ALEX 2018 Workshop: Abstracts

Variational flows in heterogeneous media

Andrea Braides

University of Rome Tor Vergata (Italy)

We consider gradient-flow type evolutions with underlying energies $F_{\epsilon }$ depending on a small parameter $\epsilon$. An effective evolution for such energies as $\epsilon \to 0$ can be constructed either as the limit of gradient flows at fixed $\epsilon$ or directly using the minimizing-movement approach as in the book of Ambrosio, Gigli and Savaré [1, 2]. If some conditions introduced by Colombo and Gobbino are satisfied then this effective motion can be characterized as a curve of maximal slope of the $\mathrm{\Gamma }$-limit of $F_{\epsilon }$ [4] (see also related works [11] and [10] in different contexts). Such conditions are not satisfied if the pattern of local minima is lost in the passage to the limit. This is a usual case for inhomogeneous surface energies, and in particular when dealing with the passage from discrete to continuum.

We will examine some situations when the limit geometric flow can be characterized and shows a different behaviour than the approximating flows. This characterization can be obtained in a simpler way when dealing with crystalline surface energies, in which case we reduce our analysis to a system of ODEs. Examples when the limit of motions by crystalline curvature can be studied is when the heterogeneity is derived from an highly oscillating forcing term [6, 9].

We will also examine geometric flows arising as a limit of simple spin systems defined on lattices of spacing $\epsilon$, whose corresponding energies converge to crystalline energies. In this case, the gradient flow at fixed $\epsilon$ loses meaning and the minimizing-movement approach must be followed. The limit flow may show pinning by local minima [5, 7], development of bulk microstructure [8], or oscillations of interfaces [3]. In all these cases the limit flow is not easily derived as a geometric flow for some effective energy.