Italian (yet both instructors are fluent in English)
Course Content
Introduction to the Calculus of Variations (CoV) and Perspectives in Partial Differential Equations (PDE):
i) classical methods in the CoV, Euler-Lagrange equations; direct methods for integral functionals; some applications;
ii) functional-analytical approach to evolutionary PDE: fundamental methods and tools (operator semigroups theory), a choice of associated questions/problems and applications (stability properties, optimal control of PDE).
Some bibliographical references (alphabetical order in either thematic area):
i) Calculus of Variations:
- G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-dimensional Variational Problems, Oxford University Press, 2008.
- B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, 2015.
- B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, 2007.
- E. Giusti, Metodi diretti nel calcolo delle variazioni, UMI, 1994.
ii) Evolutionary PDE and Infinite Dimensional Systems, Semigroup Theory, Mathematical Control Theory:
- P. Acquistapace, Appunti di teoria dei semigruppi, https://people.dm.unipi.it/~acquistp/teosg.pdf
- 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/
- 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 variational or analytical-functional methods for the study of linear partial differential equations.
Skills acquired at the end of the course: ability to analyze certain minimum problems, 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
Basic elements of functional analysis. Introductory elements to partial differential equations. Fundamentals of the theory of ordinary differential equations.
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
CALCULUS of VARIATIONS
Introductions to the problems of the Calculus of Variations. Fundamental lemmas. Euler-Lagrange equations. Minimality and convexity.
Methods in the Calculus of Variations and weak solutions: Dirichlet principle; the bounded slope condition; direct methods; semicontinuity results.
Applications to continuum mechanics.
BOCHNER INTEGRAL for functions f: I ---> X, where I is a real interval and X is a Banach space. L^p(I,X) and W^{1,p}(I,X) spaces.
OPERATOR THEORY. Linear operators in Banach spaces: bounded, closed, preclosed (or closable) operators. Respective characterizations, illustrative examples.
OPERATOR SEMIGROUPS THEORY. Introduction. Strongly continuous semigroups, asymptotic properties of the semigroup, the "sharp growth bound". The semigroup generator, classical generation results (Hille-Yosida Theorem). Dissipative operators, the Lumer-Phillips theorem; groups of operators, Stone's theorem. Reformulation of initial and boundary value problems for evolution equations as Cauchy problems in infinite-dimensional spaces. Strict, strong, mild, weak solutions. Illustration of the use of the aforementioned results. (Sectorial operators and analytical semigroups.)*
A glimpse of MATHEMATICAL CONTROL THEORY. Linear control systems. Optimal control and feedback control. The linear-quadratic case and (differential and algebraic) Riccati equations. The general case, the Hamilton-Jacobi-Bellman equation (outline). Emphasis on the interplay between functional-analytical methods and PDE methods. Multipliers/energy methods for the study of uniform stability.