Noticias en español
Asymptotic behavior and rigidity in one-phase free boundary problems.
Abstract: In this talk, we will present some results concerning the behavior of solutions to the one-phase Bernoulli problem that are modeled –either at infinitesimal or at large scales–by one-homogeneous solutions with isolated singularity. We address the uniqueness of blow-up and blow-down limits provided that the one homogeneous solution has additional symmetries (integrability through rotations), and establish a rigidity type theorem, in the spirit of Simon-Solomon, given suitable conditions on the linearized operator around the one-homogeneous solution.
Read MoreClearing-out of dipoles for minimisers of 2-dimensional discrete energies with topological singularities.
Abstract: A key question in the analysis of discrete models for material defects, such as vortices in spin systems and superconductors or isolated dislocations in metals, is whether information on boundary energy for a domain can be sufficient for controlling the number of defects in the interior. We present a general combinatorial dipole-removal argument for a large class of discrete models including XY systems and screw dislocation models, allowing to prove sharp conditions under which controlled flux and boundary energy guarantee tohave minimizers with zero or one charges in the interior....
Read MoreWell-posedness for 2D non-homogeneous incompressible fluids with general density-dependent odd viscosity
Abstract: Viscosity in fluids is often related to the dissipation of energy. However, in physical systems where the microscopic dynamics do not obey time-reversal symmetry, a non-dissipative viscosity can emerge, often referred to as “odd viscosity”. In this talk, we will consider the initial value problem for a system of equations describing the motion of two-dimensional non homogeneous incompressible fluids exhibiting odd viscosity effects. We will prove the local existence and uniqueness of strong solutions in sufficiently regular Sobolev spaces. Differently from previous...
Read MoreNonexistence of positive supersolutions for semilinear fractional elliptic equations in exterior domains.
Abstract: In this talk, our goal is to investigate the nonexistence of positive solutions to nonlinear fractional elliptic inequalities in exterior domains of Rn, n ≥ 1. Our results extend the classical Liouville-type theorems of Gidas–Spruck [3] for semilinear elliptic equations, as well as the framework of Armstrong–Sirakov [1] for supersolutions of elliptic equations, to the nonlocal setting. They are also closely related to the fundamental solution approach of Felmer–Quaas [2] for nonlinear integral operators, although our arguments require substantial modifications to address the...
Read MoreDeformation to positive scalar curvature on complete manifolds with boundary.
Abstract: We will talk about conformal deformations of the scalar curvature and mean curvature on complete Riemannian manifolds with boundary. We establish sufficient conditions for the existence of conformal deformations to complete metrics with positive scalar curvature and mean convex boundary.
Read MoreNew advances on the regularity of solutions to equations ruled by the $\infty$-Laplacian
Abstract: The theory of regularity, beyond its theoretical relevance in the study of partial differential equations, plays a central role in modeling various natural phenomena, including those arising in biology, materials science, fluid dynamics, and mathematical physics. In this talk, we will present results concerning the regularity of solutions to equations governed by the infinity Laplacian, with particular emphasis on infinity-harmonic functions, as well as recent advances on singular problems and Hénon-type equations.
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