# zbMATH — the first resource for mathematics

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
Full Text: