Seminario de Análisis y Geometría
Los seminarios de Análisis y Geometría se llevan a cabo los días jueves a las 16:10 en la Sala 2 de la Facultad de Matemáticas, Pontificia Universidad Católica de Chile.Organizadores: Pedro Gaspar y Nikola Kamburov
2025-01-21
11:00hrs.
11:00hrs.
Tomás Sanz Perela. Universitat de Barcelona
Stable cones in the Alt-Phillips free boundary problem
Sala 1
Abstract:
Stable cones in the Alt-Phillips free boundary problem
Sala 1
Abstract:
In this talk I will describe a recent result, obtained in collaboration with Aram Karakhanyan, in which we obtain for the first time a stability condition for the Alt-Phillips free boundary problem. Then, I will discuss how do we use it to classify global stable axially-symmetric solutions in dimensions 3, 4, and 5.
2025-01-16
17:10hrs.
17:10hrs.
Ederson Moreira Dos Santos. Universidade de São Paulo (São Carlos)
Standing waves for nonlinear Hartree type equations: existence and qualitative properties
Sala multiuso primer piso Edificio Villanueva
Abstract:
In this talk I will report some results on standing wave solutions for nonlinear Hartree type equations, regarding existence and some qualitative properties, such as definite sign, radial symmetry and sharp asymptotic decay.
Joint work with with Eduardo Böer (ICMC-USP).
Standing waves for nonlinear Hartree type equations: existence and qualitative properties
Sala multiuso primer piso Edificio Villanueva
Abstract:
In this talk I will report some results on standing wave solutions for nonlinear Hartree type equations, regarding existence and some qualitative properties, such as definite sign, radial symmetry and sharp asymptotic decay.
Joint work with with Eduardo Böer (ICMC-USP).
2025-01-16
16:10hrs.
16:10hrs.
Dennis Kriventsov. Department of Mathematics, Rutgers University
Rectifiability of interfaces with positive Alt-Caffarelli-Friedman limit via quantitative stability
Sala multiuso primer piso Edificio Villanueva
Abstract:
The Alt-Caffarelli-Friedman (ACF) monotonicity formula captures fine behavior of pairs of nonnegative harmonic functions which vanish on the mutual boundary of complementary domains. I will describe the following theorem: the set of points with positive limit for the ACF formula, corresponding roughly to where both functions have linear growth away from the interface, is countably (n-1)-rectifiable. The proof leverages a new quantitative stability property for the ACF formula, which in turn is based on a new quantitative stability result for the Faber-Krahn inequality. Indeed, we show that the square of the L2 distance between the first Dirichlet eigenfunctions of some domain and some ball of the same volume is controlled by the difference of their first eigenvalues. Our proof of this relies on regularity theory for some “critical" modifications of Bernoulli-type free boundary problems. This is based on joint work with Mark Allen and Robin Neumayer.
Rectifiability of interfaces with positive Alt-Caffarelli-Friedman limit via quantitative stability
Sala multiuso primer piso Edificio Villanueva
Abstract:
The Alt-Caffarelli-Friedman (ACF) monotonicity formula captures fine behavior of pairs of nonnegative harmonic functions which vanish on the mutual boundary of complementary domains. I will describe the following theorem: the set of points with positive limit for the ACF formula, corresponding roughly to where both functions have linear growth away from the interface, is countably (n-1)-rectifiable. The proof leverages a new quantitative stability property for the ACF formula, which in turn is based on a new quantitative stability result for the Faber-Krahn inequality. Indeed, we show that the square of the L2 distance between the first Dirichlet eigenfunctions of some domain and some ball of the same volume is controlled by the difference of their first eigenvalues. Our proof of this relies on regularity theory for some “critical" modifications of Bernoulli-type free boundary problems. This is based on joint work with Mark Allen and Robin Neumayer.
2025-01-09
16:10hrs.
16:10hrs.
