Special Session 114: Recent Advances in Partial Differential Equations and Harmonic Analysis

p-Ellipticity for Elliptic Systems with Complex Coefficients

Andrea Carbonaro University of Genoa Italy Co-Author(s):

Abstract: We extend the concept of p-ellipticity for scalar elliptic operators---introduced by us in 2016---to the case of elliptic systems with complex coefficients. Although our condition turns out to be equivalent to a condition for systems (equally named) by Dindos, Li and Pipher (2021), the two approaches are different and yield different results. Our primary goal is to develop an algebraic and geometric framework for studying p-ellipticity in the systems setting. We investigate how the structural properties of fourth-order coefficient tensors determine the range of p-ellipticity, and how this range encodes the anisotropic and asymmetric features of the tensor. We also aim to explicitly determine or estimate the range of p-ellipticity for significant concrete systems and notable classes of tensors. As an application of our theory, we establish contractivity on L^p of the semigroup generated by the associated elliptic operators, under Dirichlet, Neumann, and mixed boundary conditions.

Shifted Maximal Functions and the Boundedness of Rough Singular Integral Operators

Andrew Haar Charles University Czech Rep Co-Author(s):

Abstract: A classical problem at the intersection of harmonic analysis and PDE is the boundedness of the so-called Calderon commutators, which, not being of convolution type, require creative tools to analyze. Indeed, we will see a new approach to these pioneered by C. Muscalu in 2014 through shifted variants of certain square and maximal functions, which have gone on to become rather fashionable tools in the last decade, especially in the theory of multilinear singular integrals. We will see in the context of some recent results on multilinear Fourier multipliers how these shifted operators naturally appear. Then, we will conclude by briefly investigating how these results can be applied to prove boundedness for a general class of rough multilinear singular integrals, which include the Calderon commutators as a special case.

Inequalities of the electromagnetism for H\{o}lder continuous functions

Massimo Lanza de Cristoforis Dipartimento di Matematica `Tullio Levi-Civita` Italy Co-Author(s):

Abstract: We plan to prove some classical inequalities of the electromagnetism for H\{o}lder continuous functions that may have infinite Dirichlet integral. Thus for functions that do not belong to the classical variational setting.

Low regularity solutions of the nonlinear Schr\{o}dinger equation on the spatial quarter-plane

Dionyssis Mantzavinos University of Kansas USA Co-Author(s):    Turker Ozsari

Abstract: We will discuss the Hadamard well-posedness (existence, uniqueness and continuous dependence on the data) of the nonlinear Schr\{o}dinger equation with power nonlinearity formulated on the spatial quarter-plane with Sobolev initial data and nonhomogeneous Dirichlet boundary data in appropriate Bourgain-type spaces. We will work in a low regularity setting, where Strichartz estimates are necessary in addition to the standard Sobolev estimates. An interesting feature of this problem when compared to related works in the literature is that both of the spatial variables are restricted to the half-line. For this reason, a new approach is needed than the one previously used for the well-posedness of other initial-boundary value problems. An additional challenge is posed by the fact that the spatial domain (quarter-plane) has a corner at the origin, thus requiring one to expand the validity of certain Sobolev extension results to the case of a domain with a non-smooth (Lipschitz) and non-compact boundary. This is joint work with Turker Ozsari.

Transmission Problems in Planar Domains

Marius Mitrea Baylor University USA Co-Author(s):

Abstract: In this talk, I will present recent advances in the analysis of transmission boundary value problems on rough domains, with data prescribed from within broad scales of function spaces. The central result provides a purely algebraic condition that fully characterizes when the problem is (Fredholm) solvable.

Maximal operators of Zygmund type

Laurent Moonens Universite Paris Saclay France Co-Author(s):    Galia Dafni, Emma D`Aniello, Giorgi Oniani

Abstract: We will present recent results about maximal operators of Zygmund-type, i.e. of the form: $$ Mf(x):=\sup_{B\in B(x)} \frac{1}{|B|} \int_B |f|, $$ where, for each $x$ (in the $n$-th dimensional Euclidean space), $B(x)$ is a collection of parallelepipeds parallel to the coordinate axes (or, more generally, of convex sets) containing, or being ``close to'' $x$. Depending on the geometry of sets in the collections $B(x)$, one may wish to understand the optimal weak-type inequality $M$ enjoys. The talk will detail a few famous cases, and provide some new results on this front. If time permits, we shall also discuss some issues related to their behavior in BMO or ``moderated'' BMO spaces.

