ANALISI SUPERIORE

Basics on Functional Analysis an applications to differential operators.

Elements of distributions theory. Distribution functions and distribution measures. Relationships. Sequences of distributions. Derivative of a distribution. Product between a regular function and a distribution. Comparison with the classical derivative.

Sobolev Spaces. Definitions and elementary properties. The spaces H¹(]a,b[) and H¹₀(Ω). Poincaré inequality. Dual space of H¹₀(Ω). Local and global approximations. Traces*. Rellich theorem*. Sobolev embedding theorems*.

Elements of Calculus of Variations. Some typical problems. Functionals of the calculus of variations. First and second variations of a functional. Du Bois-Reymond lemma, Eulero-Lagrange equation. Particular cases. Wirtinger inequality. The brachistochrone problem. Functionals of many variables.

Direct methods. Global minimum principle. Compacteness criteria: Ascoli-Arzelà theorem, weak compactness in Lebesgue spaces* and in Sobolev spaces*. Carathéodory functions. Lower semi-continuity and existence theorems of Tonelli. Regularity problems.

Elements of critical points theory. Differential calculus in Banach spaces. Nemitski operator. Potential operators. Critical points via minimization methods. Weak solutions of the Dirichlet problem associated to a semi linear elliptic equation.

Fixed points in metric spaces. Contractions theorem. Global existence and uniqueness theorem for the Cauchy problem for an ordinary differential equation. Caristi theorem*. Non expansive applications, Browder-Kirk theorem. Brouwer theorem*. Extensions.

Schauder-Tychonoff theorem. Partition of the unity*. Basics on locally convex spaces. Schauder-Tychonoff theorem. Consequences: Schauder theorem, zeroes of a function on a Banach space, a sufficient condition for the surjectivity of a function on a Hilbert space. Leray-Schauder condition, Schaefer theorem. Application to an abstract elliptic problem.

The proofs signed with * can be omitted.

1. H. BREZIS, Analisi Funzionale, Liguori Editore, Napoli, 1986.

2. G. BUTTAZZO – M. GIAQUINTA – S. HILDEBRANDT, One-dimensional variational pro-blems, Claredon Press, Oxford, 1998.

3. L.V. KANTOROVICH – G.P. AKILOV, Analisi Funzionale, Editori Riuniti, Roma, 1980.

4. C.D. PAGANI – S. SALSA, Analisi matematica 2, Zanichelli, Bologna, 2016.