Interior gradient estimate for 1-D anisotropic curvature flow

We establish the interior gradient estimate for general 1-D anisotropic curvature flow. The estimate depends only on the height of the graph and not on the gradient at initial time.


Introduction
Let Ω be a bounded domain in R n . A surface given as a graph u : Ω → R is a minimal surface when u satisfies (1.1) div ∇u For this equation, the following interior gradient estimates are well-known ( [6,8,9]): Given a constant M andΩ ⊂⊂ Ω, there exists a constant C depending only on M andΩ such that if sup Ω |u| ≤ M, then supΩ |∇u| ≤ C. The standard elliptic theory ( [5]) subsequently gives all the interior C k,α (Ω) estimates of the graph u which depends only on M andΩ. The similar estimates are also known for the mean curvature flow equation ( [3]). That is, if u : Ω × (0, T ) → R satisfies and sup Ω×[0,T ] |u| ≤ M,Ω ⊂⊂ Ω, 0 < T 0 < T , then there exists C such that supΩ ×[T 0 ,T ] |∇u| ≤ C. Again, C is a constant depending only on M,Ω and T 0 . Note that C is independent of the gradient at t = 0.
One direction to extend those results are to consider general anisotropic curvature problem, namely, to consider the variational problem corresponding to the energy functional where ν = (∇u, −1)/ 1 + |∇u| 2 is the unit normal vector to the graph of u and the function a : S n−1 → R + is the surface energy density and should satisfy certain convexity property. The Euler-Lagrange equation is div p a p (ν) = 0, and the curvature flow equation is For both elliptic and parabolic problems in general dimensions, it is not known if the similar interior estimates for (1.1) and (1.2) hold equally for the anisotropic equations (1.3) and (1.4) so far. The main reason for the difficulty to extend the results is the lack of monotonicity formula for the mass ratio, which plays an important role in the measure-theoretic treatment of minimal submanifolds. On the other hand, if one allows the interior gradient estimate to depend on the gradient at t = 0, then the argument of [7] gives the interior gradient estimate.
In this paper, we show the interior gradient estimates for general anisotropic curvature flow for one-dimensional case which is independent of the initial time gradient. The proof utilizes the result of Angenent ([2]), which says that the number of zeros of the solution of parabolic equations is nonincreasing as time increases. We compare the solution to those for a suitable heat equation and use this result. We utilize the fact that the equation is invariant under the rotation here. We remark that the method we use is valid only for 1 dimensional case.

Theorem
Let r > 0 be given. The graph u : [−r, r] × [0, T ] → R is said to be an anisotropic curvature flow if smooth function u satisfies where a : R 2 → [0, ∞) is an anisotropic surface energy density function satisfying the following assumptions: The left-hand side of the equation corresponds to the normal velocity of the curve (x, u(x, ·)) while the right-hand side is the weighted anisotropic curvature. This is a gradient flow of the anisotropic surface energy functional Under these assumptions, we show Given 0 < s < r and 0 < t 0 < T , there exists a constant C > 0 depending only on δ 0 , M, t 0 , s, r such that Note that the estimate is independent the gradient of the initial data. Also we point out that the dependence of C on a is only through the lower bound of the uniform convexity δ 0 , but not on the upper bound (such as C 1 bound). Thus, the result in this paper can be extended equally to the nonsmooth anisotropic curvature flow problem [4] by approximations. Before the proof, we cite the following theorem due to Angenent [2] which is crucial in the proof: Here, a, b, c are smooth functions of (x, t) and a > 0. Then for all t ∈ (0, T ], the zero set of x → u(x, t) will be finite, even when counted with multiplicity. The number of zeros of x → u(x, t) counted with multiplicity is nonincreasing function of t.
Proof of Theorem. Given 0 < s < r and 0 < t 0 < T , we construct a solution v for (2.1) on [−s, s] × (0, T ] with the following properties: The property (c) means that v has an initial data which is vertical at x = −s. We show that the function v has a gradient bound 0  ,t), there has to be at least another pointx <x < λ + s such that u(x,t) = v λ (x,t). Thus u − v λ has at least two zeros at t =t on λ − s < x < λ + s. Function u − v λ satisfies the equation of the type (2.2) on [λ − s, λ + s] × (0, T ], with non-zero boundary values for all t > 0 due to sup |u| ≤ M and (a). Thus we may use Lemma 1 and conclude that u − v λ has at least two zeros in x variable for allt > t > 0. Since v λ > M for x away from λ − s and all small t, and since we assume that u is a smooth function up to t = 0, this is impossible to satisfy for all small enough t.
Thus it remains to prove the existence of such v. To do this, we invert the role of independent variable x and dependent variable y = v(x, t). Let y = w(x, t) be the inverse function of v with respect to the space variables, i.e., w satisfies y = v(w(y, t) Since ∂y ∂x = 1/ ∂x ∂y , we need to show that there exists a constant C > 0 such that w Let w be the unique smooth solution of (2.3) with the initial data Γ and the boundary data (a'). Since any functions c 1 + c 2 x are solutions of (2.3), one obtains the gradient estimate , by using these functions as barriers and the standard maximum principle applied to w x . Also, note that the convexity of w is preserved, i.e., w xx ≥ 0. This is seen by differentiating the equation with respect to t and then applying the maximum principle to w t . w t = 0 on the boundary and w t = a qq w xx ≥ 0 for t = 0 imply w t ≥ 0. The equation Now, (2.4) implies that a qq (−1, w x ) ≥ c(s, δ 0 )(call this δ)> 0 by assumption (c). We claim that the solution of   . This is because of the following combined with the standard maximum principle: In the last line, we used z xx ≥ 0, which follows by the same reason for w xx ≥ 0 before, and a qq (−1, w x ) ≥ δ. We next claim that for t 0 ≤ t, there exists c = c(t 0 , s, δ) > 0 such that z x ≥ c on [−M − 1, M + 1] × [t 0 , T ]. z x satisfies again the heat equation with non-negative initial data and the homogeneous Neumann data, and thus by the strong maximum principle (or extending the solution to R by a suitable reflection argument and then using the representation formula with the heat kernel) we have such c. Since w xx ≥ 0, for (x, t) with t ≥ t 0 , we have as the result. Note that we are using w ≥ z and w = z on the boundary x = −M − 1. This completes the proof.

Remark 1
After completing the manuscript, we learned that Julie Clutterbuck at Australian National University has recently considered the problem and obtained the similar interior estimates.