Planelike minimizers in periodic media

Abstract

We show that given an elliptic integrand J in R d that is periodic under integer translations, and given any plane in R d , there is at least one minimizer of J that remains at a bounded distance from this plane. This distance can be bounded uniformly on the planes. We also show that, when folded back to R d /Z d , the minimizers we construct give rise to a lamination. One particular case of these results is minimal surfaces for metrics invariant under integer translations.The same results hold for other functionals that involve volume terms (small and average zero). In such a case the minimizers satisfy the prescribed mean curvature equation. A further generalization allows the formulation and proof of similar results in manifolds other than the torus provided that their fundamental group and universal cover satisfy some hypotheses.

Level set methods to compute minimal surfaces in a medium with exclusions (voids)

“…The recent results in [10] concern a generalization of the problem of minimal surfaces in periodic media and show that, given a metric with periodic coefficients, there exists a number M so that one can find a minimizer in any strip of width M. The width M is independent of the orientation of the strip. Moreover, the minimizers constructed in [10] have the property that, when folded to the fundamental domain, they are laminations.…”

“…Moreover, the minimizers constructed in [10] have the property that, when folded to the fundamental domain, they are laminations.…”

“…In [59], the results of [10] are extended to a situation where the medium has exclusions, i.e., regions for which the metric vanishes. In both papers, the surfaces are considered as boundaries of sets and the perimeter is defined in a weak sense (see [27]).…”

“…To avoid this, we use a smooth weight w 1, with 1, where w has value inside the exclusions and increases smoothly to 1 on the outside. Since w is smooth, the existence of minimizers is contained in [10], as well as a discussion on the regularity of minimizers. Since mean curvature is a local property, we can restrict ourselves to an open set Ω ⊂ R n−1 , where the minimal surface is the graph of the smooth function f (x ), x ∈ R n−1 .…”

$\Gamma$-convergence of the Allen-Cahn energy with an oscillating forcing term

“…From [8,Section 11] (which can be adapted to the case g ∈ L ∞ ) it follows that there exist a set E ⊂ R N and a constant k = k(g) > 0 such that sup x∈∂E dist(x, ∂H ν,0 ) k, j ∈ N, (5.23) and for any compact set K ⊆ R N ,…”