Aims and scope -

Problem solving for optimization problems and minimization of integral functionals: deduction, manipulation and solution of Euler equations and variational problems connnected to conditions of extremality.

Basic knowledge of variational inequalities and their formulation aiming to deal with constrained problems. 

These topics will be presented in the perspective of their relevant applications to problems in Mathematical Physics and Engineering.

The teaching activity of this course includes lecture hours and training sessions.

Expectations of achievements and abilities resulting from a successful attendance of this course are those associated with the Dublin Descriptors DD1, DD2, DD3, DD4.

The course provides an introduction to modern techniques in Calculus of Variations and Geometric Measure Theory. Minimization of integral functionals is studied in connection with variational formulations of boundary value problems for differential equations and problems with unilateral constraints. Some applications to continuum mechanics and image analysis are presented.

Topics presentation is focused on the acquisition and critical use of main tools in: Calculus of Variations, Functional Analysis, optimization of integral functional with and without constraint, Euler equations, interplay of integral minimization with variational formulation of PDEs and variational inequalities.

Lecture hours will allow students to know and understand the topics above (DD1) and to apply their knowledge and understanding (DD2, DD3).

Training sessions will allow students to become familiar with the theoretical concepts presented during lecture hours (DD1, DD2, DD3), and help developing ability to make judgements, and improve communication skills (DD3, DD4) to be tested by an oral examination.

PROGRAM

1. Dido's problema. Steiner's problem. First variation of an integral functional. 

2. Classical Calculus of Variations. Euler-Lagrange equations. First integrals. Null-lagrangian integrals. The direct method in Calculus of Variations.

3. Isoperimetric inequality. Wirtinger inequality. Laplace equation. Mean value property. Bounded slope condition. Hilbert Theorem concerning Lipschitz extremals. 

4. Friedrichs mollifiers. Sobolev spaces. Poincaré inequality. Traces. Convexity of integral functionals. Semicontinuity of integral functionals. Sobolev embeddings.

5. Reflexivity. Separability. Weak coinvergence. Weak-star coinvergence. Banach-Alaoglu-Bourbaki Theorem. Elliptic equations in divergence form. Clamped plate. Elasticity system.

6. Variational inequalities in Hilbert spaces. Lions-Stampacchia Theorem. Minimum problems with convex constraints. Noncoercive problems. Recession cone. Obstacle problem for membrane and plate.

7. Measures. BV functions of one or several variables. Absolutely continuous functions.

8. Hausdorff measure. Hausdorff dimension. Hutchinson's self-similar fractal sets. Rectifiable sets. Federer-Vol'pert Theorem. Cantor-Vitali function.

9. Finite perimeter sets. First variation of area functional and mean curvature. Mumford-Shah functional: strong and weak formulation. 

10. Gamma-convergence. Definitions and examples. 

Basic Mathematical Analysis  courses of Laurea Triennale in Ingegneria.

The final proof consists in an oral exam about basic concepts of the program, with closer examination on two chapters to be chosen in the program. The choice of two chapters must be communicated before the exam.

The examination consists in answering theoretical questions and solving exercises about topics of the course, in order to assess the knowledge of theorems and the ability to exploit the mathematical tools presented in the frontal lessons: direct method in calculus of variations, variational formulation of partial differential equations, functional analysis tools, Sobolev spaces and distributions, variational inequalities.

Referring to the Dublin Descriptors, the exam is aimed at evaluating acquired skills, understanding of mathematical tools and the ability to apply knowledge (DD1, DD2, DD3, DD4).

Second Edition