AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/315768 315768 Research Article Mean Curvature Type Flow with Perpendicular Neumann Boundary Condition inside a Convex Cone Guo Fangcheng http://orcid.org/0000-0003-2491-5232 Li Guanghan Wu Chuanxi Apreutesei Narcisa C. 1 School of Mathematics and Statistics Hubei University Wuhan 430062 China hubu.edu.cn 2014 1772014 2014 09 01 2014 30 06 2014 17 7 2014 2014 Copyright © 2014 Fangcheng Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the evolution of hypersurfaces with perpendicular Neumann boundary condition under mean curvature type flow, where the boundary manifold is a convex cone. We find that the volume enclosed by the cone and the evolving hypersurface is invariant. By maximal principle, we prove that the solutions of this flow exist for all time and converge to some part of a sphere exponentially as t tends to infinity.

1. Introduction

Let Nn+1(K) be a space form of sectional curvature K=1,0, or -1. It is well known that the Riemannian metric of Nn+1(K) can be defined as (1)ds2=dρ2+ϕ2(ρ)dz2, where dz2 is the standard induced metric of unit sphere Sn in Euclidean space Rn+1, and (2)ϕ(ρ)={sinρK=1ρK=0sinhρK=-1.

Recently, Guan and Li  introduced a new type flow in the above space form which was called mean curvature type flow, as follows: (3)Ft=(nϕ(ρ)-Hu)ν, where H and ν are the mean curvature and the outward unit normal vector of the evolving hypersurfaces, respectively, and u is the support function of the evolving hypersurfaces defined by u=ϕ(ρ)(/ρ),ν. 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 Rn+1. Precisely, we suppose that ΣRn+1 is a convex cone with outward unit normal vector μ and Mn is a smooth n-dimensional hypersurface with boundary M defined by an initial embedding F0:MnRn+1. We will study how Mn evolves under the flow (3) with boundary conditions: F0(M)Σ, and μ,ν=0, where ν is the outward unit normal vector to Mn. Namely, we will consider the following mean curvature type flow with perpendicular Neumann boundary condition: (4)dFdt=(n-Hu)ν(y,t)Mn×[0,T)F(·,0)=M0F(y,t)Σ(y,t)Mn×[0,T)μ,ν(y,t)=0(y,t)Mn×[0,T), where u is the Euclidean support function to the hypersurface Mt in Rn+1 defined by u=F,ν.

We take the origin on Rn+1 at the vertex of the cone, and |F(y)| denotes the norm of the position vector at some point yMn. In this paper, we obtain the following main result.

Theorem 1.

Suppose Σ is a convex cone and M0 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 Sn centered at the vertex of the cone; then, a solution to (4) exists for all time and stays always between two spheres with radii R1=maxyM0|F0(y)| and R2=minyM0|F0(y)|. Furthermore, the solution converges exponentially fast to the intersection of the interior of the cone and the sphere with radius (5)R=(n+1)|V||M|, where |M| denotes the area of the intersection of the limit sphere and the interior of the cone and |V| 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  considered the evolution of a graph over a bounded domain ΩRn 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  founded boundary monotonicity formulae and classified Type I boundary singularities for H>0 with a perpendicular Neumann boundary condition. Recently, Lambert in  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  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 Mn be a tubular hypersurface between two parallel planes in Rn+1, which can be represented as a graph over some cylinder inside it and meets the parallel planes perpendicularly. Recently, Hartley  studied the motion of Mn 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 Ck topology for any kN as time t 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  considered the evolution of 2-dimensional graph over compact convex domain in R2 by mean curvature flow under this boundary condition and proved this flow exists for all time and converges to translating solutions at last. Guan  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.