Luciano Sciaraffia. Albert-Ludwigs-Universität Freiburg
Minimal networks and curvature flow singularities
Sala 1
Abstract:
The object of our study is networks, that is, unions of a finite number of curves. The talk will address two topics: the existence of certain minimal configurations and the curvature flow.
I will begin by reviewing some of the literature on minimal networks on surfaces, with particular emphasis on the case of the 2-sphere. Inspired by a conjecture of Hass and Morgan regarding the existence of theta-shaped minimal networks in convex 2-spheres, I will then consider such networks on spheres of any dimension, endowed with a Riemannian metric close to the standard one. Using a finite-dimensional reduction method in conjunction with the Lusternik–Schnirelmann category, we establish results on existence and multiplicity.
Next, I will introduce the network flow and review some of its key properties, focusing on the formation of singularities and the main differences from the shortening flow of simple curves. I will discuss the so-called type-0 singularities, after which it is possible to extend the flow. In the specific case of symmetric initial data with two triple junctions, we show that the set of singular times is finite.
The methods employed in the main proofs are elementary, and the talk should be accessible to master's students with a background in analysis.
Minimal networks and curvature flow singularities
Sala 1
Abstract:
The object of our study is networks, that is, unions of a finite number of curves. The talk will address two topics: the existence of certain minimal configurations and the curvature flow.
I will begin by reviewing some of the literature on minimal networks on surfaces, with particular emphasis on the case of the 2-sphere. Inspired by a conjecture of Hass and Morgan regarding the existence of theta-shaped minimal networks in convex 2-spheres, I will then consider such networks on spheres of any dimension, endowed with a Riemannian metric close to the standard one. Using a finite-dimensional reduction method in conjunction with the Lusternik–Schnirelmann category, we establish results on existence and multiplicity.
Next, I will introduce the network flow and review some of its key properties, focusing on the formation of singularities and the main differences from the shortening flow of simple curves. I will discuss the so-called type-0 singularities, after which it is possible to extend the flow. In the specific case of symmetric initial data with two triple junctions, we show that the set of singular times is finite.
The methods employed in the main proofs are elementary, and the talk should be accessible to master's students with a background in analysis.
2024-12-05
16:10hrs.
16:10hrs.
Javier Monreal. Departamento de Ingeniería Matemática, Universidad de Chile
Variational Approach for the Singular Perturbation Domain Wall Coupled System
Sala 2
Abstract:
In this talk, I will present results on a singular perturbation problem modeling domain walls. I will discuss the existence of solutions both for a nonzero perturbation parameter and in the Thomas-Fermi approximation (when the parameter is set to zero), demonstrating their continuous connection as the parameter approaches zero. Finally, I will show that the behavior of one of the variables in the limit can be described by a Painlevé II equation, obtained by the use of an appropriate change of variables.
Variational Approach for the Singular Perturbation Domain Wall Coupled System
Sala 2
Abstract:
In this talk, I will present results on a singular perturbation problem modeling domain walls. I will discuss the existence of solutions both for a nonzero perturbation parameter and in the Thomas-Fermi approximation (when the parameter is set to zero), demonstrating their continuous connection as the parameter approaches zero. Finally, I will show that the behavior of one of the variables in the limit can be described by a Painlevé II equation, obtained by the use of an appropriate change of variables.
2024-11-28
16:10hrs.
16:10hrs.
Jessica Trespalacios. Departamento de Ingeniería Matemática - Universidad de Chile
Solitons Dynamics for Einstein equations
Sala 2
Abstract:
Given the complexity of the Einstein equations, it is often a good choice to study a question of interest in the framework of a restricted class of solutions. One way to impose such restrictions is to consider solutions that satisfy a given symmetry condition. In this work, we consider the particular class of spacetimes that admit two space-like Killing vector fields. More precisely, we will focus on the Einstein vacuum model $R_{\mu \nu}(\tilde g)=0$, where $\tilde g$ is the metric tensor and $R_{\mu \nu}$ is the Ricci tensor, in the Belinski-Zakharov setting. This ansatz is compatible with the well-known Gowdy symmetry.
