Minimizers of Anisotropic Surface Tensions Under Gravity: Higher Dimensions via Symmetrization

Baer, Eric

doi:10.1007/s00205-014-0788-z

Cited by 12 publications

(16 citation statements)

References 33 publications

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“…In fact we assume that the admissible film profiles are graphs of height functions and the conditions that we need to impose on γ f , γ s , γ f s are such that the right-hand side of (1.1) belongs to [0, 1] (where, though, the boundary values can be included in our analysis). For more general conditions on the adhesion coefficient σ we refer the reader to [1] and [8], where every set of finite perimeter is an admissible drop, and the boundary regularity of optimal drops is studied also in the presence of anisotropy.…”

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confidence: 99%

Analytical validation of the Young–Dupré law for epitaxially-strained thin films

Davoli

Piovano

2019

Math. Models Methods Appl. Sci.

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We present here an analysis of the regularity of minimizers of a variational model for epitaxially strained thin-films identified by the authors in the companion paper [7]. The regularity of energetically-optimal film profiles is studied by extending previous methods and by developing new ideas based on transmission problems. The achieved regularity results relate to both the Stranski-Krastanow and the Volmer-Weber modes, the possibility of different elastic properties between the film and the substrate, and the presence of the surface tensions of all three involved interfaces: film/gas, substrate/gas, and film/substrate. Finally, geometrical conditions are provided for the optimal wetting angle, i.e., the angle formed at the contact point of films with the substrate. In particular, the Young-Dupré law is shown to hold, yielding what appears to be the first analytical validation of such law for a thin-film model in the context of Continuum Mechanics.

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confidence: 99%

Analytical validation of the Young–Dupré law for epitaxially-strained thin films

Davoli

Piovano

2019

Math. Models Methods Appl. Sci.

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“…Fix a model M Λ (x F , z) for a film-lattice center x F ∈ R 2 \ S, a wall vector z ∈ Z S , and an interaction vector Λ := (e F S , c F , c S ) ∈ (R + ) 3 . The empirical measure µ Dn associated to a configuration D n := {x 1 , .…”

Section: Mathematical Setting and Main Resultsmentioning

confidence: 99%

“…For the specific models M 0 Λ (z) and M 1 Λ (z) with x F := x 0 F introduced in Section 2.2 we consider localized energies which together contribute to the overall energy of configurations. To this end, fix a model M Λ (x 0 F , z) for a choice of z ∈ Z S and Λ := (e F S , c F , c S ) ∈ (R + ) 3 . We define the local energy E loc per site x ∈ L F (x 0 F ) with respect to a configuration D n , by…”

Section: 4mentioning

confidence: 99%

“…The strategy consists in associating to each configuration D n an auxiliary set H n such that χ H n converges in BV loc to χ D (see also [2]) in order to transfer the problem on the surface energy functionals analyzed in, e.g., [3]. Furthermore, by fixing δ > 0 we reduce the analysis to the region {x ∈ R 2 : 0 ≤ x 2 ≤ δ}, since outside of it one can simply use Reshetnyak Lower-Semiconinuity Theorem.…”

Section: 4mentioning

confidence: 99%

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Microscopical Justification of the Winterbottom problem for well-separated Lattices

Piovano¹,

Velčić²

2021

Preprint

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We consider the discrete atomistic setting introduced in [25] to microscopically justify the continuum model related to the Winterbottom problem, i.e., the problem of determining the equilibrium shape of crystalline film drops resting on a substrate, and relax the rigidity assumption considered in [25] to characterize the wetting and dewetting regimes and to perform the discrete to continuum passage. In particular, all results of [25] are extended to the setting where the distance between the reference lattices for the film and the substrate is not smaller than the optimal bond length between a film and a substrate atom. Such optimal film-substrate bonding distance is prescribed together with the optimal film-film distance by means of two-body atomistic interaction potentials of Heitmann-Radin type, which are both taken into account in the discrete energy, and in terms of which the wetting-regime threshold and the effective expression for the wetting parameter in the continuum energy are determined.

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“…Although this kind of analysis was recently carried out in the anisotropic setting under suitable symmetry assumptions on Φ = Φ(ν), see [Bae12], it is clear that the approach itself is intrinsically limited to the case when Ω is an half-space, σ is a constant, and g is a function of the vertical variable x n only.…”

Section: 2mentioning

confidence: 99%

Regularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law

Philippis

Maggi

2014

Arch Rational Mech Anal

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Abstract. Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a closed negligible set, showing in particular the validity of Young's law at almost every point of the free boundary. Our regularity results are not specific to capillarity problems, and actually apply to sets of finite perimeter (and thus to codimension one integer rectifiable currents) arising as minimizers in other variational problems with free boundaries.

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Minimizers of Anisotropic Surface Tensions Under Gravity: Higher Dimensions via Symmetrization

Cited by 12 publications

References 33 publications

Analytical validation of the Young–Dupré law for epitaxially-strained thin films

Analytical validation of the Young–Dupré law for epitaxially-strained thin films

Microscopical Justification of the Winterbottom problem for well-separated Lattices

Regularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law

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