From discrete to continuum: a young measure approach

The passage from atomistic to continuum models is usually done via Γ-convergence with respect to the weak topology of some Sobolev space; the obtained continuum energy, in a one dimensional model, is then convex. These kind of results are not optimal for problems related to materials which may undergo to phase transitions.We present here a new simple way for dealing with these problems. Our method consists in rewriting the discrete energy in terms of particular measures and taking the Γ-limit with respect to the weak convergence of measures.The continuum energy arising from a linear chain of discrete mass points interacting with only the nearest neighbours turns out to be written in terms of Young measures. While, if the discrete mass points interact not only with the nearest neighbours but also with the second nearest neighbours we obtain a continuum problem in which appears a "multiple Young measure" representing multiple levels of interaction. In this way we obtain a novel continuum problem which is able to capture the "microstructure" at two different levels.