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

Abstract

In this work we considerwhich 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 with hidden singularity. In particular, (1) holds almost everywhere when w hh is replaced by its absolutely continuous part.

Gradient flow approach to an exponential thin film equation: global existence and latent singularity

“…Even if (5) has a nice variational structure, and V has Banach space structure, the non-reflexivity of V imposes extra technical difficulties. Instead of arguing with maximal monotone operator like in [13], we try to use the result [2, Theorem 4.0.4] by Ambrosio, Gigli and Savaré. After defining the energy functional rigorously, we take the counterintuitive approach of ignoring the variational structure of (5) and the Banach space structure of W 2,1 (Ω).…”

“…While in the attachment-detachment-limited (ADL) case, i.e. the dominant processes are the attachment and detachment of atoms at step edges and the mobility function [17] takes the form M (∇u) = |∇u| −1 , we refer readers to [17,1,12,13] for analytical results. Note that the simplifed version of PDE (3), which linearizes the Gibbs-Thomson relation, does not distinguish between convex and concave parts of surface profiles.…”

Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects

“…To the best of our knowledge, for arbitrarily large times, whether the solution to (1) remains strictly monotone is still an open question. We also refer to [11,12,13,14,22,10,9,7] and the references therein for some related 4th order degenerate equation but evolving only local derivatives coming from nearest-neighbor interactions between steps.…”

Strong solutions to a fourth order exponential PDE describing epitaxial growth

“…It was first observed in [11] that one had to allow the possibility that the exponent be a measure-valued function. Later, the idea of "exponential singularity" was employed in [2,4,5,14,18]. However, measure exponents do not arise in the one-dimensional case.…”