  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 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 based on the
underlying physics. This modification fixes the problem of possible
illposedness due to the nonconvexity of the energy functional. We prove the
existence and uniqueness of both the static and dynamic solutions and derive a
minimum energy scaling law for them. 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.
