Expansion of embedded curves with turning angle greater than −π

On a simple maximum principle technique applied to equations on the circle

“…As a consequence we have short time existence for a unique solution w(x, t), which is smooth and is defined on S 1 × [0, T ) for some finite T > 0, and satisfies 0 < 1 λ A w(x, t) λ (2) for all (x, t) ∈ S 1 × [0, T ), for some positive constant λ > 0. We assume that w is bounded on S 1 × [0, T ).…”

Non-Trapping sets and Huygens Principle

“…In a recent paper [2], Chow, Liou and Tsai have defined a class of regular curves, the curves which have turning angle greater than −π. For this class of curves: the expansion by a positive strictly decreasing function of the curvature has good properties: in particular if the curve is initially embedded it remains embedded at all times, eventually becomes convex and tends to a circle in a C 2 norm.…”

“…The evolution according to the Huygens principle can be considered as a particular case of the motion by mean curvature, but for the fact that generally, in the literature, the velocity is a non constant function of the curvature. Nevertheless the two problems may have similar features: in particular it is interesting to characterize class of sets for which the evolution has good properties (see [2], and references therein).…”

“…In [2], Chow, Liou and Tsai define the class of regular curves whose turning angle is greater than −π. With this condition, they are able to prove that the expansion of a curve, by a positive strictly decreasing function of the curvature, has good properties.…”

“…We define a class of sets (non-trapping sets) for which the perimeter is a continuous function of t, and we give an algorithm to approximate the evolution. Finally we restrict our attention to the class of sets for which the turning angle of the boundary is greater than −π (see [2]). For this class of sets we prove that the perimeter is a Lipschitz-continuous function of t. This evolution problem is relevant for the applications because it is used as a model for solid fuel combustion.…”