Societat filial de:

Interactions between analysis, probability, and partial differential equations

ORG: Juan Carlos Cantero, UAB - Odí Soler i Gibert, UPC

GENERAL DESCRIPTION

The thematic session Analysis, Probability, and Partial Differential Equations aims to showcase the connections between mathematical analysis and partial differential equations, highlighting their intersection with probability theory. We propose a structure of six short talks: two by local researchers and four by international speakers. This format is designed to enrich the collaborative network among universities in the Barcelona metropolitan area while providing an excellent platform to present our current research.

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SCHEDULE

 

Thursday October 8th (Prat de la Riba)

16:00h Caloric measure and parabolic rectifiability, by Olli Saari (Universitat Politècnica de Catalunya)

 

16.40h A duality approach to gradient Hölder estimates for linear divergence form elliptic equations, by Yuanlin Sun (Universitat Politècnica de Catalunya / University of Jilin)

 

17.20h Spider’s webs and weak type inequalities for maximal functions on Gromov hyperbolic spaces, by Nikos Chalmoukis (Università di Milano Bicocca)

 

Friday October 9th (Pere i Joan Coromines)

16:00h On Payne-Weinberger inequalities, by Tomàs Sanz (Universitat de Barcelona)

 

16.40h Geometric effects on the persitence of vortex structures, by Émeric Roulley (Università degli Studi di Milano)

 

17.20h A segregation problem with nonlinear diffusion, by Maria Soria (Universidad Autònoma de Madrid)

 

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ABSTRACTS

Caloric measure and parabolic rectifiability, by Olli Saari
Abstract. In this talk, I will recall Dahlberg’s theorem on quantitative absolute continuity of the harmonic measure and the surface measure of a Lipschitz domain and then discuss its parabolic counterpart. The parabolic Lipschitz graphs where the correct anisotropic Hausdorff measure is absolutely continuous with respect to the caloric measure are characterized in terms of a beta number condition and a parabolic version of rectifiability. This based on joint work with Simon Bortz, Moritz Egert and Sandra Ferris.

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A duality approach to gradient Hölder estimates for linear divergence form elliptic equations, by Yuanlin Sun
Abstract. This talk presents a duality approach to gradient regularity for linear elliptic equations in divergence form. We first consider \(\alpha\)-Hölder continuous coefficients and explain how an iteration argument, together with local Hardy–Hölder duality, yields \(C^{1,\alpha}\) estimates from constant-coefficient theory. We then derive a sparse bound for the gradient involving local Hardy norms. Finally, we adapt the same iteration scheme to uniformly continuous coefficients by replacing local Hardy–Hölder duality with Lebesgue duality. This yields gradient reverse Hölder inequalities for every exponent \(q\in(2,\infty)\), and consequently the corresponding sparse bounds.
This is joint work with Olli Saari, Hua-Yang Wang, and Yuanhong Wei.
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Spider’s webs and weak type inequalities for maximal functions on Gromov hyperbolic spaces, by Nikos Chalmoukis

Abstract. Let (X, d) be a Gromov hyperbolic space equipped with a locally finite Borel measure that is locally doubling and of exponential growth. In this setting, we establish sharp weak type bounds for the centered Hardy–Littlewood maximal operator. Our results aim to clarify how the mapping properties of the Hardy–Littlewood maximal operator depend on the geometry of a space which is globally negatively curved. In particular, the bounds apply to Cartan-Hadamard manifolds of pinched negative curvature and symmetric spaces of noncompact type and rank one. A key ingredient in the proof is a discretization procedure, which allows us to approximate in a convenient way an arbitrary Gromov hyperbolic space by a class of graphs which we call spider’s webs.
This is a joint work with S. Meda and F. Santagati.

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A segregation problem with nonlinear diffusion, by Maria Soria-Carro
Abstract. In this talk, I will introduce a segregation problem describing the interaction between species competing for the same resources. In the strong-competition limit, the populations segregate and occupy different regions, and the interface between them becomes a central object of study. I will first discuss a simple two-species model, where this interface can be understood through a harmonic function, and then turn to a model with nonlinear diffusion, where the interface becomes part of the unknown and must be studied as a free boundary.
This is joint work with Emily Casey (U. Minnesota), Cornelia Mihaila (S. Michael’s College), and Clara Torres-Latorre (ICMAT).
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Geometric effects on the persitence of vortex structures, by Émeric Roulley
Abstract. Vortex patches are weak solutions of the Euler equations whose vorticity is constant on a localized region of the fluid. A particularly important class is formed by V-states, namely vortex patches that undergo a rigid rotation. We prove a rigidity result showing that if a non-stationary V-state is contained in a simply connected fluid domain with real-analytic boundary, then the ambient domain must necessarily be a disc.
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On Payne-Weinberger inequalities, by Tomàs Sanz
Abstract. In this talk, I will first review the classical Payne–Weinberger inequality, which provides an upper bound on the difference between the first Dirichlet eigenvalue of a domain and that of the disk with the same volume. The bound depends only on the isoperimetric deficit of the domain. I will briefly discuss the main ideas behind its proof and explain how the same approach yields analogous inequalities for other operators, including Robin Laplacians and quantum dot Dirac operators.