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Zbl 1173.35047
Wolansky, G.
Minimizers of Dirichlet functionals on the $n$-torus and the weak KAM theory. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 2, 521-545 (2009). ISSN 0294-1449

This paper deals with the minimization of a Dirichlet-type functional $F$ on the $n$-torus ${\Bbb T}^n$. It is first argued that the supremum of this functional is not attained in $L^1({\Bbb T}^n)$, but in the set $\overline{\cal M}$ of Borel probability measures on ${\Bbb T}^n$. This motivates the author to extend the domain of $F$ from the set of nonnegative densities in $L^1({\Bbb T}^n)$ to $\overline{\cal M}$. First objective of the present paper is to define a generalized minimizer of $F$. This is mainly done by means of the effective Hamiltonian, in the framework of the weak KAM theory. A second objective of this paper is to relate the generalized minimizer of $F$ to the minimal Mather measure. Essentially, the minimal Mather measures of a given Lagrangian is connected with the measure which minimizes a certain optimal transportation plan. That is why the third main objective of the present paper is to approximate $F$ by an optimal transportation function and to establish a combinatorial search algorithm.
[Vicenţiu D. Rădulescu (Craiova)]

MSC 2000:: *35J20 Second order elliptic equations, variational methods
37K55 Perturbations, KAM for infinite-dimensional systems
58E05 Abstract critical point theory
49J35 Minimax problems (existence)

Keywords: Monge-Kantorovich; optimal mass transport; periodic Lagrangian; effective Hamiltonian; rotation vector; Dirichlet functional; KAM theory

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