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Course Program and Bibliography Marco
Bramanti's personal room: |
The aim of this course is to introduce the students to the research area of elliptic-parabolic degenerate operators of Hörmander type. We will present a general overview of this area, with some more specific insight on selected topics.
COURSE PROGRAM
In this course, we will consider second-order linear differential
operators with smooth coefficients and positive semi-definite quadratic
forms (thus, elliptic-parabolic, generally degenerate) that fall within
the class of Hörmander operators. We will introduce some important
notions, objects and theorems in this field, in particular introducing
to the literature about Hörmander operators on homogeneous groups.
The style of the course will balance technical parts, with the proof of
some important results, and informative talks, aiming to give a wider
picture to this deep subject.
By attending this course, the student is expected to familiarize with
the notions of Hörmander operators, Hörmander vector fields,
hypoellipticity, fundamental solutions, a priori estimates, geometry
related to a differential operator, appreciating the analogies and
differences between uniformly elliptic (or parabolic) operators and
these more general ones.
A tentative table of contents of the course is the following.
Introduction to Hörmander operators.
Linear differential operators with smooth coefficients in the context
of distribution theory: the notions of hypoellipticity and solvability.
The case of second-order operators; the language of vector fields,
commutators, Lie algebras. Hörmander's hypoellipticity theorem;
Hörmander operators. Notable examples: sub-Laplacians on Heisenberg
groups, Kolmogorov-Fokker-Planck operators. Motivations to study these
operators and connections with stochastic physical models. Subelliptic
estimates: their role to prove Hörmander's theorem, and their use.
Geometric properties of Hörmander operators.
Exponential of a vector field, relation between the exponential of the
commutator of two vector fields and the commutator of their exponential
maps. Control distance induced by a system of vector fields, Chow
connectivity properties; maximum principles. The role of weighted
control distance to study operators with drift.
Homogeneous groups and homogeneous left invariant Hörmander operators.
Generalities on homogeneous groups, sub-Laplacians on Carnot groups, homogeneous invariant operators on homogeneous groups.
Fundamental solutions of Hörmander operators.
Motivation: the role of the fundamental solution of the Laplacian in
the quest of a priori estimates in Sobolev spaces. Existence of a good
fundamental solution for homogeneous Hörmander operators on homogeneous
groups. Properties of this fundamental solution, and its role to prove
natural estimates in Sobolev spaces for these operators. Some remarks
on the extension of the previous theory to general Hörmander operators:
natural estimates on the fundamental solution, and relation with the
geometry of vector fields; local estimates in Sobolev spaces for
general Hörmander operators.
Kolmogorov-Fokker-Planck operators.
Generalities on the classes of KFP operators studied by
Lanconelli-Polidoro. Construction of an explicit fundamental solution
of Gaussian type. Some properties and uses of this fundamental solution.
Depending on the remaining time and the interests of participants, we could develop other related subjects.
BIBLIOGRAPHY
M. Bramanti, L. Brandolini: Hörmander operators. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2023.
M. Bramanti: An invitation to hypoelliptic operators and Hörmander's vector fields. SpringerBriefs Math., 2014.
S. Biagi, A. Bonfiglioli: An introduction to the geometrical analysis
of vector fields-with applications to maximum principles and Lie
groups. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.
A. Bonfiglioli, E. Lanconelli, F. Uguzzoni: Stratified Lie groups and
potential theory for their sub-Laplacians, Springer Monogr. Math., 2007.
The lessons will be held inside the math department (unless otherwise specified), that is:
4th floor: room of "effediesse" lab;
3rd floor: seminars' room
6th floor: MOX seminars' room.
Pay attention to the schedule, since the lessons are not held at fixed days and hours.
May, Monday 4th, 11 a.m-1 p.m., 4th floor
Tuesday 5th, 2-4 p.m, 4th floor
Wednesday 6th, 2-4 p.m, 3rd floor
Monday 11th, 11 a.m-1 p.m., 4th floor
Tuesday 12th, 2-4 p.m, 6th floor
Wednesday 13th, 2-4 p.m, 3rd floor
Thursday 14th, 11 a.m-1 p.m., 4th floor
Wednesday 20th, 3-5 p.m., 4th floor
Friday 22th, 10-12 a.m., 4th floor
Monday 25th, ore 11 a.m-1 p.m., 4th floor
Tuesday 26th, ore 3-5 p.m., 4th floor