Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lamellar phase solutions for diblock copolymers with nonlocal diffusions   For a diblock copolymer with total chain length $\gamma>0$ and mass ratio
$m\in(-1,1)$, we consider the problem of minimizing the doubly nonlocal free
energy $$ \mathcal{E}_{\varepsilon}(u) =\mathcal{H}(u)
+\frac{1}{\varepsilon^{2s}}
  \int_{\Omega}W(u)\,dx +\frac{1}{2}\int_{\Omega}
  \left|(-\gamma^{2}\Delta)^{-\frac{1}{2}}(u-m)\right|^2\,dx $$ in a domain
$\Omega$, where $\mathcal{H}(u)$ is a fractional $H^s$-norm with
$s\in(0,\frac12)$, and $W$ is a double-well potential. This arises in the study
of micro-phase separation phenomena for diblock copolymers with nonlocal
diffusions.
  On the unit interval, we identify the $\Gamma$-limit as $\varepsilon\to0^+$,
and also find explicit isolated local minimizers associated the lamellar
morphology phase in the case $m=0$, provided that the chain is sufficiently
short or the nonlocal interaction is sufficiently strong (i.e. as $s\to0^+$).
We stress that such extra condition is new for the nonlocal case and is not
present in the classical model. The proof, while elementary, requires a careful
analysis of the nonlocal integrals.

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The categories of this paper are: hep-ex
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph