34676 - Advanced Analysis 2

Course contents

FIRSTPART:

The Brouwer degree. The Axioms and their consequences. The Brouwer fixed point Theorem. The Rouchè Theorem. The Borsuk Theorem. The Borsuk-Ulam Theorem. The open map Theorem. The Perron-Frobenius Theorem. Applications to mathematical analyisis, to geometry and to the study of differential equations. Construction of the Brouwer degree.

The Leray-Schauder degree in Banach spaces. Construction and main properties. The Schauder fixed point Theorem. Applications to the study of differential equations.

SECOND PART (lectured by Dr. Andrea Bonfiglioli):

The second part of the course (lectured by dr. A. Bonfiglioli) consists of an introduction to stratified Lie groups and their sub-Laplacians. The course is a good occasion to get in touch with a very recent topic of investigation in partial differential equations (id est, the study of sub-Laplacians on Carnot groups), by using only elementary Calculus, basic (Linear) Algebra, and to see exemplifications of important topics in Lie group theory.

1st Lecture:
Introductory stuff, definitions, notation:
vector fields in R^N, integral curves, Lie algebras (of vector fields and abstract Lie algebras), composition of vector fields, "exponential-type" maps for vector fields, the "second order" CBHD (Campbell-Baker-Hausdorff-Dynkin) formula; Lie group structures on R^N.

2nd:
Characterizations of the Lie algebra of a Lie group on R^N. "Jacobian" bases. Notion of the H"ormander rank condition (possible geometrical meanings). Completeness of the vector fields in the Lie algebra of a Lie group. The Exponential Map: definition, well-posedness and first properties.

3rd:
Explicit computations of exponential maps in significant cases (possibly, for non-nilpotent groups; examples of non-invertible exponential maps, etc). Definition of homogeneous Lie group. Examples: The Heisenberg groups (plus H-type groups?); the Kolmogorov operator; the Engel group.

4th:
Properties of the homogeneous groups: nilpotency, global invertibility of Exp, polynomiality and ?pyramid-shape? of the group law. Further remarks on the CBHD Formula (and on the Connectivity Theorem of Chow).

5th:
Dilations on the homogenous Lie groups and on their Lie algebras. "Canonical" forms. Definition of (homogeneous) Carnot group and of sub-Laplacian. Examples.

6th:
Further basic properties of the sub-Laplacians (divergence form, selfadjointedness, hypoellipticity, etc). "Intrinsic" gradient with some properties (geometrical/"subelliptic" meaning).

7th:
Picone's Weak Maximum Principle with complete proofs.

8th:
Propagation of Maxima; the Hopf Lemma; the Theorem of Nagumo-Bony (all with complete proofs).

9th:
Corollaries of the weak maximum principle. The Dirichlet problem for a sub-Laplacian (uniqueness of the solution). Some comparison principles. Notion of fundamental solution and homogeneous norms: basic properties. Mollifiers on Carnot groups.

10th:
Properties of the fundamental solution (with complete proofs): homogeneity, symmetry, positivity, pole.

11th:
Mean value formulas (surface and solid) with complete proofs.
[Some recalls on the Hausdorff measures and on the divergence theorem may be needed here: these may be given during an auxiliary lecture].

12th:
Corollaries (without proofs) of the mean value formulas: Harnack and Liouville Theorems. Gauss-Koebe Theorem. Brelot Axiom.

Readings/Bibliography

Lloyd N.G., Degree Theory, Cambridge University Press.

Deimling K., Nonlinear Functional Analysis, Springer.

Pini B., Lezioni di Analisi Matematica di II livello - Parte I, Clueb.

Bonfiglioli, A.; Lanconelli, E.; Uguzzoni, F. Stratified Lie groups and potential theory for their sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin, 2007.

Assessment methods

oral examination

Office hours

See the website of Francesco Uguzzoni

See the website of Andrea Bonfiglioli