In:
Communications on Pure and Applied Mathematics, Wiley, Vol. 75, No. 2 ( 2022-02), p. 422-446
Abstract:
Let K be a convex polyhedron and ℱ its Wulff energy, and let denote the set of convex polyhedra close to K whose faces are parallel to those of K . We show that, for sufficiently small ε , all ε ‐minimizers belong to As a consequence of this result we obtain the following sharp stability inequality for crystalline norms: There exist γ = γ ( K , n ) 〉 0 and σ = σ ( K , n ) 〉 0 such that, whenever ∣ E ∣ = ∣ K ∣ and ∣ E Δ K ∣ ≤ σ , then urn:x-wiley:00103640:media:cpa21928:cpa21928-math-0003 In other words, the Wulff energy ℱ grows very fast (with power 1) away from The set appearing in the formula above can be informally thought as a sort of “projection” of E onto Another corollary of our result is a very strong rigidity result for crystals: For crystalline surface tensions, minimizers of ℱ( E ) + ∫ E g with small mass are polyhedra with sides parallel to the those of K . In other words, for small mass, the potential energy cannot destroy the crystalline structure of minimizers. This extends to arbitrary dimensions a two‐dimensional result obtained in [9]. © 2020 Wiley Periodicals LLC.
Type of Medium:
Online Resource
ISSN:
0010-3640
,
1097-0312
Language:
English
Publisher:
Wiley
Publication Date:
2022
detail.hit.zdb_id:
1468142-0
detail.hit.zdb_id:
1568-4
detail.hit.zdb_id:
220318-2
SSG:
17,1
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