# Surface energies in multi phase systems with diffuse phase boundaries

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ProcFBP05.dvi Universita¨t Regensburg Mathematik Surface energies in multi phase systems with diffuse phase boundaries Bjo¨rn Stinner Preprint Nr. 13/2005 Surface energies in multi phase systems with diffuse phase boundaries Bjo¨rn Stinner Abstract. A Ginzburg–Landau type functional for a multi–phase system in- volving a diffuse interface description of the phase boundaries is presented with the following calibration property: Prescribed surface energies (possibly anisotropic) of the phase transitions are correctly recovered in the sense of a Γ–limit as the thickness of the diffuse interfaces converges to zero. Possible applications are grain boundary motion and solidification of alloys on which numerical simulations are presented. 1. Introduction Consider a domain Ω ⊂ Rd, d ∈ {1, 2, 3}, which is subdivided into several (not necessarily connected) regions Ωα, α ∈ {1, . . . ,M}, M ∈ N, separated by hy- persurfaces Γαβ , 1 ≤ α < β ≤ M. The interfaces are supposed to carry energy obtained by integrating surface densities which may depend on the orientation of the hypersurface. The total energy of the system has the form (1.1) FSI = ∑ α<β ∫ Γαβ σαβ(ναβ)dH d−1. To avoid wetting effects is is assumed that σαβ + σβδ > σαδ > 0 for each triple of phases α, β, δ. Energies as in (1.1) can be approximated by Ginzburg–Landau energies of the form (1.2) FPF = ∫ Ω ( εa(φ,∇φ) + 1 ε w(φ) ) dLd. Here, φ = (φ1, . . . , φM) : Ω→ ΣM with ΣM := { v ∈ RM : M∑ α=1 vα = 1 and 0 ≤ vα ≤ 1 for all α = 1, . . . ,M } Received by the editors September 9, 2005. 1991 Mathematics Subject Classification. 82B26, 74N20, 74E10. Key words and phrases. phase transitions, phase field model, sharp interface model, surface energy, anisotropy. 2 Bjo¨rn Stinner is a set of phase field variables. For each α, φα is assigned to one of the phases (labelled α and occupying Ωα) and describes its presence in a point of Ω. The function a(φ,∇φ) is a non–negative gradient term, and w(

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