No abstract

We study the 2+1 dimensional continuum model for long-range elastic interaction on stepped epitaxial surface proposed by Xu and Xiang [28]. The long-range interaction term and the two length scales in this model present challenges for the PDE analysis. In this paper, we prove the existence of both the static and dynamic solutions and derive the minimum energy scaling law for this 2+1 dimensional continuum model. We show that the minimum energy surface profile is attained by surfaces with step meandering instability, which is essentially different from the energy scaling law for the 1+1 dimensional epitaxial surfaces under elastic effects [17], which is 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.