The main goal of this talk is to describe rigorously the conditions for the global existence of small solutions, and their decay in the light cone, as well as the stability of a first set of solitonic solutions (gravisolitons), for the so-called reduced Einstein equation, viewed as an identification of the Principal Chiral Field (PCF) model.
Solitons Dynamics for Einstein equations
Sala 2
Abstract:
Given the complexity of the Einstein equations, it is often a good choice to study a question of interest in the framework of a restricted class of solutions. One way to impose such restrictions is to consider solutions that satisfy a given symmetry condition. In this work, we consider the particular class of spacetimes that admit two space-like Killing vector fields. More precisely, we will focus on the Einstein vacuum model $R_{\mu \nu}(\tilde g)=0$, where $\tilde g$ is the metric tensor and $R_{\mu \nu}$ is the Ricci tensor, in the Belinski-Zakharov setting. This ansatz is compatible with the well-known Gowdy symmetry.
The main goal of this talk is to describe rigorously the conditions for the global existence of small solutions, and their decay in the light cone, as well as the stability of a first set of solitonic solutions (gravisolitons), for the so-called reduced Einstein equation, viewed as an identification of the Principal Chiral Field (PCF) model.
2024-11-14
16:10hrs.
16:10hrs.
Matías Díaz. Facultad de Matemáticas, PUC de Chile
The Ginzburg-Landau model for inhomogeneous type-II superconductors
Sala 2
Abstract:
The Ginzburg-Landau model for inhomogeneous type-II superconductors
Sala 2
Abstract:
Superconductivity is a phenomenon characterized by the possibility of nondissipating electric currents and the expulsion of applied external magnetic fields. The behavior of superconductors can be studied through the Ginzburg-Landau model of superconductivity, which is a phenomenological model. Type-II superconductors are characterized by the occurence of vortices in the presence of an external magnetic field, which are magnetic defects coexisting with the superconducting phase. In this talk, we will discuss the occurrence of vortices in an inhomogeneous superconductor and present results concerning the existence and stability of vortexless configurations as we increase the intensity of the applied external magnetic field. This is joint work with Carlos Román (PUC).
2024-11-07
16:10hrs.
16:10hrs.
Almir Silva Santos. Universidade Federal de Segipe, Brasil
On constant Q-curvature metrics with Delaunay-type ends
Sala 2
Abstract:
The Q-curvature of a Riemannian metric is a function that depends on the metrics up to fourth order. Over the past few decades, numerous mathematicians have devoted significant attention to investigating this quantity within conformal and non-conformal geometric frameworks. The equation that relates the Q-curvatures of two conformal metrics is a fourth-order nonlinear PDE with a critical exponent. It is similar to the Yamabe equation, although it has its particularities. The primary objective of this talk is to investigate constant Q-curvature metrics in the conformal setting, focusing on the intriguing case of isolated singularities. Joint work with R. Caju (Universidad de Chile) and J. Ratzkin (Universität Würzburg/German).
On constant Q-curvature metrics with Delaunay-type ends
Sala 2
Abstract:
The Q-curvature of a Riemannian metric is a function that depends on the metrics up to fourth order. Over the past few decades, numerous mathematicians have devoted significant attention to investigating this quantity within conformal and non-conformal geometric frameworks. The equation that relates the Q-curvatures of two conformal metrics is a fourth-order nonlinear PDE with a critical exponent. It is similar to the Yamabe equation, although it has its particularities. The primary objective of this talk is to investigate constant Q-curvature metrics in the conformal setting, focusing on the intriguing case of isolated singularities. Joint work with R. Caju (Universidad de Chile) and J. Ratzkin (Universität Würzburg/German).
2024-09-26
16:10hrs.
16:10hrs.
