This reference book gives to the reader a complete but comprehensive presentation of the foundations of convex analysis and presents applications to significant situations in engineering. The presentation of the theory is self-contained and the proof of all the essential results is given.
The examples consider meaningful situations such as the modeling of curvilinear structures, the motion of a mass of people or the solidification of a material. Non convex situations are considered by means of relaxation methods and the connections between probability and convexity are explored and exploited in order to generate numerical algorithms.
PART 1. MOTIVATION: EXAMPLES AND APPLICATIONS 1. Curvilinear Continuous Media.
2. Unilateral System Dynamics.
3. A Simpliﬁed Model of Fusion/Solidiﬁcation.
4. Minimization of a Non-Convex Function.
5. Simple Models of Plasticity.
PART 2. THEORETICAL ELEMENTS 6. Elements of Set Theory.
7. Real Hilbert Spaces.
8. Convex Sets.
9. Functionals on a Hilbert Space.
11. Variational Problems.