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
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
2023-11-21
16:10hrs.
16:10hrs.
Vanderson Lima. Universidade Federal Do Rio Grande Do Sul
Eigenvalue problems and free boundary minimal surfaces in spherical caps
Sala 2
Abstract:
In a recent work with Ana Menezes (Princeton University), we introduced a family of functionals on the space of Riemannian metrics of a compact surface with boundary, defined via eigenvalues of a Steklov-type problem. In this talk I will present the ideas behind the following results: each such functional is uniformly bounded from above, and the maximizing metrics are induced by free boundary minimal immersions in some geodesic ball of a round sphere; the maximizer in the case of a disk is a spherical cap of dimension two; free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions are rotationally symmetric.
Eigenvalue problems and free boundary minimal surfaces in spherical caps
Sala 2
Abstract:
In a recent work with Ana Menezes (Princeton University), we introduced a family of functionals on the space of Riemannian metrics of a compact surface with boundary, defined via eigenvalues of a Steklov-type problem. In this talk I will present the ideas behind the following results: each such functional is uniformly bounded from above, and the maximizing metrics are induced by free boundary minimal immersions in some geodesic ball of a round sphere; the maximizer in the case of a disk is a spherical cap of dimension two; free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions are rotationally symmetric.
2023-11-07
16:10hrs.
16:10hrs.
Ricardo Freire. Dim-Fcfm, Universidad de Chile
The Cauchy problem for nonlinear dispersive models of long internal waves in the presence of the Coriolis force
Sala 2
Abstract:
We are engaged in an examination of two models concerning long internal waves, taking into consideration the influence of rotation. Mathematically, the rotational effect will be incorporated into the Benjamin-Ono and Intermediate Long Wave equations as an additional nonlocal term, symbolizing the Coriolis Force. This force induces a deviation in the course of the fluid's trajectory. Its salience is notably pronounced in investigations pertaining to atmospheric and oceanic phenomena, wherein the rotational effects of the Earth hold substantial sway. Specifically, the goal is to establish the global well-posedness of this models in the energy space and to obtain an unconditional uniqueness result for the solution in appropriate Sobolev-type spaces. Through the application of the method of space-frequency-localized energy estimates, we can recover the 'Energy spaces', within which we can demonstrate outcomes such as the propagation of regularity, asymptotic stability, decay, and other results where energy-based methods are imperative.
The Cauchy problem for nonlinear dispersive models of long internal waves in the presence of the Coriolis force
Sala 2
Abstract:
We are engaged in an examination of two models concerning long internal waves, taking into consideration the influence of rotation. Mathematically, the rotational effect will be incorporated into the Benjamin-Ono and Intermediate Long Wave equations as an additional nonlocal term, symbolizing the Coriolis Force. This force induces a deviation in the course of the fluid's trajectory. Its salience is notably pronounced in investigations pertaining to atmospheric and oceanic phenomena, wherein the rotational effects of the Earth hold substantial sway. Specifically, the goal is to establish the global well-posedness of this models in the energy space and to obtain an unconditional uniqueness result for the solution in appropriate Sobolev-type spaces. Through the application of the method of space-frequency-localized energy estimates, we can recover the 'Energy spaces', within which we can demonstrate outcomes such as the propagation of regularity, asymptotic stability, decay, and other results where energy-based methods are imperative.
2023-10-24
16:10hrs.
16:10hrs.
Maria Fernanda Espinal Florez. Facultad de Matemáticas, PUC de Chile
Constant sigma_2-curvature metrics with non isolated singularities
Sala 2
Abstract:
We study construction of a complete non-compact Riemannian metrics with positive constant $\sigma_2$-curvature, on the sphere $\mathbb{S}^n$ with a prescribed singular set $\Lambda$ given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than $\frac{n-\sqrt{n}-2}{2}$. A necessary condition is that $2 \leq 2k < n$.
Joint work with M. Del Mar González and Lorenzo Mazzieri.
Constant sigma_2-curvature metrics with non isolated singularities
Sala 2
Abstract:
We study construction of a complete non-compact Riemannian metrics with positive constant $\sigma_2$-curvature, on the sphere $\mathbb{S}^n$ with a prescribed singular set $\Lambda$ given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than $\frac{n-\sqrt{n}-2}{2}$. A necessary condition is that $2 \leq 2k < n$.
Joint work with M. Del Mar González and Lorenzo Mazzieri.
2023-10-10
16:10hrs.
16:10hrs.
Leonelo Iturriaga. Departamento de Matemática, Universidad Técnica Federico Santa María
Semilinear elliptic equations involving nonlinearities with zeros
Sala 2
Abstract:
In this talk we review some results concerning with the existence and multiplicity of positive solutions for semilinear elliptic problems resembling the following form
where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $N\geq 3$, $f$ is a locally Lipschitz function defined in $[0,+\infty)$, which is nonnegative with a positive zero, and $\lambda$ is a positive parameter. We will also explore how these results can be extended to the fractional operators.
Semilinear elliptic equations involving nonlinearities with zeros
Sala 2
Abstract:
In this talk we review some results concerning with the existence and multiplicity of positive solutions for semilinear elliptic problems resembling the following form
$$
\begin{cases}
-\Delta u=\lambda f(u), &\textrm{in }\Omega\\ u=0, &\textrm{on }\partial\Omega,
\end{cases}
$$
2023-09-26
16:10hrs.
16:10hrs.
