The paper is a transcript of the lecture given at the European Symposium on Well-Posedness in Optimization in Warsaw. It contains a complete theory of variational problems with integrands not depending on x, including existence and relaxation theorems, a complete description of solutions and the connection between variational convergences of functionals and convergence of value functions and solutions of associated variational problems with the main emphasis on functionals that lack coercivity.
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We consider an elastic thin film as a bounded open subset ω of R2. First, the effective energy functional for the thin film ω is obtained, by Γ-convergence and 3D-2D dimension reduction techniques applied to the sequence of re-scaled total energy integral functionals of the elastic cylinders (…) as the thickness ε goes to 0. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function for the elastic cylinders has the growth prescribed by an Orlicz convex function M. Here M is assumed to be non-power-growth-type and to satisfy the conditions (…) and (…) (that is equivalent to the reflexivity of Orlicz and Orlicz–Sobolev spaces generated by M). These results extend results of H. Le Dret and A. Raoult for the case M(t) = (…) for some (…).
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