Maximal monotone operator in non-reflexive Banach space and the application to thin film equation in epitaxial growth on vicinal surface

摘要

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 vicinalsurface. We formulate the problem as the gradient flow of a suitably-definedconvex functional in a non-reflexive space. Then by restricting it to a Hilbertspace and proving the uniqueness of its sub-differential, we can apply theclassical maximal monotone operator theory. The mathematical difficulty is dueto the fact that $w_{hh}$ can appear as a positive Radon measure. We prove theexistence of a global strong solution. In particular, the equation holds almosteverywhere when $w_{hh}$ is replaced by its absolutely continuous part.