2. Reparametrization and Notations

Let Mn be a compact hypersurface inside an n-dimensional convex cone Σ with boundary condition MΣ, given by the embedding F:ΩSnRn+1, where Ω is the intersection of the interior of the cone and unit sphere Sn centered at the vertex o of the cone; that is, Mn can be expressed as a graph over Ω. Precisely, for any point xΩSn, there is only one ray from the vertex o through x intersecting the hypersurface Mn at some point F(x); the position vector to Mn can be expressed as (6)F(x)=ρ(x)x,ρ(x)R+,xΩSn.

Let {x1,,xn} be the local normal coordinates on Sn; eij denotes the standard spherical metric under the coordinates; the covariant derivative and divergent operator on Sn with respect to the metric eij are denoted by and div, respectively. Then, tangent vectors and the outward unit normal vector on Mn can be expressed as in  (see also P28 in ): (7)iF=iρx+ρix,ν=1ρ2+|ρ|2(ρx-eijiρjx), where eij is the inverse of eij. Thus, the support function, induced metric, and second fundamental form can be given by straightforward calculation as following: (8)u=ρ2ρ2+|ρ|2,gij=ρ2eij+iρjρ,gij=1ρ2(eij-iρjρρ2+|ρ|2),hij=1ρ2+|ρ|2(-ρijρ+2iρjρ+ρ2eij).

Furthermore, the system (4) is equivalent to the following parabolic PDE defined on Ω×[0,T): (9)dρdt=(n-Hu)ρu=div(1σρ)+nρσ|ρ|2dρdt=(n-Hu)ρu=vv(x,t)Ω×[0,T),ρ·μ=0(x,t)Ω×[0,T),ρ(x,0)=ρ0(x), where σ=ρ2+|ρ|2.

The system (9) is a quasi-linear parabolic equation in divergence form, whose long-time existence is equivalent to the uniform parabolicity, and C1 bound on ρ by the classical theory of nonlinear parabolic equations (see, e.g., Chapter 12 in ).

For simplicity, let r=lnρ; then, (10)irρ=iρ,|r|2ρ2=|ρ|2,u=ρω, where ω=1+|r|2. Then, the geometric quantities in (8) can be represented as (11)ν=1ω(x-eijirjx),gij=ρ2(eij+irjr),gij=1ρ2(eij-irjrω2),hij=ρω(-ijr+irjr+eij), and (9) can be rewritten as (12)drdt=nu-H=div(1ρωr)+(n+1)|r|2ρω(x,t)Ω×[0,T),r·μ=0(x,t)Ω×[0,T),r(x,0)=r0(x).

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 . Let S~:Sn-1Bn(0)Rn 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 Rn into Rn+1 at height 1, defining Σ to be the set of all rays going through the origin and some point (S~(z),1)Rn+1, where {z1,,zn-1} denotes the (n-1)-dimensional coordinate for S~. So we can parametrize the cone by (13)FΣ(l,S~(z))=l1+|S~|2(S~(z)+en+1), where en+1 is the (n+1)th standard coordinate vector in Rn+1. The second fundamental form of the boundary cone has the following characterization.

Proposition 2.

For the second fundamental form of the boundary cone, one has

AΣ(/l,·)=0,

AΣ(S~/zi,S~/zj)=(l/(1+|S~|2))AS~(S~/zi,S~/zj), i=1,,n-1,

where AΣ and AS~ denote the second fundamental forms, respectively, of Σ in Rn+1 and S~ in Rn.

Proof.

From the parametrization for the cone, it is easy to check that (14)S~zi,μ=0,S~+en+1,μ=0.

Since μ is the outward unit normal vector to the boundary cone, calculating by Gauss equation directly we have (15)AΣ(l,l)=hllΣ=-2FΣl2,μ=0,μ=0,AΣ(l,S~zi)=hliΣ=-11+|S~|2S~zi-(11+|S~|2)immmmm×(S~(z)+en+1),μ(11+|S~|2)i=0. This proves the first identity.

Similarly, using (14) again we have (16)AΣ(S~zi,S~zj)=-2FΣzizj,μ=-2S~zizj,μl1+|S~|2.

