Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model

In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 dimensions via a continuum model below the roughening transition temperature. The model invokes a highly degenerate parabolic partial differential equation (PDE) for surface diffusion, which is related to the weighted-$H^{-1}$ (nonlinear) gradient flow of a convex, singular surface free energy in homoepitaxy. The PDE is motivated both by an atomistic broken-bond model and a mesoscale model for steps. By constructing an explicit solution to the PDE, we demonstrate the lack of symmetry in the evolution of top and bottom facets in periodic surface profiles. Our explicit, analytical solution is compared to numerical simulations of the PDE via a regularized surface free energy.Comment: 23 pages, 5 figures, comments welcome! Text slightly modified, references updated in Version 2. Referee comments addresse

A vicinal surface model for epitaxial growth with logarithmic free energy

We study a continuum model for solid films that arises from the modeling of one-dimensional step flows on a vicinal surface in the attachment-detachment-limited regime. The resulting nonlinear partial differential equation, $u_t = -u^2(u^3+\alpha u)_{hhhh}$, gives the evolution for the surface slope $u$ as a function of the local height $h$ in a monotone step train. Subject to periodic boundary conditions and positive initial conditions, we prove the existence, uniqueness and positivity of global strong solutions to this PDE using two Lyapunov energy functions. The long time behavior of $u$ converging to a constant that only depends on the initial data is also investigated both analytically and numerically.Comment: 18 pages, 7 figure

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

In this work, we study a fourth order exponential equation, $u_t=\Delta e^{-\Delta u},$ derived from thin film growth on crystal surface in multiple space dimensions. We use the gradient flow method in metric space to characterize the latent singularity in global strong solution, which is intrinsic due to high degeneration. We define a suitable functional, which reveals where the singularity happens, and then prove the variational inequality solution under very weak assumptions for initial data. Moreover, the existence of global strong solution is established with regular initial data.Comment: latent singularity, curve of maximal slope. arXiv admin note: text overlap with arXiv:1711.07405 by other author

Scale-Invariant Extinction Time Estimates for Some Singular Diffusion Equations

(typos corrected 7/15/10 and 2/10/11) In honor of Louis Nirenberg’s 85th birthday We study three singular parabolic evolutions: the second-order total variation flow, the fourth-order total variation flow, and a fourth-order surface diffusion law. Each has the property that the solution becomes identically zero in finite time. We prove scale-invariant estimates for the extinction time, using a simple argument which combines an energy estimate with a suitable Sobolev-type inequality. YG is grateful to Professor Yoshie Sugiyama for informative remarks. Much of this work was done while YG visited the Courant Institute in Fall 2009; its hospitality is gratefully acknowledged, as is support from the Japa

Existence theorems for a crystal surface model involving the p-Laplace operator

The manufacturing of crystal films lies at the heart of modern nanotechnology. How to accurately predict the motion of a crystal surface is of fundamental importance. Many continuum models have been developed for this purpose, including a number of PDE models, which are often obtained as the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. In this paper we offer an analytical perspective into some of these models. To be specific, we study the existence of a weak solution to the boundary value problem for the equation - \Delta e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right)}+au=f, where $p>1, a>0$ are given numbers and $f$ is a given function. This problem is derived from a crystal surface model proposed by J.L.~Marzuola and J.~Weare (2013 Physical Review, E 88, 032403). The mathematical challenge is due to the fact that the principal term in our equation is an exponential function of a p-Laplacian. Existence of a suitably-defined weak solution is established under the assumptions that $p\in(1,2], \ N\leq 4$, and $f\in W^{1,p}$. Our investigations reveal that the key to our existence assertion is how to control the set where -\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right) is $\pm\infty$

A C0 interior penalty discontinuous Galerkin method for fourth‐order total variation flow I: Derivation of the method and numerical results

We consider the numerical solution of a fourth‐order total variation flow problem representing surface relaxation below the roughening temperature. Based on a regularization and scaling of the nonlinear fourth‐order parabolic equation, we perform an implicit discretization in time and a C0 Interior Penalty Discontinuous Galerkin (C0IPDG) discretization in space. The C0IPDG approximation can be derived from a mixed formulation involving numerical flux functions where an appropriate choice of the flux functions allows to eliminate the discrete dual variable. The fully discrete problem can be interpreted as a parameter dependent nonlinear system with the discrete time as a parameter. It is solved by a predictor corrector continuation strategy featuring an adaptive choice of the time step sizes. A documentation of numerical results is provided illustrating the performance of the C0IPDG method and the predictor corrector continuation strategy. The existence and uniqueness of a solution of the C0IPDG method will be shown in the second part of this paper

PDE and Materials

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Characterization of subdifferentials of a singular convex functional in Sobolev spaces of order minus one

Subdifferentials of a singular convex functional representing the surface free energy of a crystal under the roughening temperature are characterized. The energy functional is defined on Sobolev spaces of order -1, so the subdifferential mathematically formulates the energy's gradient which formally involves 4th order spacial derivatives of the surface's height. The subdifferentials are analyzed in the negative Sobolev spaces of arbitrary spacial dimension on which both a periodic boundary condition and a Dirichlet boundary condition are separately imposed. Based on the characterization theorem of subdifferentials, the smallest element contained in the subdifferential of the energy for a spherically symmetric surface is calculated under the Dirichlet boundary condition.Comment: 26 page