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

## TL;DR

This paper develops a mathematical framework using maximal monotone operator theory to analyze a thin film equation in epitaxial growth, proving the existence of global strong solutions despite measure-valued second derivatives.

## Contribution

It formulates the thin film equation as a gradient flow in a convex functional setting and proves global existence of solutions using maximal monotone operator theory.

## Key findings

- Established a gradient flow formulation for the thin film equation.
- Proved the existence of global strong solutions.
- Addressed mathematical challenges with measure-valued second derivatives.

## 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.