Mean Curvature Type Flow with Perpendicular Neumann Boundary Condition inside a Convex Cone

and Applied Analysis 3 the uniform parabolicity, and C1 bound on ρ by the classical theory of nonlinear parabolic equations (see, e.g., Chapter 12 in [13]). For simplicity, let r = ln ρ; then, ∇ i rρ = ∇ i ρ, |∇r| 2 ρ 2 = 󵄨󵄨󵄨󵄨 ∇ρ 󵄨󵄨󵄨󵄨 2 , u = ρ ω , (10) where ω = √1 + |∇r|2. Then, the geometric quantities in (8) can be represented as ] = 1 ω (x − e ij ∇ i r∇ j x) , g ij = ρ 2 (e ij + ∇ i r∇ j r) , g ij = 1 ρ 2 (e ij − ∇ i r∇ j r ω 2 ) ,


Introduction
Let N +1 () be a space form of sectional curvature  = 1, 0, or −1.It is well known that the Riemannian metric of N +1 () can be defined as where  2 is the standard induced metric of unit sphere S  in Euclidean space R +1 , and Recently, Guan and Li [1] introduced a new type flow in the above space form which was called mean curvature type flow, as follows: where  and ] are the mean curvature and the outward unit normal vector of the evolving hypersurfaces, respectively, and  is the support function of the evolving hypersurfaces defined by  = ⟨()(/), ]⟩.They proved that this flow evolves closed star-shaped hypersurfaces in space form into some sphere.A natural feature of this flow is that, along the mean curvature type flow, the volume enclosed by the evolving hypersurface is a constant and its area is always decreasing as long as the solution exists.
Inspired by this result, we focus on the corresponding problem with perpendicular Neumann boundary condition inside a convex cone in R +1 .Precisely, we suppose that Σ ⊂ R +1 is a convex cone with outward unit normal vector  and   is a smooth -dimensional hypersurface with boundary  defined by an initial embedding  0 :   → R +1 .We will study how   evolves under the flow (3) with boundary conditions:  0 () ⊂ Σ, and ⟨, ]⟩ = 0, where ] is the outward unit normal vector to   .Namely, we will consider the following mean curvature type flow with perpendicular Neumann boundary condition: where  is the Euclidean support function to the hypersurface   in R +1 defined by  = ⟨, ]⟩.We take the origin on R +1 at the vertex of the cone, and |()| denotes the norm of the position vector at some point  ∈   .In this paper, we obtain the following main result.

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Abstract and Applied Analysis Theorem 1. Suppose Σ is a convex cone and  0 is the initial hypersurface which can be represented as a graph over the intersection of the interior of the cone Σ and a unit sphere S  centered at the vertex of the cone; then, a solution to (4) exists for all time and stays always between two spheres with radii  1 = max ∈ 0 | 0 ()| and  2 = min ∈ 0 | 0 ()|.Furthermore, the solution converges exponentially fast to the intersection of the interior of the cone and the sphere with radius where | ∞ | denotes the area of the intersection of the limit sphere and the interior of the cone and || denotes the volume of the domain enclosed by the cone and the evolving hypersurface which is a constant under this flow.
The MCF with boundary conditions has been extensively studied by many mathematicians.Huisken in [2] considered the evolution of a graph over a bounded domain Ω ⊂ R  with perpendicular Neumann boundary condition and proved that the solution exists for all time and converges to a plane domain at last.More generally, for the hypersurfaces not necessarily represented as graphs, Stahl [3,4] studied this problem and proved that the flow converges to a round point on the condition that the boundary manifold was umbilic and the initial surface was convex.Buckland in [5] founded boundary monotonicity formulae and classified Type I boundary singularities for  > 0 with a perpendicular Neumann boundary condition.Recently, Lambert in [6] considered this problem in a Minkowski space with a timelike cone boundary condition and proved that this flow converges to a homothetically expanding hyperbolic solution.Subsequently, he [7] also considered this problem inside a rotational tori.
However, little is known about the modified MCF, such as volume or area preserving MCF, with perpendicular Neumann boundary condition.Let   be a tubular hypersurface between two parallel planes in R +1 , which can be represented as a graph over some cylinder inside it and meets the parallel planes perpendicularly.Recently, Hartley [8] studied the motion of   under the volume preserving mean curvature flow by the center manifold analysis and proved that the solution exists for all time and converges exponentially fast to a cylinder in the   topology for any  ∈ N as time  tends to infinity.
Generally, for the prescribed contact angle which is not necessarily a right angle, this problem seems more hard.Altschuler and Wu in [9] considered the evolution of 2dimensional graph over compact convex domain in R 2 by mean curvature flow under this boundary condition and proved this flow exists for all time and converges to translating solutions at last.Guan [10] extended Altschuler-Wu's result to graphs of high dimensions.
The rest of this paper is organized as follows.In Section 2, we reparametrize the system (4) as a graph and give some primary facts.In Section 3, the evolution equations and boundary derivatives for some useful geometric quantities will be derived.In Section 4, a maximal principle will be introduced and some basic estimates will be given.In the last section, we prove the convergence and complete the proof of the main theorem.

