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 5 de la Facultad de Matemáticas, Pontificia Universidad Católica de Chile.Organizadores: Pedro Gaspar y Nikola Kamburov
2024-05-23
16:10hrs.
16:10hrs.
Carolina Rey. Departamento de Matemática, Universidad Técnica Federico Santa María
TBA
Sala 5
Abstract:
TBA
TBA
Sala 5
Abstract:
TBA
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.
2023-06-20
16:00hrs.
16:00hrs.
Nicolás Valenzuela. Dim, Universidad de Chile
Una nueva visión para el Laplaciano fraccionario via redes neuronales profundas
Sala 1
Abstract:
Esta charla se enfocará en el estudio del problema de Dirichlet con Laplaciano fraccionario y su relación con redes neuronales profundas. En particular, a partir de una representación estocástica
de la solución del problema de Dirichlet fraccionario, demostraremos que existe una red neuronal profunda que aproxima la solución del problema a una precisión arbitraria y más aún, la cantidad de parámetros necesaria para de nir la red es a lo más polinomial en la dimensión, el recíproco de la precisión y el tamaño del conjunto. Además se realizarán simulaciones numéricas en Python para encontrar redes neuronales que aproximan las soluciones del problema de Dirichlet fraccionario para distintas configuraciones del problema.
Una nueva visión para el Laplaciano fraccionario via redes neuronales profundas
Sala 1
Abstract:
Esta charla se enfocará en el estudio del problema de Dirichlet con Laplaciano fraccionario y su relación con redes neuronales profundas. En particular, a partir de una representación estocástica
de la solución del problema de Dirichlet fraccionario, demostraremos que existe una red neuronal profunda que aproxima la solución del problema a una precisión arbitraria y más aún, la cantidad de parámetros necesaria para de nir la red es a lo más polinomial en la dimensión, el recíproco de la precisión y el tamaño del conjunto. Además se realizarán simulaciones numéricas en Python para encontrar redes neuronales que aproximan las soluciones del problema de Dirichlet fraccionario para distintas configuraciones del problema.
2023-06-13
16:00hrs.
16:00hrs.
Hanne Van Den Bosch. Dim & Cmm, Universidad de Chile
A Keller-Lieb-Thirring Inequality for Dirac operators
Sala 1
Abstract:
The classical Keller-Lieb-Thirring inequality bounds the ground state energy of a Schrödinger operator by a Lebesgue norm of the potential. This problem can be rewritten as a minimization problem for the Rayleigh quotient over both the eigenfunction and the potential. It is then straightforward to see that the best potential is a power of the eigenfunction, and the optimal eigenfunction satisfies a nonlinear Schrodinger equation. This talk concerns the analogous question for the smallest eigenvalue in the gap of a massive Dirac operator. This eigenvalue is not characterized by a minimization problem. By using a suitable Birman-Schwinger operator, we show that for sufficiently small potentials in Lebesgue spaces, an optimal potential and eigenfunction exists. Moreover, the corresponding eigenfunction solves a nonlinear Dirac equation.
A Keller-Lieb-Thirring Inequality for Dirac operators
Sala 1
Abstract:
The classical Keller-Lieb-Thirring inequality bounds the ground state energy of a Schrödinger operator by a Lebesgue norm of the potential. This problem can be rewritten as a minimization problem for the Rayleigh quotient over both the eigenfunction and the potential. It is then straightforward to see that the best potential is a power of the eigenfunction, and the optimal eigenfunction satisfies a nonlinear Schrodinger equation. This talk concerns the analogous question for the smallest eigenvalue in the gap of a massive Dirac operator. This eigenvalue is not characterized by a minimization problem. By using a suitable Birman-Schwinger operator, we show that for sufficiently small potentials in Lebesgue spaces, an optimal potential and eigenfunction exists. Moreover, the corresponding eigenfunction solves a nonlinear Dirac equation.
2023-06-06
16:00hrs.
16:00hrs.
Alexander Quaas. Departamento de Matemática, Universidad Técnica Federico Santa María
Large harmonic functions for fully nonlinear fractional operators
Sala 1
Large harmonic functions for fully nonlinear fractional operators
Sala 1
2023-05-30
16:00hrs.
16:00hrs.
