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Direct Methods in the Calculus of Variations

Lecturer: 
Sandro Zagatti
Course Type: 
PhD Course
Academic Year: 
2013-2014
Period: 
November-January
Duration: 
20 h
Description: 
  1. Elements of convex analysis, polar and bipolar function and their properties, convex envelopes. Semiclassical theory, Euler-Lagrange equations and relation with elliptic PDE’s. Regularity of minimizers.
  2. Direct method, quasiconvexity, polyconvexity, rank-one convexity and their relations. Semicontinuity theorems for scalar and vectorial functionals; existence of minimizers.
  3. Relaxation theorems, representation of relaxed functionals; convex, quasiconvex, polyconvex and rank-one convex envelopes.
  4. Non semicontinuous problems. Hamilton-Jacobi equations, differential inclusions and applications to non convex problems. 

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