2012
DOI: 10.1007/978-3-642-25361-4_11
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Surface Relaxation Below the Roughening Temperature: Some Recent Progress and Open Questions

Abstract: We discuss two recent projects concerning the evolution of a crystal surface below the roughening temperature. One addresses the evolution of a monotone one-dimensional step train (joint work with Hala Al Hajj Shehadeh and Jonathan Weare). The other addresses the finite-time flattening predicted by a fourth-order PDE model (joint work with Yoshikazu Giga). For each project we begin with a discussion of the mathematical model; then we summarize the recent results, the main ideas behind them, and some related op… Show more

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Cited by 13 publications
(21 citation statements)
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References 28 publications
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“…They all considered this nonlinear fourth order parabolic equation, which comes from the same step flow model on vicinal surface. The aim is to answer the two questions in Section 1.1, which also are stated as open questions in [13]. The nonlinear structure of this equation, the key for both previous and current works, is important for the positivity of solution because it is known that the sign changing is a general property for solutions to linear fourth order parabolic equations.…”
Section: Definitionmentioning
confidence: 98%
“…They all considered this nonlinear fourth order parabolic equation, which comes from the same step flow model on vicinal surface. The aim is to answer the two questions in Section 1.1, which also are stated as open questions in [13]. The nonlinear structure of this equation, the key for both previous and current works, is important for the positivity of solution because it is known that the sign changing is a general property for solutions to linear fourth order parabolic equations.…”
Section: Definitionmentioning
confidence: 98%
“…If ∆u ∈ M + (Ω), taking µ = ∆u in the definition (17), we can see from Lemma 2 that the two definitions are equivalent. If ∆u + C ∈ M + (Ω), then we can take µ = ∆u + C in Lemma 2 and definition (17).…”
Section: Remarkmentioning
confidence: 99%
“…The dual pair ·, · H ′ ,H is the usual integration so we just use ·, · in the following article. Recall the definition of φ in (17). Proof.…”
Section: Corollary 14mentioning
confidence: 99%
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“…In the DL regime, Giga and Kohn [14] rigorously showed that with periodic boundary conditions on h, finite-time flattening to a spatially-uniform solution, h ≡ C, occurs for α = 0. A heuristic argument provided by Kohn [11] indicates that the flattening dynamics is linear in time. However, in the ADL regime with the nonlinear mobility given by (1.2), the dynamics of the surface height equation (1.1) is still an open question [11].…”
mentioning
confidence: 99%