By M. Aizenman (Chief Editor)
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Additional info for Communications in Mathematical Physics - Volume 299
13) 52 C. B. Muratov Fig. 2. 11. 2. , we have E N < min E N + O(ε5/3−σ ). Nevertheless, to prove closeness of minimizers of E to those of E N we also need some coercivity of the energy E N . e. the choice of xi which minimize E N with fixed ri . This becomes a difficult problem in the case of interest, since we generally expect N 1 (for a numerical solution at a few values of N and κ = 2 see Fig. 2). It would be natural to conjecture that at small enough ε the minimizing droplets will arrange themselves into a periodic lattice close to a hexagonal (close-packed) lattice.
11). 14) for some C > 0 independent of ε 1. Our next step is to show that the area of each connected component of Ω¯ + = ∅ is uniformly bounded above and below independently of ε. 2. Let Ω¯ + = ∪i=1 i i + disjoint connected components of Ω¯ . 15) 1. Proof. 1 the set ∂ Ω¯ + is of class C 3,α we have N < ∞. 15) holds, we first write E¯ as | ln ε| E¯ = N ¯ −2 |Ω¯ i+ | |∂ Ω¯ i+ | − 2δκ i=1 +2| ln ε|−1 Ω¯ i+ Ω¯ i+ j =i Ω¯ i+ +2| ln ε|−1 G(ε1/3 | ln ε|−1/3 (x¯ − y¯ )) d xd ¯ y¯ ⎞ Ω¯ +j G(ε1/3 | ln ε|−1/3 (x¯ − y¯ )) d xd ¯ y¯ ⎠ .
Let us note that, since the case d = 1 is now well-understood [29–31,42], the remaining open case of physical interest is that of d = 3. Before turning to the analysis, let us briefly mention a perfect example of an experimental system in which the regimes studied by us could be easily realized, which is inspired by the beautiful Nobel Lecture of Prof. G. Ertl [43,44]. Consider molecules which undergo adsorption and desorption to and from a crystalline surface. On the surface, the atoms may hop around and reversibly stick to each other to form monolayer aggregates .
Communications in Mathematical Physics - Volume 299 by M. Aizenman (Chief Editor)