Observing that μ can be decomposed as (17)μ=μ,nn+μ,en+1en+1 and 2S~/zizj,en+1=0, we, then, have (18)AΣ(S~zi,S~zj)=-lcos(μ,n)1+|S~|2n,2S~zizj=l1+|S~|2AS~(S~zi,S~zj), where we use the fact that cos(μ,n)=1/1+|S~|2.

3. Evolution Equations and Boundary Derivatives

In this section, we will derive evolution equations for some useful geometric quantities by straightforward calculation. Let {y1,,yn} be the local normal coordinates of the evolving hyperserface and let gij be the corresponding induced metric; let ¯ and Δ¯ be, respectively, the covariant derivative and Laplace operator with respect to the induced metric gij; and |F| and |A| denote the norm of the position vector and the second fundamental form to evolving hypersurfaces Mn in Rn+1, respectively.

Lemma 3.

Under the mean curvature type flow (4), we have

(d/dt-uΔ¯)|F|2=0,

dν/dt=-¯(n-Hu),

(d/dt-uΔ¯)u=n-2Hu+|A|2u2+H¯u,F.

Proof.

(i) We can calculate directly as in  (19)d|F|2dt=2dFdt,F=2(n-Hu)ν,F=2(n-Hu)u.

On the other hand, (20)Δ¯|F|2=gij¯i¯jF,F=2gij¯iFyj,F=2n-2gijhijν,F=2(n-Hu). This proves (i).

(ii) As ν,ν=1, we have dν/dt,ν=0, and then, (21)dνdt=dνdt,FyigijFyj=-ν,yiFtgijFyj=-yi(n-Hu)gijFyj=-¯(n-Hu).

(iii) Similarly, as in (i), (22)dudt=ddtF,ν=dFdt,ν+F,dνdt=(n-Hu)-F,¯(n-Hu),Δ¯u=gij¯i¯jF,ν=gij¯i(Fyj,ν+F,hjkFyk)=gijgikhjk+F,¯H-hjkhilgijgklu=H+F,¯H-|A|2u. Then, (iii) follows by combining the above two formulae.

The following relationship between AΣ and A was proved by Stahl in .

Lemma 4 (see [<xref ref-type="bibr" rid="B13">3</xref>]).

For pM×[0,T), one has (23)AΣ(ξ,ν)=-A(ξ,μ), where ξTpMTpΣ.

In order to apply the Hopf maximal principle to obtain the basic estimates, we also need the following boundary derivatives.

Lemma 5.

For (y,t)M×[0,T), one has

¯|F|2,μ=0,

¯u,μ=uAΣ(ν,ν).

Proof.

(i) Denote by ~ the Euclidean covariant derivative in Rn+1. Obviously, ~|F|2TpΣ; combining the boundary condition in (4), we have (24)¯|F|2,μ=~|F|2-~|F|2,νν,μ=~|F|2,μ=0.

(ii) Calculating directly, we obtain (25)¯u=F,νyigijyj=A(F,yi)gijyj,¯u,μ=A(F,yi)gijyj,μ=A(F,μ). Observing F=F-uν, combination of Lemma 4 and (i) in Proposition 2 yields (26)¯u,μ=-AΣ(F-uν,ν)=-AΣ(F,ν)+uAΣ(ν,ν)=uAΣ(ν,ν).

Let aij(y,t)L(Mn×[0,T)) be a positive definite matrix such that (27)Lf=dfdt-aij¯i¯jf is a parabolic operator. We have the following maximal principle from Hopf Lemma  or Stahl’s corresponding result in .

Theorem 6.

Suppose f:Mn×[0,T)R satisfies (28)dfdt-aij¯i¯jf(y,t)0(y,t)Mn×[0,T)s.t.  ¯f(y)=0,¯f,μ0(y,t)Mn×[0,T), and then f(y,t)supyMnf(y,0) for all (y,t)Mn×[0,T).

