Knowledge and understanding:
Know the basic concepts presented in the course.
Ability to apply knowledge and understanding:
Know how to apply the theoretical elements in the resolution of specific problems,
such as maximum and / or minimum problems, ordinary or partial differential equations.
Autonomy of judgment:
Know how to locate the most appropriate techniques in solving assigned problems or applications, even outside the specific context of the field of mathematical analysis.
Self-compiling mathematical proofs; introduce, in an oral and written way, a subject, or a mathematical theory, from those learned when attending the course.
Study independently, starting with the recommended bibliography.
Address the proposed problems by selecting the most meaningful ones independently.
In details, the program of the course may include the following arguments (whenever not yet treated in other courses):
– Functional analysis: regularization and approximation methods by convolution of integrable functions; distribution theory; Sobolev spaces; bounded variation functions.
– Calculus of Variations: the direct method, lower semicontinuity and coercivity for integral functionals; first variation and Euler equations; constrained minimization and Lagrange multipliers; relaxation and variational convergence.
– Optimal control: well posedness and optimality necessary conditions; epidemic optimal control; relaxation and variational convergence of optimal control problems.
– Differential calculus in normed vector spaces: Fréchet derivative, Gateaux derivative, local inversion theorem, global inversion theorem, applications to differential equations.
– Fixed point theorems: Brouwer fixed point theorem, Rothe theorem, Bohl theorem, Poicarè-Miranda theorem, Schauder theorem, applications to differential equations.
– Introduction to topological degree and bifurcation theory, with applications to ecological and celestial mechanics models.