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A comparison theorem for crystalline evolution in the plane. (English) Zbl 0862.35047
Crystal surfaces are often endowed with energies whose dependence on orientation \(n\) (outward unit normal) displays “low energy cusps” at a finite set \(\mathcal N\) of orientations; such energies lead to crystal shapes that are fully faceted, the orientations of the facets being the orientations \(n\in{\mathcal N}\). A possible model for the planar evolution of such crystals is based on an evolution equation relating the normal velocity \(V_i(t)\) and crystalline curvature \(K_i(t)=\chi_iL_i(t)^{-1}\) of each facet \(F_i\), where \(L_i(t)\) is the length of \(F_i\), while \(\chi_i\) has constant value \(-1\), \(+1\), or \(0\) according as the crystal is strictly convex, strictly concave, or neither near \(F_i\); this evolution equation has the form \[ \beta(n_i)V_i(t)=l(n_i)K_i(t)-U, \] where \(\beta(n_i)>0\), the kinetic modulus, and \(l(n_i)>0\), the Wulff modulus, depend only on the (fixed) orientation \(n_i\in{\mathcal N}\) of \(F_i\), where \(U\) is the constant bulk energy of the crystal relative to its exterior.
In many respects this evolution equation exhibits behavior typical of a parabolic PDE, and it seems reasonable to ask whether it has an associated comparison principle. What makes this question especially important is that comparison can form the basis for weak formulations of the underlying evolution problem. We here establish such a comparison principle: We show that if \(\mathcal C\) and \(\overline{\mathcal C}\) are admissible evolving crystals with \({\mathcal C}(0)\) contained in \(\overline{\mathcal C}(0)\), then \({\mathcal C}(t)\) is contained in \(\overline{\mathcal C}(t)\) as long as both evolutions are well defined.

35K55 Nonlinear parabolic equations
80A22 Stefan problems, phase changes, etc.
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