Now we apply the above theorem to |F|2. Combining (i) in Lemma 3 and (i) in Lemma 5, we immediately have the following estimates.

Corollary 7.

Let F(y,t) be a solution to system (4); then, one has (29)minyM0n|F0(y)||F(y,t)|maxyM0n|F0(y)| for any (y,t)Mn×[0,T).

This result means that the evolving hypersurface always stays between two spheres with radii R1=maxyM0n|F0(y)| and R2=minyM0n|F0(y)|.

In order to obtain the gradient estimate for system (12), we only need to estimate the lower bound for u=ρ/ω=ρ/1+|r|2. From (iii) in Lemma 3, (30)(ddt-uΔ¯)u=n-2Hu+|A|2u2+H¯u,F(n-Hu)-Hun(n-Hu)+H¯u,F=1n(n-Hu)2+H¯u,F.

On the other hand, noticing μ,ν=0 by assumption, ν must be in the tangent space of the boundary cone; that is, νTΣ. Combining (ii) in Lemma 5, Proposition 2, and the convexity of the boundary cone, we have (31)¯u,μ=uAΣ(ν,ν)0. Equivalently, (32)(ddt-uΔ¯)(-u)-1n(n-Hu)20(y,t)Mn×[0,T)s.t.¯u(y,t)=0,¯(-u),μ=-uAΣ(ν,ν)0(y,t)Mn×[0,T).

By the maximal principle we have the following.

Corollary 8.

For all (y,t)Mn×[0,T), the support function u(y,t) satisfies (33)u(y,t)minyMnu(y,0).

Combining Corollaries 7 and 8, we obtain long time existence for the system (12) by the standard argument for divergence PDE (cf. ), and then, the first part of Theorem 1 follows by the equivalence of (4) and (12).

Let V be the domain enclosed by the interior of the cone and the evolving hypersurface. The volume element for V is denoted by dV and dV for its boundary. We will also denote the area elements for the evolving hypersurface and its boundary by dv and dv, respectively. Flow (4) has the following interesting property.

Proposition 9.

Along flow (4), the volume of V, denoted by |V|, is a constant.

Proof.

For the function Φ(y,t)=(1/2)|F(y,t)|2 defined on the evolving hypersurface Mt, we have (34)Δ¯Φ=n-Hu from the proof of Lemma 3(i). Integrating the above equation on Mt and taking into consideration Lemma 5(i) yield (35)Mt(n-Hu)dv=MtΔ¯Φdv=Mt¯Φ,μdv=12Mt¯|F|2,μdv=0.

Denote by div~X the divergence of the position vector X for V in Rn+1; we have (36)Vdiv~XdV=(n+1)|V|.

Taking derivative with respect to time t and combining divergence theorem yield (37)(n+1)ddt|V|=Vdiv~(dXdt)dV=VdXdt,ηdV=ΣdXdt,μdvΣ+MtdXdt,νdv=Mt(n-Hu)dv=0, where η is the outward unit normal vector to the (n+1)-dimensional region V, and dvΣ is the area element on boundary cone Σ. Hence, |V| is a constant and the result follows.

5. Convergence

In this section, we use an idea of Guan and Li in  to obtain our estimate and prove the exponential convergence. For that purpose, the system (9) or (12) is convenient for us. Now, we choose a local coordinate {x1,,xn} on ΩSn. and Δ are again the covariant derivative and Laplace operator with respect to the standard metric eij on Sn. Let L be a parabolic operator defined by (38)Lf=dfdt-1ρω(eij-irjrω2)fij on Ω×[0,T) for a smooth function f:Ω×[0,T)R. From , we have the following evolution equation for |r|2/2 at the critical point (39)L(|r|22)=-nρω|r|4-n-1ρω|r|2-1ρωΔr|r|2-1ρω|2r|2.

