Italian or English (if non-italian students attend the class)
Course Content
Calculus of variations. Sobolev spaces and elliptic partiial differential equations of second order. Unique continuation properties for partial differential equations
1. L.C. Evans, Mathematical Methods for Optimization. Dynamic Optimization. https://math.berkeley.edu/~evans/math%20195%20notes.pdf
2. I. Gelfand, Fomin S. Calculus of Variations, Dover, 2000.
3. M.L.Krasnov et al, Problems and exercises in the calculus of variations, Mir, 1975.
4. G.Talenti, Colesanti A., Salani P., Un’introduzione al calcolo delle variazioni, Unione Matematica Italiana, 2016.
PART II
1. L. C. Evans. Partial Differential Equations. AMS 1998.
2. F. John. Partial Differential Equations. Springer (4th edition).
3. Notes provided by the teacher.
Learning Objectives
The course aims to acquaint the students with the basic and advanced knowledge and techniques that are necessary to understand and study the modern theory of (elliptic and parabolic) partial differential equations, linear and nonlinear, and some related questions in the field of calculus of variations. At the end of the course the student will learn most of the modern techniques in the study of PDEs and CV. One of the goals is to let the students develop the basic technical skills and the critical thinking needed to apply the acquired knowledge to modeling and solving of problems in different settings, not purely mathematical. Special attention will be paid to help the students to develop communication skills necessary for teamwork. The course covers topics and provides learning skills that are important, or strongly suggested, to undertake a research career in the field of partial differential equations.
Prerequisites
BASICS of Calculus, ODEs, Linear Algebra and Geometry, Real and Complex Analysis.
Teaching Methods
Lectures: Presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.
Training sessions: training of the students to solving a selection of problems with particular emphasis on the study of geometric properties of the solutions of PDEs. The training sessions are conducted so as to help the students to apply and communicate the theoretical knowledge, to improve their ability of independent reasoning.
Moodle learning platform: online teacher-student interaction, posting of additional notes and bibliography, supplementary exercise sheets.
Remark: The suggested reading includes supplementary material that may be useful for further personal studies on the subject.
Further information
Even if, for administrative reasons, this class is in the list of choices in Analysis designed for the Indirizzo Applicativo, the class is a good choice also for the students of the Indirizzo Generale, especially for those who are interested in a career in reaserch.
Type of Assessment
Oral examination: more specifically, seminars on a topic agreed with
each and every student
Course program
PART I CALCULUS OF VARIATIONS
1.1. Introduction. Basic problem of the calculus of variations (CV): path of minimal length, minimal surface of rotation, brachistochrone, Newton’s problem of minimal resistence.
1.2. Functionals, integrands. Types of the minima of the functionals. First variation and first-order minimality conditions. Euler-Lagrange (E-L) equation.
1.3. Integration of the E-L equation. Symmetries, reduction of order of the E-L equation and conservation laws. Noether’s theorem.
1.4. Problems of the CV with various boundary conditions. Transversality conditions.
1.5. Integral constraints and isoperimetric problem of the CV.
1.6. Solution of various classical problems of the CV by the use of E-L equation.
1.7. Second variation and second-order minimality conditions. Legendre condition.
1.8. Weierstrass condition for strong minima.
1.9 Positivity of the second variation and Jacobi’s equation. Conjugate points.
1.10. Fields of extremals and strong minimality for the problems of the CV.
1.11. Existence of minimizers in the problems of the CV. Tonelli’s Theorem.
1.12. Multidimensional CV. Problem setting; types of minima.
1.13.First variation and multidimensional E-L equation.
1.14. Classical partial differential equations as multidimensional E-L equations. Variational principles.
1.15. Multidimensional isoperimetric problem.
1.16. Second variation and second-order minimality conditions for the multidimensional CV.
1.17. Solution of some problems of the multidimensional CV with the use of the first and second variations.
PART II . PARTIAL DIFFERENTIAL EQUATIONS.
1. Definition of Sobolev spaces with integer exponent. Density Theorems. The dual space of H_0^1. Notes about Sobolev spaces with noninteger exponent. Sobolev spaces and Fourier trasform. Trace. Embedding Sobolev Theorems. Morrey inequality. Compactness theorems.
2. Variational formulation of some boundary value problems for second order elliptic partial differential equations. Dirichlet and Neumann problem. Lax - Milgram Theorem. L^2 regularity theory for weak solutions: (i) interior regularity, (ii) boundary regularity for Dirichlet boundary value problem. Dirichlet to Neumann map. Inclusion inverse problem.
3. Unique continuation properties and Cauchy problems for partial differential equations.
a) Equations in the analytical field. Cauchy Kovalevski Theorem. Holmgren Theorem.
b) Hadamard’s definition of well-posed problem
c) Uniqueness and continuous dependence of solutions to equations with nonanalytic coefficients. Stability estimates for elliptic equations of second order. Introduction to the method of Carleman estimates. Three sphere inequality.