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.

Abstract :  In this talk, I will present results on a singular perturbation problem modeling domain walls. I will discuss the existence of solutions both when the perturbation parameter is non-zero and when it is set to zero (Thomas-Fermi approximation), demonstrating their continuous connection as the parameter approaches zero. Finally, I will show that the behavior of one of the variables can be modeled by a Painlevé II equation in the limit, by the use of an appropriate change of variables.

Abstract :Fourier sharp restriction theory has been a topic of interest over the last decades. On the other hand, efforts have been made in order to develop the theory of Fourier restriction over finite fields. In this talk, we will present some recently made developments (in a joint work with Diogo Oliveira e Silva) in the intersection of these two topics.

Abstract : In this talk we present a comprehensive framework for deriving rigorous and efficient bounds on the approximation error of deep neural networks in PDE models characterized by branching mechanisms, such as waves, Schrödinger equations, and other dispersive models. This framework utilizes the probabilistic setting established by Henry-Labordère and Touzi. We illustrate this approach by providing rigorous bounds on the approximation error for both linear and nonlinear waves in physical dimensions d = 1, 2, 3, and analyze their respective computational costs starting from time zero. We investigate two key scenarios: one involving a linear perturbative source term, and another focusing on pure nonlinear internal interactions. This is joint work with Claudio Muñoz (U. Chile).

Abstract: In this talk,  we will show a  proof of the nonexistence of breather solutions for NLS equations. By using a suitable virial functional, we are able to characterize the nonexistence of breather solutions by only using their inner energy and the power of the corresponding nonlinearity of the equation. We extend this result for several NLS models with different power nonlinearities and even the derivative NLS equation.

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

Chair: Gabrielle Nornberg

Abstract: —

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

Chair: Gabrielle Nornberg

About Manuel del Pino

Manuel del Pino currently holds the position of Professor in the Department of Mathematical Sciences at the University of Bath, UK. In 2018, he was honored with a Royal Society Research Professorship, the Society’s top research award, allowing exceptional scientists to focus on research by relieving them of teaching and administrative duties.

Previously, Professor del Pino worked at the Universidad de Chile, and has made significant contributions to the theory of asymptotic patterns in nonlinear partial differential equations. He is also a member of the Chilean Academy of Science and was the recipient of Chile’s National Prize of Science in 2013.

Research interests:

• Analysis of nonlinear partial differential equations
• Blow-up patterns in nonlinear evolution problems
• Singular limits in variational problems with loss of compactness

Manuel’s work deals with the analysis of singularity formation in non-linear partial differential equations. Of central interest is the construction of families of solutions with blow-up patterns in evolution problems in geometry, fluid dynamics and chemotaxis. A closely connected topic in his research is the analysis of singular limits in variational problems with loss of compactness.