On some singular integrals of Brascamp-Lieb type

Camil Muscalu Cornell University USA Co-Author(s):    Cristina Benea

Abstract: The goal of the presentation is to describe some recent results obtained in joint work with Cristina Benea about some natural singular integrals of Brascamp-Lieb type.

A Geometric Perspective on Poincar\`e-Sobolev Inequalities thru Harmonic Analysis and without derivatives

Carlos Perez Moreno Ikerbasque at BCAM and University of The Basque Country Spain Co-Author(s):    Iker Gardeazabal-Emiel Lorist and Alejandro Claros-Linfei Zheng

Abstract: In this lecture, we present a modern perspective on function smoothness and regularity by employing sharp tools from Harmonic Analysis. The first part of the talk moves beyond classical pointwise analysis to explore functional smoothness through the lens of local averages. This approach provides a more flexible and robust framework for understanding analytical behavior, particularly in settings where traditional derivatives are not or cannot be considered. In the second part, we apply these harmonic analysis techniques to derive improved local Poincar\`e-Sobolev estimates, which are instrumental in proving the celebrated De Giorgi regularity theorem.This framework also provides a proof of the well known John-Nirenberg theorem for $BMO$ functions. The central theme of the discussion will be the self-improving property, a remarkable phenomenon where modest local control over oscillations leads to significantly stronger global integrability. As another application, we present an extension of the celebrated Nash inequality (which yields another proof of De Giorgi`s theorem), as well as an improved generalization of the Gagliardo,Nirenberg,Sobolev theorem using Campanato spaces.

Stein-Weiss type inequality in L1 norm for vector fields and applications

Tiago Picon University of Sao Paulo Brazil Co-Author(s):

Abstract: In this talk, we investigate the endpoint case $p = 1$ of the Stein-Weiss inequality for the Riesz potential. Our main result provides a characterization of this inequality for a special class of complex vector fields associated with cocanceling operators. As an application, we recover and extend several classical inequalities, and we also establish new solvability results for equations involving canceling and elliptic differential operators acting on measures. This work is based on joint research with Jorge Hounie (UFSCar, Brazil), Pablo De Napoli (Universidad de Buenos Aires, Argentina), Victor Biliatto (USP, Brazil), and Joel Coacalle (USP, Brazil)

Layer potential operators for transmission problems on extension domains

Anna Rozanova-Pierrat CentraleSup\`elec, University Paris-Saclay France Co-Author(s):    G. Claret, M. Hinz, A. Teplyaev

Abstract: We use the well-posedness of transmission problems on classes of two-sided Sobolev extension domains to give variational definitions for (boundary) layer potential operators and Neumann-Poincar\`e operators. These classes of domains contain Lipschitz domains, and also domains with fractal boundaries. Although our variational formulation does not involve any measures on the boundary, we recover the classical results in smooth domains by considering the surface measure on the boundary. We discuss properties of these operators and generalize basic results in imaging beyond the Lipschitz case.

The inhomogeneous six-wave kinetic equation

Luisa M Velasco University of Texas at Austin USA Co-Author(s):

Abstract: Six-wave interactions arise in several physically relevant systems, including photon cascades in optical wave turbulence and Kelvin-wave interactions in superfluids in quantum wave turbulence. While four-wave interactions are typically dominant in systems of nonlinearly interacting waves, the resonant interactions corresponding to the dispersion relation $\omega = k^2$ are trivial in one dimension. As a result, these interactions do not appear in the statistical description of the one-dimensional system. In this setting, higher-order interactions must be considered, and the six-wave kinetic equation provides the relevant model. We initiate the analysis of the Cauchy problem for the spatially inhomogeneous six-wave kinetic equation, which is derived from the quintic nonlinear Schrodinger equation. More precisely, we prove the existence and uniqueness of nonnegative mild solutions. We also analyze their long-time behavior, proving scattering and bijectivity of the corresponding wave operators. This is joint work with N. Pavlovic and M. Taskovic.