LieLowerBounds · Lower bounds for partial differential operators on compact Lie groups
„Хоризонт 2020“ — Действия „Мария Склодовска-Кюри“
- Период
- 2019-06-01 → 2021-05-31
- Финансиране от ЕС
- 166 320 €
- Участници
- 1
- Схема
- MSCA-IF-EF-ST
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Накратко на български
Математическите неравенства за псевдодиференциални оператори върху компактни групи на Ли се изследват чрез примери като уравненията на Шрьодингер. Те помагат за разбирането на физически процеси в нелинейната оптика и при кондензацията на Бозе-Айнщайн.
Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.
Резултати накратко
Lower bounds for partial differential operators on compact Lie groups
The problem we addressed in this project, which is of theoretical nature, concerns the validity of some fundamental lower bounds and other a priori estimates for pseudo-differential operators on Lie groups primarily of compact type. Lower bounds are inequalities involving pseudo-differential operators with suitable properties. In the Euclidean setting such inequalities have been proved to be powerful tools to study a wide range of problems arising in mathematical analysis, especially problems related to partial differential equation. Some problems to which the aforementioned bounds apply are, for instance, the study of hypoellipticity, solvability and well-posedness of initial value problems for evolution equations. In the manifold setting not much is known about the validity of these inequalities for non-elliptic operators, therefore the understanding of the questions listed above is much limited with respect to the Euclidean case. More in detail, we are interested in the application of a priori estimate in the resolution of problems concerning degenerate operators on compact Lie groups, in particular time-degenerate Schrödinger operators. These equations have recently attracted the attention of the mathematicians and physicists and they naturally appear in the study of Bose-Einstein condensations and nonlinear optics. Among manifolds Lie groups are certainly of high importance, not only because they have nice geometric properties, but also because we encounter such structures in many physical situations. Our objective here is to derive some unknown fundamental lower bounds and other inequalities for pseudo-differential operators of non-elliptic type in the Lie group setting. The final scope is to use these results to solve solvability and well-posedness problems for degenerate partial-differential operators.
Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз
Цел на проекта
The theory of partial differential operators is one of the most important branches of mathematics with several consequences in many other mathematical fields and with applications in other sciences. This project, which is of theoretical nature, intends to investigate the validity of the Fefferman-Phong, the Hörmander and the Melin inequalities for partial differential operators, and in general for pseudo-differential operators, on compact Lie groups, and apply them to the problem of solvability of degenerate partial differential operators. The analysis of partial differential operators requires the study of geometric quantities attached to the operators, in particular, the (total) symbol, the principal symbol and the subprincipal symbol. However, in the context of compact Lie groups, the principal symbol is globally well-defined but it is not the same for the other symbols mentioned above. Our goal is to define in a suitable way the other geometric quantities needed in the analysis of the problem and use them to obtain lower bounds for partial differential operators on compact Lie groups (i.e. the Fefferman-Phong, the Hörmander and the Melin inequalities). These lower bounds will be used to treat the problem of solvability of partial differential operators on compact Lie groups. We remark that the validity of these inequalities will yield the development of several results in the theory of partial differential equations on compact Lie groups, as, for instance, in the problems related to solvability, hypoellipticity, and well-posedness of the (weakly-hyperbolic) Cauchy problem.
Оригинален текст от CORDIS (на английски).
Участници
- UNIVERSITEIT GENT · GentКоординаторБелгия
Връзки
Данни: CORDIS, © Европейски съюз