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Flat \(\phi \) curvature flow of convex sets. (English) Zbl 1241.53055
This paper deals with the flat \(\phi \) curvature flow, introduced by F. Almgren, J. E. Taylor and L. Wang [SIAM J. Control Optimization 31, No. 2, 387–438 (1993; Zbl 0783.35002)] for modelling the \(\phi\)-weighted mean curvature flow [J. E. Taylor, J. Geom. Anal. 8, No. 5, 859–864 (1998; Zbl 0968.53046)]. For an arbitrary initial compact convex subset \(K_0\subset \mathbb R^n\), \(n\geq 2\), the author constructs a flat \(\phi \) curvature flow \(K(t)\) such that \(K(t)\) remains compact convex throughout the evolution, the proof is base on a discretization approach, cf. R. J. McCann [Commun. Math. Phys. 195, No. 3, 699–723 (1998; Zbl 0936.74029)], G. Bellettini, V. Caselles, A. Chambolle and M. Novaga [J. Math. Pures Appl. (9) 92, No. 5, 499–527 (2009; Zbl 1178.53066); Arch. Ration. Mech. Anal. 179, No. 1, 109–152 (2006; Zbl 1148.53049)]. Moreover, a new Hölder continuity estimate for the flow is established.
MSC:
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
49N60 Regularity of solutions in optimal control
49Q20 Variational problems in a geometric measure-theoretic setting
35J60 Nonlinear elliptic equations
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