Differential Equations

Continuity for maximal operators at the derivative level.

Event Date: Jan 06, 2022 in Differential Equations, Seminars

Abstract: Maximal operators are central objects in harmonic analysis. The oscillatory behavior of such objects has been an important topic of study over the last decades. However, even in the dimension one there are 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.

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The fibering method applied to the level sets of a family of functionals.

Event Date: Dec 16, 2021 in Differential Equations, Seminars

Abstract:  Given an one-parameter family of C1-functionals, Φμ : X →R, defined on an uniformly convex Banach space X, we describe a method that permit us find critical points of Φμ at some energy level c ∈ R. In fact, we show the existence of a sequence μ(n,c), n ∈N, such that Φμ(n,c) has a critical level at c ∈ R, for all n ∈ N. Moreover, we show some good properties of the curves μ(n,c), with respect to c (for example, they are Lipschitz), and as a consequence of this analysis, we recover many know results on the literature concerning bifurca- tions of elliptic partial differential...

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Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation.

Event Date: Dec 09, 2021 in Differential Equations, Seminars

Abstract: In this talk we will discuss recent results regarding local well-posedness for the generalized Zakharov–Kuznetsov equation. We prove a high-dimensional version of the Strichartz estimates for the unitary group associated with the free Zakharov-Kuznetsov equation. As a by-product, we deduce maximal estimates which allow us to prove local well-posedness for the generalized Zakharov-Kuznetsov equation in the whole subcritical case whenever d\ge 4, k\ge 4 complementing the recent results of Kinoshita and Herr-Kinoshita. Finally, we use some of those maximal estimates in order to...

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Aharonov–Casher theorem on manifolds with boundary and APS boundary condition.

Event Date: Dec 02, 2021 in Differential Equations, Seminars

Abstract: The Aharonov–Casher (AC) theorem is a result from 1979 on the number of the so-called zero modes of a system described by the magnetic Pauli operator in $\R^2$. In this talk I will address the same problem for the Dirac operator when $\R^2$ is exchanged by a certain two dimensional manifold with a boundary. More concretely I consider a plane, disc and sphere with a finite number of circular holes cut out. The magnetic field consists of two contributions; a smooth compactly supported field on the manifold, and Aharonov—Bohm solenoids generating magnetic field inside the holes....

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Perturbations of interpolations formulas.

Event Date: Nov 25, 2021 in Differential Equations, Seminars

Abstract: In this talk we will discuss some problems related to the theory of Fourier interpolation. The goal is to talk about the general problem of how to obtain new interpolation formulas from a previously known one by some perturbation argument, and also mentions some recent developments in joint work with João Pedro Ramos (ETH Zürich). This talk is meant for a broad audience with basic knowledge in analysis.

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Symmetric positive solutions to nonlinear Choquard equations with potentials.

Event Date: Nov 11, 2021 in Differential Equations, Seminars

Abstract: I will present some existence results for a class of Choquard equations in which the potential has a positive limit at infinity and satisfies suitable decay assumptions. Also, it is taken invariant under the action of a closed subgroup of linear isometries of RN. As a consequence, the positive solution found is invariant under the same action. I will mainly focus on the physical case involving a quadratic nonlinearity. Joint work together with Liliane Maia and Benedetta Pellacci.

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