimization problem with volume constraint involving the $$\Phi$$-Laplacian in Orlicz-Sobolev spaces is considered for the case where $$\Phi$$ does not satisfy the natural condition introduced by Lieberman. A minimizer $$u_\Phi$$ having non-degeneracy at the free boundary is proved to exist and some important consequences are established like the Lipschitz regularity of $$u_\Phi$$ along the free boundary, that the set $$\{u_\Phi >0\}$$ has uniform positive density, that the free boundary is porous with porosity $$\delta>0$$ and has finite $$(N-\delta)$$-Hausdorff measure. Under a geometric compatibility condition set up by Rossi and Teixeira, it is established the behavior of a $$\ell$$-quasilinear optimal design problem with volume constraint for $$\ell$$ small. As $$\ell \to 0^+$$, we obtain a limiting free boundary problem driven by the infinity-Laplacian operator and find the optimal shape for the limiting problem. The proof is based on a penalization technique and a truncated minimization problem in terms of the Taylor polynomial of $$u_\Phi$$.

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

# Integrability, gravitation and field theories

### Date:  April 22, 2022 at 12 Santiago time

Abstract: The fruitful relationship between physics and integrability has been widely tested by different approaches. They appear, for example, in areas like gravitation and condensed matter physics using holographic or generating solution techniques. In this talk I will review some recent developments connecting integrable models, gravitation, non-Hermitian physics and field theories. The relationship between the AKNS hierarchy and anti-de Sitter three dimensional gravity will be discussed as well as the stability of complex solitons having real conserved quantities.

Based on:

• M. Cárdenas, F. C, K. Lara, and M. Pino, Physical Review Letters, 127, 161601, 2021.
• F. C, A. Fring, T. Taira, Nuclear Physics B, 971, 115516, 2021.
• F. C, A. Fring, T. Taira, Nuclear Physics B, 2022.
• J. Cen, F.C, A. Fring, T. Taira, Physics Letters A, 435, 128060 2022.

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

# Branch points for (almost-)minimizers of two-phase free boundary problems

### Date:  April 19, 2022 at 12 Santiago time

Abstract: In this talk, we will discuss minimizers and almost-minimizers of Alt-Caffarelli-Friedman type functionals. In particular, we will consider branch points in their free boundary. This is based on recent joint work with Guy David, Max Engelstein, and Tatiana Toro.

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

# Extremals in Hardy-Littlewood-Sobolev inequalities for stable processes

### Date:  April 13, 2022 at 12 Santiago time

Abstract: In this talk, we prove the existence of an extremal function in the Hardy-Littlewood-Sobolev inequality for the energy associated with an stable operator, via a concentration-compactness principle for stable processes. This result can be seen as the first step to study existence of positive solutions of the corresponding nonlocal equations with critical nonlinearities perturbed by lower order terms.

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

# Finite point blowup for the critical generalized Korteweg-de Vries equation

### Date: April 5, 2022 at 12 Santiago time

Abstract: In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de~Vries (gKdV) equation, including the determination of sufficient conditions for blowup,the stability of blowup in a refined topology and the classification of minimal mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and multi-point blow-up solutions were also constructed. However, all these results, as well as numerical simulations, involve the bubbling of a solitary wave going at infinity at the blow-up time, which means that the blow-up dynamics and the residue are eventually uncoupled. Even at the formal level, there was no indication whether blowup at a finite point could occur for this equation.

After reviewing the theory of blow-up for the critical gKdV equation in the first part of the talk, we will answer this question by constructing solutions that blow up in finite time under the form of a single-bubble concentrating the ground state at a finite point with an unforeseen blow-up rate.
Finding a blow-up rate intermediate between the self-similar rate and other rates previously known also reopens the question of which blow-up rates are actually possible for this equation.

This talk is based on a joint work with Yvan Martel (École Polytechnique/France).

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

# Global existence and long time behavior in the 1+1 dimensional principal Chiral model with applications to solitons

### Date:  March 29, 2022 at 12 Santiago time

Abstract: We consider the 1+1 dimensional vector valued Principal Chiral Field model (PCF) obtained as a simplification of the Vacuum Einstein Field equations under the Belinski-Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition. We also construct virial functionals which provide a clear description of decay of smooth global solutions inside the light cone. Finally, some applications are presented in the case of PCF solitons, a first step towards the study of its nonlinear stability.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Hanne Van Den Bosch