Introduzione al Calcolo delle Variazioni
Roberto Monti
Matematica – Anno Accademico 2016-17
Versione Preliminare degli Appunti del Corso – 15 Marzo 2017 E-mail address: monti@math.unipd.it
3
Indice
(1) Metodo diretto del Calcolo delle variazioni p.1 (2) Funzionali classici
(a) Equazioni di Eulero-Lagrange p.6
(b) Equazione di Du Bois-Reymond p.11
(c) Metodo di convessit`a (metodi indiretti) p.12 (d) Principio di Fermat per l’ottica geometrica p.14
(e) Problema della brachistocrona p.18
(f) Funzionali del solo gradiente. Condizione di pendenza limitata p.23 (3) Funzionali sugli spazi di Sobolev
(a) Elementi essenziali sugli spazi di Sobolev p.36 (b) Convessit`a e semicontinuit`a inferiore in W1,p p.44
(c) Esistenza dei minimi in W1,p p.49
(d) Esempi p.51
(4) Funzioni a variazione limitata
(a) Definizione e Teorema di Riesz p.62
(b) Decomposizione della misura gradiente distribuzionale p.71 (c) Semicontinuit`a inferiore e approssimazione p.74 (d) Teorema di compattezza e disuguaglianza di Poincar´e p.78
(e) Tracce ed estensioni p.83
(f) Propriet`a fini e funzioni SBV p.85
(g) Funzionale di Mumford-Shah p.89
(5) Insiemi di perimetro finito
(a) Definizione ed esempi p.94
(b) Una soluzione del problema di Plateau p.97
(c) Frontiera ridotta e stime di densit`a p.101
(d) Blow-up della frontiera ridotta p.110
(e) Struttura della frontiera ridotta p.115
(6) Superfici minime
(a) Formula dell’area per grafici C1 p.120
(b) Formula dell’area nel caso generale p.127
(c) Superfici minime p.143
(d) Formula di rappresentazione di Weierstrass p.145 (e) Formula di monotonia per insiemi stazionari p.153 (7) Teorema isoperimetrico
(a) Riarrangiamento di Steiner p.160
(b) Propriet`a isoperimetrica della sfera p.169
(c) Riarrangiamento di Schwarz p.173
(d) Formula di coarea p.182 (8) Γ-convergenza
(a) Rilassamento p.185
(b) Γ-limiti p.187
(c) Funzionale di Modica-Mortola p.193
(9) Cenni alla teoria del trasporto ottimo
(a) Problema di Monge p.201
(b) Formulazione di Kantorovic p.204
(c) Teorema di Brenier p.212
(d) Applicazione alla disuguaglianza isoperimetrica p.214 (10) Cenni alla teoria delle correnti
(a) Correnti, massa e bordo p.217
(b) Correnti rettificabili. Problema di Plateau p.225
(c) Teorema di deformazione p.232
(d) Le variet`a olomorfe sono minime p.240
(e) Coni di Simon p.251
(11) Esercizi (12) Bibliografia
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