Andres Larrain-Hubach. Department of Mathematics, University of Dayton
Una aplicación de una técnica de Getzler a la torsión analítica
Sala 2
Abstract:
En su prueba del teorema de índice, Getzler desarrolló una técnica para extraer ciertos términos de la expansión asintótica del núcleo de calor de un laplaciano. En esta charla revisaremos el argumento original y mostraremos cómo puede ser modificado para analizar problemas relacionados con la torsión analítica holomorfa de Ray-Singer.
Una aplicación de una técnica de Getzler a la torsión analítica
Sala 2
Abstract:
En su prueba del teorema de índice, Getzler desarrolló una técnica para extraer ciertos términos de la expansión asintótica del núcleo de calor de un laplaciano. En esta charla revisaremos el argumento original y mostraremos cómo puede ser modificado para analizar problemas relacionados con la torsión analítica holomorfa de Ray-Singer.
2023-09-05
16:10hrs.
16:10hrs.
Victor Cañulef. Cmm, Universidad de Chile
On the collapse of the local Rayleigh condition for the hydrostatic Euler equations
Sala 2
Abstract:
The hydrostatic Euler equations are derived from the incompressible Euler equations by means of the hydrostatic approximation. Among the different stability criteria that arise in the study of linear stability for the incompressible Euler equations, we can mention Rayleigh's stability criterion, which gives rise to the local Rayleigh condition. Linear and nonlinear instability of the hydrostatic Euler equations around certain shear flows is well-known, as well as the finite time blow-up of certain solutions that do not satisfy the local Rayleigh condition. On the other hand, local existence, uniqueness and stability has been established in Sobolev spaces under the local Rayleigh condition. In this talk I will present new features of the $H^4$ solution to the hydrostatic Euler equations under the local Rayleigh condition; under certain assumptions, we establish the dichotomy between the breakdown of the local Rayleigh condition and the formation of singularities. Additionally, we get necessary conditions for global solvability in Sobolev spaces. As a byproduct, we show the $x$-independence of stationary solutions. Our proof relies on new monotonicity identities for the solution to the hydrostatic Euler equations under the local Rayleigh condition.
On the collapse of the local Rayleigh condition for the hydrostatic Euler equations
Sala 2
Abstract:
The hydrostatic Euler equations are derived from the incompressible Euler equations by means of the hydrostatic approximation. Among the different stability criteria that arise in the study of linear stability for the incompressible Euler equations, we can mention Rayleigh's stability criterion, which gives rise to the local Rayleigh condition. Linear and nonlinear instability of the hydrostatic Euler equations around certain shear flows is well-known, as well as the finite time blow-up of certain solutions that do not satisfy the local Rayleigh condition. On the other hand, local existence, uniqueness and stability has been established in Sobolev spaces under the local Rayleigh condition. In this talk I will present new features of the $H^4$ solution to the hydrostatic Euler equations under the local Rayleigh condition; under certain assumptions, we establish the dichotomy between the breakdown of the local Rayleigh condition and the formation of singularities. Additionally, we get necessary conditions for global solvability in Sobolev spaces. As a byproduct, we show the $x$-independence of stationary solutions. Our proof relies on new monotonicity identities for the solution to the hydrostatic Euler equations under the local Rayleigh condition.
2023-08-22
16:10hrs.
16:10hrs.
Sebastián Muñoz. Department of Mathematics, Purdue University
Boundary rigidity problem under the presence of a magnetic field and a potential
Sala 2
Abstract:
The boundary rigidity problem asks if it is possible to determine the Riemannian metric on a compact manifold, up to a boundary fixing isometry, from the knowledge of the boundary distance function. In this talk I will discuss the solution to this problem in dimension 2, and what is known for the magnetic case, and the magnetic case with potential. Finally, and if time allows it, I will briefly discuss some open problems and recent results.
Boundary rigidity problem under the presence of a magnetic field and a potential
Sala 2
Abstract:
The boundary rigidity problem asks if it is possible to determine the Riemannian metric on a compact manifold, up to a boundary fixing isometry, from the knowledge of the boundary distance function. In this talk I will discuss the solution to this problem in dimension 2, and what is known for the magnetic case, and the magnetic case with potential. Finally, and if time allows it, I will briefly discuss some open problems and recent results.
2023-08-08
16:10hrs.
16:10hrs.
Boyan Sirakov. Departamento de Matemática, PUC - Rio de Janeiro
Regularity estimates with optimized constants
Sala 2
Abstract:
We present several classical regularity estimates for general uniformly elliptic operators of second order with unbounded coefficients, giving explicit and optimal dependence of the constants in these inequalities in terms of Lebesgue norms of the lower order coefficients of the operator, and the size of the domain. Among these estimates are the interior and global Harnack inequalities, the Hopf lemma, $L^\infty$-estimates, and logarithmic gradient estimates. Applications include the Landis conjecture and the Vazquez strong maximum principle for operators with unbounded coefficients.
Regularity estimates with optimized constants
Sala 2
Abstract:
We present several classical regularity estimates for general uniformly elliptic operators of second order with unbounded coefficients, giving explicit and optimal dependence of the constants in these inequalities in terms of Lebesgue norms of the lower order coefficients of the operator, and the size of the domain. Among these estimates are the interior and global Harnack inequalities, the Hopf lemma, $L^\infty$-estimates, and logarithmic gradient estimates. Applications include the Landis conjecture and the Vazquez strong maximum principle for operators with unbounded coefficients.
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Edificio Rolando Chuaqui, Campus San Joaquín. Avda. Vicuña Mackenna 4860, Macul, Chile.Teléfono (+56) 95504 4511.
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