Measure theoretic aspects of the calculus of variations

elective monographs

A part of the calculus of variations, where the objects defined with the help of measures play a key role will be presented. Among other things we will present the necessary background from the convex analysis with the emphasis on the dual problems. As an application we will present the Monge-Kantorovich optimal transportation problem.

There is a group of optimization problems e.g. the optimal transport or the free material design, which exploits the tools of the measure theory accompanied by the convex analysis. We want to present a careful introduction into these methods and their application and finally reach the current state of art.

We will study the basic problem of the calculus of variations, in the measure theoretic context which is existence of minimizers of functionals. In order to solve this problem we have to know that the functional in question are lower semicontinuity. For this purpose we will present the Reshetniak theorems and the slicing measure theorem, which is interesting for its own sake.

The tools of the convex analysis based on the Legendre-Fenchel transform will be useful here. We shall see applications of these methods to variational problems on sets with lower dimensions than the ambient Euclidean space.

An important of the convex analysis is the 'dual problem', [ET]. It

happens that it is easier than the primary problem. It also permits to construct solutions to the primary problem, [S]. As an examples serves the Monge optimal mass transportation problem. Besides that duality may be used to find the relaxation of functionals, [ET], i.e. their lower semicontinuous envelopes.

We plan to present a part of the theory of Gamma-convergence of

functionals, which permits to study convergence of gradient flows.

[AFP] Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.

[BBS] G.Bouchitte, G.Buttazzo, P.Seppecher, Energies with respect to a measure and applications to low-dimensional structures, Calc. Var. Partial Differential Equations, 5 (1997), no. 1, 37--54.

[ET] Ekeland, Ivar; Témam, Roger Convex analysis and variational problems. Translated from the French. Corrected reprint of the 1976 English edition. Classics in Applied Mathematics, 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999

[S] Santambrogio, Filippo Optimal transport for applied mathematicians. Calculus of variations, PDEs, and modeling. Progress in Nonlinear Differential Equations and their Applications, 87. Birkhäuser/Springer, Cham, 2015.

other prosented during the lectures

1) The participant knows and understands the basic questions of the Calculus of Variations and Convex Analysis.

2) The participant knows and understands the basic minimization problems on spaces of Radon measures.

3) The participant knows and understands the dual problems of Convex Analysis.