Projects / Programmes
p-Ellipticity in Harmonic Analysis and Partial Differential Equations
| Code |
Science |
Field |
Subfield |
| 1.01.01 |
Natural sciences and mathematics |
Mathematics |
Analysis |
| Code |
Science |
Field |
| P130 |
Natural sciences and mathematics |
Functions, differential equations |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
elliptic PDE, Littlewood-Paley theory, spectral multipliers, operator semigroups, dispersive estimates, Schrödinger operators
Researchers (6)
Organisations (2)
Abstract
The content of the project primarily concerns the area of harmonic analysis and partial differential equations (PDE). Thus it lies at the junction of two very active fields of contemporary mathematics. The core of our interest is a condition named p-ellipticity, that was in a joint 2016 paper, accepted for publication in J. Eur. Math. Soc., introduced by the applicant and his collaborator from the University of Genova, A. Carbonaro. In the paper they, among the rest, establish connections between this condition and several phenomena concerning the Lp theory of the so-called elliptic differential operators in divergence form. Independently of them, M. Dindoš in J. Pipher almost simultaneously found a condition that permits extending the De Giorgi - Nash - Moser theorem (without doubt one of major achievements of the 20th-century mathematics) to operators with complex coefficients. It turned out that their condition is equivalent to p-ellipticity, i.e. it is precisely its reformulation. Likewise, it turned out that by means of p-ellipticity we may reformulate a similar (yet not the same) condition, considered in 2005 by A. Cialdea and V. Maz'ya in the context of the so-called Lp dissipativity of sesquilinear forms which give rise to elliptic operators, and Lp contractivity of the corresponding operator semigroups. All this clearly testifies about the significance of p-ellipticity. The primary aim of the project is to investigate the reach of this condition in harmonic analysis and PDE, that is, to create a uniform theory centered around the concept of p-ellipticity.
Apart from p-ellipticity, attention will be within the project devoted to other mathematical problems, as well (dispersive estimates for Schrödinger operators and the analysis of Lie grupoids).
The in-depth presentation of the project can be found in the attachment (pdf).
Significance for science
There are cases when different researchers independently arrive at the same result. In the case of p-ellipticity, Carbonaro and the applicant on one side, and Dindoš and Pipher (both accomplished mathematicians at very renowned universities in GB and the US) on the other, independently realized that precisely this condition determines whether the problems they studied were solvable or not. However they did so by studying different problems. This testifies that the condition is a fundamental one, and that its potential impact for the development of new research directions in harmonic analysis and PDE is overwhelming. Namely, we already know that the p-ellipticity identifies the common denominator of several different analytic phenomena (generalized convexity, bilinear embeddings, holomorphic functional calculus, semigroup contractivity, De Giorgi - Nash - Moser theory for complex matrices). These examples kept arriving one after another, following the introduction of the concept. We firmly believe that the list of examples, where p-ellipticity features as the key condition, is not exhausted. Among the rest we will (by extending the heat flow method) also investigate the solvability of elliptic equations and Cauchy problems with as mild assumptions on the matrix and the domain of the problem as possible, and under various boundary conditions, which is one of the central goals in PDE. It will thus clearly be of interest to investigate further the scope of this condition, which is the very goal of this project.
The above is especially important for the development of mathematics in Slovenia, where harmonic analysis and PDE are not represented in high numbers, although these fields are truly important on the international level (during the last 25 years, several recipients of the Fields medal, world's highest recognition for achievements in mathematics, were awarded for work in these areas). The implementation of the project will therefore expand and significantly strengthen the reach of Slovenian mathematics.
Harmonic analysis is a propulsive area of contemporary mathematics with active researchers at many universities in Europe and the US. Until now, we have not organized conferences or summer schools in harmonic analysis, hence the mere organization of such an event will mean a novelty and an opportunity for meeting excellent foreign mathematicians that have never before come to Slovenia. Since the organization will be carried out together with the University of Zagreb, we will in such a way also promote cross-border collaboration between the two universities.
It appears that at FMF there are not many foreign postdoctoral researchers in mathematics, especially mathematical analysis, at least in comparison with the level of development of similar programmes abroad. By attracting to FMF (or IMFM) a promising foreign candidate for the postdoctoral position, which is one of the goals of this project, the situation in this respect would improve. Possibly also the long-term trend would, following a positive example of this kind, turn upward.
Significance for the country
There are cases when different researchers independently arrive at the same result. In the case of p-ellipticity, Carbonaro and the applicant on one side, and Dindoš and Pipher (both accomplished mathematicians at very renowned universities in GB and the US) on the other, independently realized that precisely this condition determines whether the problems they studied were solvable or not. However they did so by studying different problems. This testifies that the condition is a fundamental one, and that its potential impact for the development of new research directions in harmonic analysis and PDE is overwhelming. Namely, we already know that the p-ellipticity identifies the common denominator of several different analytic phenomena (generalized convexity, bilinear embeddings, holomorphic functional calculus, semigroup contractivity, De Giorgi - Nash - Moser theory for complex matrices). These examples kept arriving one after another, following the introduction of the concept. We firmly believe that the list of examples, where p-ellipticity features as the key condition, is not exhausted. Among the rest we will (by extending the heat flow method) also investigate the solvability of elliptic equations and Cauchy problems with as mild assumptions on the matrix and the domain of the problem as possible, and under various boundary conditions, which is one of the central goals in PDE. It will thus clearly be of interest to investigate further the scope of this condition, which is the very goal of this project.
The above is especially important for the development of mathematics in Slovenia, where harmonic analysis and PDE are not represented in high numbers, although these fields are truly important on the international level (during the last 25 years, several recipients of the Fields medal, world's highest recognition for achievements in mathematics, were awarded for work in these areas). The implementation of the project will therefore expand and significantly strengthen the reach of Slovenian mathematics.
Harmonic analysis is a propulsive area of contemporary mathematics with active researchers at many universities in Europe and the US. Until now, we have not organized conferences or summer schools in harmonic analysis, hence the mere organization of such an event will mean a novelty and an opportunity for meeting excellent foreign mathematicians that have never before come to Slovenia. Since the organization will be carried out together with the University of Zagreb, we will in such a way also promote cross-border collaboration between the two universities.
It appears that at FMF there are not many foreign postdoctoral researchers in mathematics, especially mathematical analysis, at least in comparison with the level of development of similar programmes abroad. By attracting to FMF (or IMFM) a promising foreign candidate for the postdoctoral position, which is one of the goals of this project, the situation in this respect would improve. Possibly also the long-term trend would, following a positive example of this kind, turn upward.
