On the Solutions of a $$2+1$$ 2 + 1 -Dimensional Model for Epitaxial Growth with Axial Symmetry

Abstract

In this paper, we study the evolution equation derived by Xu and Xiang (SIAM J Appl Math 69(5):1393–1414, 2009) to describe heteroepitaxial growth in $$2+1$$ dimensions with elastic forces on vicinal surfaces is in the radial case and uniform mobility. This equation is strongly nonlinear and contains two elliptic integrals and defined via Cauchy principal value. We will first derive a formally equivalent parabolic evolution equation (i.e., full equivalence when sufficient regularity is assumed), and the main aim is to prove existence, uniqueness and regularity of strong solutions. We will extensively use techniques from the theory of evolution equations governed by maximal monotone operators in Banach spaces.