Parabolic equations for curves on surfaces Part I. Curves with $p$-integrable curvature

@article{Angenent1990ParabolicEF,
  title={Parabolic equations for curves on surfaces Part I. Curves with \$p\$-integrable curvature},
  author={Sigurd B. Angenent},
  journal={Annals of Mathematics},
  year={1990},
  volume={132},
  pages={451-483}
}
  • S. Angenent
  • Published 1 November 1990
  • Mathematics
  • Annals of Mathematics
This is the first of a two-part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so-called curve shortening of flow-by-mean-curvature problem, as well as to a number of models for phase transitions in two dimensions. We introduce a class of equations for which the initial value problem is solvable for initial data with p-integrable curvature, and we also give estimates for the rate at which the p-norms of the curvature must blow up, if… 
View via Publisher
math.jhu.edu
190 Citations
Highly Influential Citations
20
Background Citations
52
Methods Citations
21
Results Citations
4

190 Citations

Citation Type

Citation Type

Anisotropic flows for convex plane curves
where V and k are, respectively, the normal velocity and curvature of the interface. Equation (1) is sometimes called the “curve shortening problem” because it is the negative L2-gradient flow of the…
EVOLUTION OF PLANE CURVES DRIVEN BY A NONLINEAR FUNCTION OF CURVATURE AND ANISOTROPY
In this paper we study evolution of plane curves satisfying a geometric equation v = β(k, ν), where v is the normal velocity and k and ν are the curvature and tangential angle of a plane curve Γ. We…
Curvature driven flow of a family of interacting curves with applications
In this paper, we investigate a system of geometric evolution equations describing a curvature‐driven motion of a family of planar curves with mutual interactions that can have local as well as…
On Evolution Equations for Moving Domains
We introduce a general framework for the description of the autonomous motion of closed surfaces which are diffeomorphic images of spheres. The governing surface motion laws are in general nonlocal…
Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity
In this paper, we investigate a time‐dependent family of plane closed Jordan curves evolving in the normal direction with a velocity that is assumed to be a function of the curvature, tangential…
Sturm’s Theorems on Zero Sets in Nonlinear Parabolic Equations
We present a survey on applications of Sturm’s theorems on zero sets for linear parabolic equations, established in 1836, to various problems including reaction-diffusion theory, curve shortening and…
Moving Surfaces and Abstract Parabolic Evolution Equations
It is the purpose of this paper to give a survey over some recent developments in the theory of classical solutions to elliptic and parabolic problems involving moving surfaces. Problems of this type…
Evolution of Plane Curves Driven by a Nonlinear Function of Curvature and Anisotropy
TLDR
This paper analyzes the so-called intrinsic heat equation governing the motion of plane curves obeying a geometric equation and shows how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities.
Propagation of a Mean Curvature Flow in a Cone
We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which moves toward the opening with normal velocity equaling to the mean curvature, and the contact angle between the…
Invariant manifolds for the porous medium equation
In this paper, we investigate the speed of convergence and higher-order asymptotics of solutions to the porous medium equation posed in R N . Applying a nonlinear change of variables, we rewrite the…
...
1
2
3
4
5
...

References

SHOWING 1-5 OF 5 REFERENCES
The normalized curve shortening flow and homothetic solutions
The curve shortening problem, by now widely known, is to understand the evolution of regular closed curves γ: R/Z -> M moving according to the curvature normal vector: dy/dt = kN = -"the ZΛgradient…
Nonlinear analytic semiflows
In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems ( x ′( t ) = f ( x ( t ))), and quasilinear initial value problems ( x ′( t )= A ( x ( t…
The heat equation shrinks embedded plane curves to round points
Soit C(•,0):S 1 →R 2 une courbe lisse plongee dans le plan. Alors C:S 1 ×[0,T)→R 2 existe en satisfaisant δC/δt=K•N, ou K est la courbure de C, et N est son vecteur unite normal entrant. C(•,t) est…
A stable manifold theorem for the curve shortening equation
On presente une famille de solutions homothetiques de l'equation pour une courbe planaire ∂X/∂τ=KN et on demontre l'existence de varietes non lineaires stables et instables autour de telles solutions
A Harnack inequality for parabolic di2erential equations