46146 - Nonlinear Analysis

Course contents

The course is splitted in two units delivered by prof. Fausto Ferrari and prof. Francesco Uguzzoni.

Francesco Uguzzoni's unit:

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.

Fausto Ferrari's unit:
   
The course will introduce the viscous solution concept. The main learning objective that seeks this course is to provide the essential tools for the study of the solutions of nonlinear equations uniformly elliptic. Among them, as a particular case, we find also the linear  elliptic equations in  non-divergence form. The genuine non-linear equations that will be introduced are the ones associated with the extremal Pucci operators. Moreover, we will present Alexandoff estimates, the maximum principle in this context and the Harnack inequality. With these tools we will proceed to the discussion of the regularity of solutions of simple non-linear equations, especially in the case  convex/concave equations.

Readings/Bibliography

Unit prof. Francesco Uguzzoni:

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.

Unit prof. Fausto Ferrari:

L. Caffarelli, X. Cabré,  Fully nonlinear elliptic equations AMS Colloquim pubblications, volume 43.  
D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, classics in mathematics, reprint of the 1998 edition, Springer. 

Teaching methods

The course includes the development of theoretical lessons exercises and applications.

Assessment methods

Prof. Ferrari: the score of the modulus will be fixed during a colloquium about the subjects introduced during during the lessons.  
Prof. Uguzzoni: the score of the modulus will be fixed during a colloquium about the subjects introduced during during the lessons.
 
THE FINAL SCORE WILL BE THE AVERAGE OF THE SCORES OBTAINED IN THE TWO MODULUS.

Teaching tools

Further details concerning the unit delivered by prof. Fausto Ferrari  will be communicated during the lessons by the teacher or, possibly,  by some warning readable on his nonofficial web page .

Further details, bibliography and exercises concerning the unit of prof. Francesco Uguzzoni will be communicated during the lessons.

Links to further information

http://www.unibo.it/SitoWebDocente/default.htm?UPN=fausto.ferrari@unibo.it

Office hours

See the website of Fausto Ferrari

See the website of Francesco Uguzzoni