Mathematics

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. Direct method, quasiconvexity, polyconvexity, rank-one convexity and their relations. Semicontinuity theorems for scalar and vectorial functionals; existence of minimizers. Relaxation theorems, representation of relaxed functionals; convex, quasiconvex, polyconvex and rank-one convex envelopes. Non semicontinuous problems. Hamilton-Jacobi equations, differential inclusions and applications to non convex problems.