Reparametrization and Notations
Let   be a compact hypersurface inside an -dimensional convex cone Σ with boundary condition  ⊂ Σ, given by the embedding  : Ω ∈ S  → R +1 , where Ω is the intersection of the interior of the cone and unit sphere S  centered at the vertex  of the cone; that is,   can be expressed as a graph over Ω. Precisely, for any point  ∈ Ω ⊂ S  , there is only one ray from the vertex  through  intersecting the hypersurface   at some point (); the position vector to   can be expressed as Let { 1 , . . .,   } be the local normal coordinates on S  ;   denotes the standard spherical metric under the coordinates; the covariant derivative and divergent operator on S  with respect to the metric   are denoted by ∇ and div, respectively.Then, tangent vectors and the outward unit normal vector on   can be expressed as in [11] (see also P28 in [12]): where   is the inverse of   .Thus, the support function, induced metric, and second fundamental form can be given by straightforward calculation as following: Furthermore, the system (4) is equivalent to the following parabolic PDE defined on Ω × [0, ): where  = √  2 + |∇| 2 .The system ( 9) is a quasi-linear parabolic equation in divergence form, whose long-time existence is equivalent to the uniform parabolicity, and  1 bound on  by the classical theory of nonlinear parabolic equations (see, e.g., Chapter 12 in [13]).
For simplicity, let  = ln ; then, where  = √ 1 + |∇| 2 .Then, the geometric quantities in (8) can be represented as and ( 9) can be rewritten as System ( 12) is also a quasi-linear parabolic equation in divergence form, and the related estimates will be derived in Section 4.
For convenient calculation, we also parametrize the boundary cone as Lambert [6].Let S : S −1 → B  (0) ⊂ R  be a smooth embedding of a sphere into a topological ball centered at the origin with outward unit normal vector n.Then, we can define the boundary cone Σ by embedding R  into R +1 at height 1, defining Σ to be the set of all rays going through the origin and some point ( S(), 1) ∈ R +1 , where { 1 , . . .,  −1 } denotes the ( − 1)-dimensional coordinate for S.So we can parametrize the cone by where  +1 is the (+1)th standard coordinate vector in R +1 .
The second fundamental form of the boundary cone has the following characterization.where  Σ and  S denote the second fundamental forms, respectively, of Σ in R +1 and S in R  .
Proof.From the parametrization for the cone, it is easy to check that Since  is the outward unit normal vector to the boundary cone, calculating by Gauss equation directly we have This proves the first identity.Similarly, using ( 14) again we have Observing that  can be decomposed as and ⟨ 2 S/    ,  +1 ⟩ = 0, we, then, have where we use the fact that cos ∠(, n) = 1/ √ 1 + | S| 2 .

Evolution Equations and Boundary Derivatives
In this section, we will derive evolution equations for some useful geometric quantities by straightforward calculation.
Let { 1 , . . .,   } be the local normal coordinates of the evolving hyperserface and let   be the corresponding induced metric; let ∇ and Δ be, respectively, the covariant derivative and Laplace operator with respect to the induced metric   ; and || and || denote the norm of the position vector and the second fundamental form to evolving hypersurfaces   in R +1 , respectively.
Lemma 3.Under the mean curvature type flow (4), we have Proof.(i) We can calculate directly as in [14] On the other hand, This proves (i).
The following relationship between  Σ and  was proved by Stahl in [3].
In order to apply the Hopf maximal principle to obtain the basic estimates, we also need the following boundary derivatives.

Gradient Estimate
Let   (, ) ∈  ∞ (  × [0, )) be a positive definite matrix such that is a parabolic operator.We have the following maximal principle from Hopf Lemma [15] or Stahl's corresponding result in [4].
Now we apply the above theorem to || 2 .Combining (i) in Lemma 3 and (i) On the other hand, noticing ⟨, ]⟩ = 0 by assumption, ] must be in the tangent space of the boundary cone; that is, ] ∈ Σ.Combining (ii) in Lemma 5, Proposition 2, and the convexity of the boundary cone, we have Equivalently, By the maximal principle we have the following.
Combining Corollaries 7 and 8, we obtain long time existence for the system (12) by the standard argument for divergence PDE (cf.[13]), and then, the first part of Theorem 1 follows by the equivalence of ( 4) and (12).
Let  be the domain enclosed by the interior of the cone and the evolving hypersurface.The volume element for  is denoted by  and   for its boundary.We will also denote the area elements for the evolving hypersurface and its boundary by V and V  , respectively.Flow (4) has the following interesting property.
where  is the outward unit normal vector to the ( + 1)-dimensional region , and V Σ is the area element on boundary cone Σ.Hence, || is a constant and the result follows.

Convergence
In this section, we use an idea of Guan and Li in [1] to obtain our estimate and prove the exponential convergence.For that purpose, the system (9) or (12)