Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence, uniqueness, and energy scaling of 2+1 dimensional continuum
  model for stepped epitaxial surfaces with elastic effects   We study the 2+1 dimensional continuum model for the evolution of stepped
epitaxial surface under long-range elastic interaction proposed by Xu and Xiang
(SIAM J. Appl. Math. 69, 1393-1414, 2009). The long-range interaction term and
the two length scales in this model makes PDE analysis challenging. Moreover,
unlike in the 1+1 dimensional case, there is a nonconvexity contribution (of
the gradient norm of the surface height) in the total energy in the 2+1
dimensional case, and it is not easy to prove that the solution is always in
the well-posed regime during the evolution. In this paper, we propose a
modified 2+1 dimensional continuum model and prove the existence and uniqueness
of both the static and dynamic solutions and derive a minimum energy scaling
law for it. We show that the minimum energy surface profile is mainly attained
by surfaces with step meandering instability. This is essentially different
from the energy scaling law for the 1+1 dimensional epitaxial surfaces under
elastic effects attained by step bunching surface profiles. We also discuss the
transition from the step bunching instability to the step meandering
instability in 2+1 dimensions.

31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
6
The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th