Maximal monotone operator theory and its applications to thin film equation in epitaxial growth on vicinal surface

Abstract

In this work we consider $$ w_t=[(w_{hh}+c_0)^{-3}]_{hh},\qquad w(0)=w^0, $$ which is derived from a thin film equation for epitaxial growth on vicinal surface. We formulate the problem as the gradient flow of a suitably-defined convex functional in a non-reflexive space. Then by restricting it to a Hilbert space and proving the uniqueness of its sub-differential, we can apply the classical maximal monotone operator theory. The mathematical difficulty is due to the fact that $w_{hh}$ can appear as a positive Radon measure. We prove the existence of a global strong solution. In particular, the equation holds almost everywhere when $w_{hh}$ is replaced by its absolutely continuous part.