We address the problem of minimization of the Cahn-Hilliard energy functional under a mass constraint over two and three-dimensional cylindrical domains. Although existence is presented for a general case the focus is mainly on rectangles, parallelepipeds and circular cylinders. According to the symmetry of the domain the exact numbers of global and local minimizers are given as well as their geometric profile and interface locations; all are onedimensional increasing/decreasing and odd functions for domains with lateral symmetry in all axes and also for circular cylinders. The selection of global minimizers by the energy functional is made via the smallest interface area criterion.The approach utilizes Γ−convergence techniques to prove existence of an one-parameter family of local minimizers of the energy functional for any cylindrical domain. The exact numbers of global and local minimizers as well as their geometric profiles are accomplished via suitable applications of the unique continuation principle while exploring the domain geometry in each case and also the preservation of global minimizers through the process of Γ−convergence.

This note is a review of results concerning the number, geometric profile and one-dimensional character of local and global minimizers of the Cahn-Hilliard functional over rectangles and parallelepipeds with and without constraint. Mathematics Subject Classification (2010). Primary 35B25; Secondary 35B35.

The main objective of this paper is to prove Monsky's Theorem, that provides a beautiful application of the 2-adic valuation in order to solve a plane geometry problem. This theorem states that given any dissection of a square into finitely many nonoverlapping triangles of equal area the number of triangles must be even. In order to prove this statement, we will need some previous results from Combinatorial Topology and Algebra.Keywords: Dissection of a square into triangles of equal area, 2-adic valuation. RESUMOO objetivo principal desse trabalho é a demonstração do Teorema de Monsky, o qual fornece uma bela aplicação da valoração 2-ádica na resolução de um problema de Geometria Plana. Esse teorema afirma que dada qualquer dissecção de um quadrado em triângulos não sobrepostos e de mesma área, o número de triângulos deve ser par. Com o objetivo de demonstrar essa afirmação precisaremos de alguns resultados da Topologia Combinatória e da Álgebra.

No abstract

In this paper, we consider an inhomogeneous Allen–Cahn problem where ( ) is a bounded set with smooth boundary and is a non‐negative Lipschitz‐continuous function in . Let be an ‐dimensional hypersurface that divides into two disjoints components ( ) such that on and in . Using the variational concept of ‐convergence, we prove the existence of stable stationary solutions developing a transition layer on as .