Using Cauchy-Schwartz inequality and similar rearrangement as in , we have (40)L|r|2-3n2ρω|r|4-2(n-1)ρω|r|2-2nρω(Δr+n2|r|2)2.

Therefore, (41)L|r|2-2(n-1)ρω|r|20.

Recall that ρ=ωu and ω=1+|r|2; we have, by Lemma 5 and the Neumann boundary assumption, for any points (x,t)Ω×[0,T), (42)(|r|2),μ=(ω2-1),μ=2ρu2ρ,μ-2ρ2u3u,μ=-2ρ2u2AΣ(ν,ν).

With the assumption of convexity on the boundary cone, we obtain, by Theorem 6, (43)|r|2(x,t)maxxΩ|r|2(x,0).

By the equivalence of (4) and (9), we have (44)minxΩρ(x,0)ρ(x,t)maxxΩρ(x,0) from Corollary 7.

From the above estimate, there exists a uniform positive constant α2(n-1)/ρω depending only on the upper bound of ω and ρ such that (41) reads as (45)L|r|2-α|r|2.

Denote by g=eαt|r|2(46)Lg=L(eαt|r|2)=αeαt|r|2+eαtL|r|20,g,μ=eαtω2,μ=-2ρ2u2eαtAΣ(ν,ν)0.

Then, by Theorem 6 again, g=eαt|r|2 has uniform upper bound C1=maxxΩg(x,0); that is (47)|r|2C1e-αt,t[0,). Or, equivalently, for a different constant C2=ρmaxxΩ2(x,0)C1, (48)|ρ|2C2e-αt,t[0,). This means that Mt converges exponentially fast to some part of a sphere.

Assume the radial function ρ(x,t0) attains its minimum at a point pΩ for some t0[0,); that is, ρ(p,t0)=ρminxΩ(x,t0). Let σ:[0,s)Ω be a geodesic on the unit sphere Sn starting from p to any point xΩ with σ(s)=x. Integrating both sides of the last inequality on [0,s] and taking into account the boundedness of s, we have, for some constant C3=C3(Ω,C2,α) and β=α/2, (49)0sdρdsds0s|ρ|dsC3e-βt0, and then, (50)ρ(x,t0)-ρ(p,t0)=ρ(x,t0)-ρminxΩ(x,t0)C3e-βt0.

So we have (51)ρmaxxΩ(x,t)-ρminxΩ(x,t)C3e-βt,t[0,).

On the other hand, (52)(n+1)|V|=Vdiv~XdV=VX,ηdV=ΣX,μdvΣ+MtX,νdv=Mtudv.

Because (53)ρmaxxΩ(x,t)-C3e-βtρ(x,t)ρminxΩ(x,t)+C3e-βt,1ω=1+|r|21+C1e-αt, the support function u=ρ/ω satisfies (54)ρmaxxΩ(x,t)-C3e-βt1+C1e-αtuρminxΩ(x,t)+C3e-βt.

Denote by |Mt| the area of Mt; combining (52), we have (55)|Mt|ρmaxxΩ(x,t)-C3e-βt1+C1e-αt(n+1)|V||Mt|(ρminxΩ(x,t)+C3e-βt), or, equivalently, (56)(n+1)|V|ρminxΩ(x,t)+C3e-βt|Mt|(n+1)|V|1+C1e-αtρmaxxΩ(x,t)-C3e-βt.

Let limtρ(x,t)=ρ(x), and from the above inequality, we obtain ρ(x)=R, a constant, and therefore, (57)|Mt|(n+1)|V|R,t. That is to say, Mt converges exponentially to the intersection of the interior of the cone Σ and a sphere centered at the vertex of the cone with radius (58)R=(n+1)|V||M|, where |M| denotes the area of the limit sphere M. This finishes the proof of Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research is partially supported by NSFC (no. 11171096), RFDP (no. 20104208110002), and Funds for Disciplines Leaders of Wuhan (no. Z201051730002).

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