Gianmarco Sperone. Facultad de Matemáticas, UC Chile
On the planar Taylor-Couette system and related exterior problems
Sala 2
Abstract:
We consider the planar Taylor-Couette system for the steady motion of a viscous incompressible fluid in the region between two concentric disks, the inner one being at rest and the outer one rotating with constant angular speed. We study the uniqueness and multiplicity of solutions to the forced system in different classes. For any angular velocity we prove that the classical Taylor-Couette flow is the unique smooth solution displaying rotational symmetry. Instead, we show that infinitely many solutions arise, even for arbitrarily small angular velocities, in a larger, class of incomplete solutions that we introduce. By prescribing the transversal flux, unique solvability of the Taylor-Couette system is recovered among rotationally invariant incomplete solutions. Finally, we study the behavior of these solutions as the radius of the outer disk goes to infinity, connecting our results with the celebrated Stokes paradox. This is a joint work with Filippo Gazzola (Politecnico di Milano) and Jirí Neustupa (Institute of Mathematics of the Czech Academy of Sciences).
On the planar Taylor-Couette system and related exterior problems
Sala 2
Abstract:
We consider the planar Taylor-Couette system for the steady motion of a viscous incompressible fluid in the region between two concentric disks, the inner one being at rest and the outer one rotating with constant angular speed. We study the uniqueness and multiplicity of solutions to the forced system in different classes. For any angular velocity we prove that the classical Taylor-Couette flow is the unique smooth solution displaying rotational symmetry. Instead, we show that infinitely many solutions arise, even for arbitrarily small angular velocities, in a larger, class of incomplete solutions that we introduce. By prescribing the transversal flux, unique solvability of the Taylor-Couette system is recovered among rotationally invariant incomplete solutions. Finally, we study the behavior of these solutions as the radius of the outer disk goes to infinity, connecting our results with the celebrated Stokes paradox. This is a joint work with Filippo Gazzola (Politecnico di Milano) and Jirí Neustupa (Institute of Mathematics of the Czech Academy of Sciences).
2024-09-12
16:10hrs.
16:10hrs.
Reshmi Biswas. Universidad Técnica Federico Santa María, Valparaíso
Quasilinear Schrödinger equations with Stein-Weiss type convolution and critical exponential nonlinearity in $\mathbb R^N$
Sala 2
Quasilinear Schrödinger equations with Stein-Weiss type convolution and critical exponential nonlinearity in $\mathbb R^N$
Sala 2
2024-09-05
16:10hrs.
16:10hrs.
Carlos Román. Facultad de Matemáticas, UC Chile
Domain branching in micromagnetism
Sala 2
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.
Domain branching in micromagnetism
Sala 2
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.
2024-07-25
16:10hrs.
16:10hrs.
Rajesh Mahadevan. Departamento de Matemática, Universidad de Concepción
Asymptotic theory for a general class of short-range interaction functionals
Sala N4 de la Facultad de Ciencias
Abstract:
Asymptotic theory for a general class of short-range interaction functionals
Sala N4 de la Facultad de Ciencias
Abstract:
In models of N interacting particles in $R^d$, the repulsive cost is usually described by a two-point function $c(x,y) =l(|x-y|/\epsilon)$ where $l:[0,\infty)\to R$ is decreasing to zero at infinity and parameter $\epsilon>0$ scales the interaction distance. In this talk we explain how to deduce an asymptotic model in the short-range regime, that is, $\epsilon << 1$ together with the natural local integrability assumption on l.
This extends recent results by Hardin-Saff-Vlasiuk, Hardin-Leble-Saff-Serfaty and Lewin, obtained in the homogeneous case $l = r^{-s}$ where $s>d$.
This extends recent results by Hardin-Saff-Vlasiuk, Hardin-Leble-Saff-Serfaty and Lewin, obtained in the homogeneous case $l = r^{-s}$ where $s>d$.
2024-06-14
15:00hrs.
15:00hrs.
Renato Velozo. Department of Mathematics, University of Toronto
Two results on modified scattering for the Vlasov-Poisson system
Sala 2
Abstract:
In this talk, I will discuss modified scattering properties of small data solutions for the Vlasov-Poisson system. On the one hand, I will show a modified scattering result for the Vlasov-Poisson system with a trapping potential. On the other hand, I will show a high order modified scattering result for the classical Vlasov-Poisson system. These are joint work(s) with Léo Bigorgne (Université de Rennes) and Anibal Velozo Ruiz (PUC).
Two results on modified scattering for the Vlasov-Poisson system
Sala 2
Abstract:
In this talk, I will discuss modified scattering properties of small data solutions for the Vlasov-Poisson system. On the one hand, I will show a modified scattering result for the Vlasov-Poisson system with a trapping potential. On the other hand, I will show a high order modified scattering result for the classical Vlasov-Poisson system. These are joint work(s) with Léo Bigorgne (Université de Rennes) and Anibal Velozo Ruiz (PUC).
2024-06-06
16:10hrs.
16:10hrs.
Andrés Zúñiga. Instituto de Ciencias de la Ingeniería, Universidad de O'higgins
Geometric construction of minimizers to least gradient problems: the density case
Sala 2
Geometric construction of minimizers to least gradient problems: the density case
Sala 2
2024-05-23
16:10hrs.
16:10hrs.
Carolina Rey. Departamento de Matemática, Universidad Técnica Federico Santa María
The Constant Q-Curvature Equation within Product Manifolds
Sala 2
Abstract:
The Q-curvature generalizes the Gaussian curvature for manifolds with three or more dimensions, revealing a significant link between curvature and topology, similar to the implications of the Gauss-Bonnet theorem. In the talk, we will show the critical role of manifold topology in finding the multiplicity of solutions to the constant Q-curvature equation in product manifolds of five or more dimensions. Based on the Lusternik-Schnirelmann theory, we find a lower bound for the number of solutions to the constant Q-curvature equation.
The Constant Q-Curvature Equation within Product Manifolds
Sala 2
Abstract:
The Q-curvature generalizes the Gaussian curvature for manifolds with three or more dimensions, revealing a significant link between curvature and topology, similar to the implications of the Gauss-Bonnet theorem. In the talk, we will show the critical role of manifold topology in finding the multiplicity of solutions to the constant Q-curvature equation in product manifolds of five or more dimensions. Based on the Lusternik-Schnirelmann theory, we find a lower bound for the number of solutions to the constant Q-curvature equation.
2024-05-16
16:10hrs.
16:10hrs.
Rodrigo Lecaros. Departamento de Matemática, Universidad Técnica Federico Santa María
Propiedad de continuación única para sistemas discretizados. Aplicación a Control y Problemas Inversos
Sala 2
Abstract:
Propiedad de continuación única para sistemas discretizados. Aplicación a Control y Problemas Inversos
Sala 2
Abstract:
El estudio de problemas de control o problemas inversos asociados a una Ecuación en Derivadas Parciales (EDP) se apoya en propiedades de continuación única (PCU) de las soluciones. Estas propiedades son fundamentales para obtener resultados de controlabilidad o estabilidad de sistemas. Sin embargo, no siempre está claro si las soluciones de una discretización del sistema cumplen estas propiedades o cuáles son las condiciones que deben cumplir las mallas para garantizar la convergencia de la PCU.
En esta presentación, se explorará el concepto de PCU y su relevancia en problemas inversos y de control. Se abordarán las principales dificultades de los problemas discretos, junto con una breve introducción al enfoque utilizado para estudiar estos casos. Para concluir, se expondrán una serie de ejemplos donde se ha investigado la PCU en sistemas discretos o semi-discretos.
En esta presentación, se explorará el concepto de PCU y su relevancia en problemas inversos y de control. Se abordarán las principales dificultades de los problemas discretos, junto con una breve introducción al enfoque utilizado para estudiar estos casos. Para concluir, se expondrán una serie de ejemplos donde se ha investigado la PCU en sistemas discretos o semi-discretos.
2024-04-18
16:10hrs.
16:10hrs.
