Italian (yet both instructors are fluent in English)
Course Content
Perspectives in linear partial differential equations (PDE):
i) geometric approach to boundary value problems for elliptic and parabolic PDE (study of geometric properties of solutions, such as convexity /stellarity of level lines);
ii) functional-analytical approach to initial-boundary value problems for evolutionary PDE (operator semigroups theory, with possible application to the study of related control problems).
Some bibliographical references (alphabetical order in either section):
i) PDE and geometric properties:
- H.J. Brascamp, E.H. Lieb, On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366-389.
- A. Colesanti, Brunn-Minkowski inequalities for variational functionals and related problems, Adv. Math. 194 (2005), no. 1, 105-140.
- L.C. Evans, Partial differential equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.
- B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, 1150, Springer-Verlag, Berlin, 1985.
- D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, ristampa dell'edizione del 1998, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
ii) PDE and semigroup theory, mathematical control theory:
- P. Acquistapace, Appunti di teoria dei controlli, http://people.dm.unipi.it/acquistp/teocon.pdf
- F. Alabau-Boussouira, P. Cannarsa, Control of non-linear partial differential equations, Mathematics of complexity and dynamical systems. Vols. 1-3, 102-125, Springer, New York, 2012.
- A. Bensoussan, G. Da Prato, M. Delfour, S. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd edition, Birkhauser, Boston, 2007.
- A. Lunardi, Introduzione alla teoria dei semigruppi, http://people.dmi.unipr.it/alessandra.lunardi/
- R. Nagel, K.-J. Engel, One-parameter semigroups for linear evolution equations, Springer, New York, 2000.
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, 1983.
- J. Zabczyk, Mathematical control theory. An introduction, Birkhauser, Boston 2008.
Learning Objectives
Knowledge objectives: acquisition of knowledge of the main themes and issues addressed in the course (see "Extended program")
Competence objectives: understanding of the specific mathematical language and of some geometric or analytical-functional methods for the study of linear partial differential equations
Skills acquired at the end of the course: ability to analyze certain qualitative properties of the solutions to boundary value problems for elliptic or parabolic equations, to understand and/or formulate simple problems inherent to evolution equations, to propose pertinent methods for their resolution, to interpret outcomes.
Ability to access the scientific literature for further information.
Prerequisites
Fundamentals of the theory of ordinary differential equations. Introductory elements to partial differential equations. Basic elements of functional analysis.
Teaching Methods
Lectures and discussion of exercises /problems in the classroom. Seminar activity on material provided by the instructors.
Further information
9 CFU (225 hours of workload, of which 72 hours in the classroom)
Type of Assessment
Oral exam, with a prevalent part in form of a seminar focused on a topic agreed by each student with the instructors.
Course program
GEOMETRIC PROPERTIES of PDE. Basic notions about elliptic PDE's boundary value problems. Archetypal problems: the first Dirichlet eigenvalue of the Laplace operator; the torsion problem; the capacity problem. Methods to deduce geometric properties of the solutions, from information on the domain. In particular: log-concavity of the first eigenfunction; power concavity of the torsion function; quasi concavity of the capacitary potential, in convex domains.
OPERATOR SEMIGROUPS THEORY. Introduction. Strongly continuous semigroups, the semigroup generator. Generation theorems (Hille-Yosida, Lumer-Phillips, Stone). Reformulation of initial-boundary value problems for evolutionary PDE as Cauchy problems in infinite-dimensional spaces. Strict, strong, weak, mild solutions. Semigroup formulation of PDE problems. Asymptotic properties C_0-semigroups. Sectorial operators, analytic semigroups.
MATHEMATICAL CONTROL THEORY. Linear control systems. Controllability, stability, optimal control: definitions, a few significant results (in finite dimension). The infinite-dimensional context: the interplay between functional-analytical methods and PDE methods, the role of appropriate inequalities (direct and inverse ones).