Juan Carlos Pozo. Departamento de Matemáticas, Universidad de Chile
On local energy decay for large solutions of the Zakharov-Kuznetsov equation
Sala 1
Abstract:
On local energy decay for large solutions of the Zakharov-Kuznetsov equation
Sala 1
Abstract:
In this talk, we will consider the Zakharov-Kuznetsov equation, which is a nonlinear wave equation posed in $R^d$ with $d=2,3$. Concretely, we consider
$u_t+(\Delta u)_x+u u_x=0.$
We will present a recent result that sheds light on the large time behavior of their solutions. Specifically, we will explain how these solutions exhibit local decay in a region that does not contain the soliton region.
2023-05-30
17:00hrs.
17:00hrs.
Julio Rossi. Departamento de Matemática, Universidad de Buenos Aires
Convexity and Partial Differential Equations
Sala 1
Abstract:
In this talk we will introduce different notions of convexity that interpolate between classical convexity and quasiconvexity and that, moreover, have a natural fractional extension. For these notions of convexity we also characterize the convex envelope inside a domain of a boundary datum in terms of being the unique viscosity solution to an associated differential equation.
Convexity and Partial Differential Equations
Sala 1
Abstract:
In this talk we will introduce different notions of convexity that interpolate between classical convexity and quasiconvexity and that, moreover, have a natural fractional extension. For these notions of convexity we also characterize the convex envelope inside a domain of a boundary datum in terms of being the unique viscosity solution to an associated differential equation.
2023-05-23
16:00hrs.
16:00hrs.
Duvan Henao. Instituto de Ciencias de la Ingeniería, Universidad de O'higgins
Dipolos armónicos en elasticidad
Sala 1
Abstract:
Whenever the stored energy density of a hyperelastic material has slow growth at infinity (below |F|^p with p less than the space dimension), it may undergo cavitation (the nucleation and sudden growth of internal voids) under large hydrostatic tension [Ball, 1982; James & Spector, 1992]. This constitutes a failure of quasiconvexity and, hence, a challenge for the existence theory in elastostatics [Ball & Murat, 1984]. The obstacle has been overcome under certain coercivity hypotheses [Müller & Spector, 1995; Sivaloganathan & Spector, 2000] which, however, fail to be satisfied by the paradigmatic example in elasticity: that of 3D neo-Hookean materials. A joint work with Marco Barchiesi, Carlos Mora-Corral, and Rémy Rodiac will be presented, where this borderline case was solved for hollow axisymmetric domains. Partial results leading to a solution when the axis of rotation is contained (where the dipoles found by [Conti & De Lellis, 2003] must be proved to be non energy-minimizing) will also be discussed.
Dipolos armónicos en elasticidad
Sala 1
Abstract:
Whenever the stored energy density of a hyperelastic material has slow growth at infinity (below |F|^p with p less than the space dimension), it may undergo cavitation (the nucleation and sudden growth of internal voids) under large hydrostatic tension [Ball, 1982; James & Spector, 1992]. This constitutes a failure of quasiconvexity and, hence, a challenge for the existence theory in elastostatics [Ball & Murat, 1984]. The obstacle has been overcome under certain coercivity hypotheses [Müller & Spector, 1995; Sivaloganathan & Spector, 2000] which, however, fail to be satisfied by the paradigmatic example in elasticity: that of 3D neo-Hookean materials. A joint work with Marco Barchiesi, Carlos Mora-Corral, and Rémy Rodiac will be presented, where this borderline case was solved for hollow axisymmetric domains. Partial results leading to a solution when the axis of rotation is contained (where the dipoles found by [Conti & De Lellis, 2003] must be proved to be non energy-minimizing) will also be discussed.
2023-05-16
16:00hrs.
16:00hrs.
Rafael Benguria. Instituto de Física, PUC de Chile
A variational formulation for Dirac Operators in bounded domains and applications to spectral geometric inequalities
Sala 1
Abstract:
A variational formulation for Dirac Operators in bounded domains and applications to spectral geometric inequalities
Sala 1
Abstract:
In this talk I will present spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. A non-linear variational formulation to characterize its principal eigenvalue will be presented. This characterization allows for a simple proof of a Szegö type inequality as well
as a new formulation of a Faber-Krahn type inequality for this operator. Moreover, strong numerical evidence supporting the existence of a Faber-Krahn type inequality, will be given.
This talk is based on joint work with Pedro Antunes, Vladimir Lotoreichik, and Thomas Ourmières-Bonafos.
Contáctenos
Edificio Rolando Chuaqui, Campus San Joaquín. Avda. Vicuña Mackenna 4860, Macul, Chile.Teléfono (+56) 95504 4511.
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