Calculus of variations

Cours Fondamental 1 (EDP, HFE)

Enseignant : Cosmin Burtea

Contact : cosmin.burtea à imj-prg.fr

Langue du cours : english

Présentation

The aim of theese notes is to introduce the reader to the calculus of variations, namely techniques that allow to study the existence of extrema of functions defined on a Banach space. These notes are largely inspired from classical refferences such as those signed by Bernard Dacorogna, Filip Rindler or Filippo Santambrogio or the "polycopie" of J.-F. Babadjian. The more "analysis"-oriented chapters are inspired by the books of H. Brezis, L. Grafakos, G. Leoni and H. Royden.

Contenu

• First week : I. Introduction : a few problems from mathematical physics and the Euler-Lagrange equations. II. Abstract Direct Method.
  ☐ Notes ☐ TD 1
• Second week : III. Measure theoretical results : Egoroff, Lusin. IV. Weak Lp-spaces and interpolation.
  ☐ Notes ☐ TD 2
• Third week : V. Sobolev spaces : approximation by smooth functions. VI. Sobolev spaces : embeddings. Rellich-Kondrachov, Poincaré's Lemma.
  ☐ Notes ☐ TD 3
• Fourth week : VII. and VIII. Existence of minimizers for convex problems.
  ☐ Notes ☐ TD 4
• Fourth week : IX. and X. Generalized notions of convexity.
  ☐ Notes ☐ TD 5

Prérequis

Connaissances d'Analyse Fonctionnelle : espace de Banach, dual d'un espace de Banach, convergence faible, espaces de Lebesgue.

Bibliographie

• Bernard Dacorogna. Introduction to the Calculus of Variations.
• Hansjörg Kielhöfer. Calculus of Variations.
• L. C. Evans. Partial Differential Equations.