Two-scale homogenization of a stationary mean-field game

In this paper, we characterize the asymptotic behavior of a first-order stationary mean-field game (MFG) with a logarithm coupling, a quadratic Hamiltonian, and a periodically oscillating potential. This study falls into the realm of the homogenization theory, and our main tool is the two-scale convergence. Using this convergence, we rigorously derive the two-scale homogenized and the homogenized MFG problems, which encode the so-called macroscopic or effective behavior of the original oscillating MFG. Moreover, we prove existence and uniqueness of the solution to these limit problems.Comment: 36 page

New Proposals on Portuguese Vegetation (II)

As a consequence of the geobotanical studies that have been developed in west and southern areas of the Iberian Peninsula, four new phytosociological associations are proposed, integrated in the perennial mesophytic grasslands of Stipo giganteae-Agrostietea castellanae (Serratulo flavescentis-Celticetum giganteae, Armerio macrophyllae-Celticetum giganteae, Centaureo rothmaleranae-Celticetum giganteae, and Centaureo coutinhoi-Dactyletum lusitanici). For each proposed unit, besides its syntaxonomical fitting, we present its ecological diagnosis and the serial and catenal context, as well as its patrimonial value