Rayssa Caju. Dim-Cmm, Universidad de Chile
Large conformal metrics with prescribed gauss and geodesic curvatures
Sala 5
Abstract:
In this talk, our goal is to study the Kazdan-Warner problem in surfaces with boundary and discuss the existence of at least two distinct conformal metrics with prescribed gaussian curvature and geodesic curvature respectively, $K_{g}= f + \lambda$ and $k_{g}= h + \mu$, where $f$ and $h$ are nonpositive functions and $\lambda$ and $\mu$ are positive constants. Utilizing Struwe's monotonicity trick, we investigate the blowup behavior of the solutions and establish a non-existence result for the limiting PDE, eliminating one of the potential blow-up profiles.
Large conformal metrics with prescribed gauss and geodesic curvatures
Sala 5
Abstract:
In this talk, our goal is to study the Kazdan-Warner problem in surfaces with boundary and discuss the existence of at least two distinct conformal metrics with prescribed gaussian curvature and geodesic curvature respectively, $K_{g}= f + \lambda$ and $k_{g}= h + \mu$, where $f$ and $h$ are nonpositive functions and $\lambda$ and $\mu$ are positive constants. Utilizing Struwe's monotonicity trick, we investigate the blowup behavior of the solutions and establish a non-existence result for the limiting PDE, eliminating one of the potential blow-up profiles.
2024-03-21
16:10hrs.
16:10hrs.
Jean Dolbeault. Ceremade, Université Paris-Dauphine
Nonlinear diffusions, entropies and stability in functional inequalities
Sala 5
Abstract:
Entropy methods coupled to nonlinear diffusions are powerful tools to study some functional inequalities of Sobolev type. Self-similar solutions can indeed be reinterpreted as optimal solutions of Aubin-Talenti type. A notion of generalized entropy is the key tool which relates the nonlinear regime to the linearized problem around the asymptotic profile and reduces the analysis to a spectral problem. Estimates can be made constructive. This gives quantitative stability results with explicit constants. Entropy methods will be compared with other direct methods, intended for instance to obtain bounds on the stability constant in the Bianchi-Egnell stability result for the Sobolev inequality.
Nonlinear diffusions, entropies and stability in functional inequalities
Sala 5
Abstract:
Entropy methods coupled to nonlinear diffusions are powerful tools to study some functional inequalities of Sobolev type. Self-similar solutions can indeed be reinterpreted as optimal solutions of Aubin-Talenti type. A notion of generalized entropy is the key tool which relates the nonlinear regime to the linearized problem around the asymptotic profile and reduces the analysis to a spectral problem. Estimates can be made constructive. This gives quantitative stability results with explicit constants. Entropy methods will be compared with other direct methods, intended for instance to obtain bounds on the stability constant in the Bianchi-Egnell stability result for the Sobolev inequality.
2024-01-04
16:10hrs.
16:10hrs.
Cristian González Riquelme. Instituto Superior Tecnico, Lisbon
How regular is the maximal function of a given function?
Sala Multiuso 1° piso, edificio Felipe Villanueva
Abstract:
Maximal operators are a central object in harmonic analysis. The regularity theory of such objects has been an object of study formany authors over the last decades. However, even in the one dimensional case, there are still interesting questions that remain open. In this talk, we will discuss recent developments and open questions about this topic, particularly about the boundedness and continuity for such operators at the derivative level and regularity improving properties of these operators.
How regular is the maximal function of a given function?
Sala Multiuso 1° piso, edificio Felipe Villanueva
Abstract:
Maximal operators are a central object in harmonic analysis. The regularity theory of such objects has been an object of study formany authors over the last decades. However, even in the one dimensional case, there are still interesting questions that remain open. In this talk, we will discuss recent developments and open questions about this topic, particularly about the boundedness and continuity for such operators at the derivative level and regularity improving properties of these operators.
2023-11-28
16:10hrs.
16:10hrs.
Noemi Wolanski. Departamento de Matemática, Universidad de Buenos Aires
Large time behavior for time/space nonlocal diffusion equations in R^N
Sala 2
Large time behavior for time/space nonlocal diffusion equations in R^N
Sala 2
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