On the heat flow equation of surfaces of constant mean curvature in higher dimensions

Acta Mathematica Scientia 2011,31B(5):1741–1748 http://actams.wipm.ac.cn

ON THE HEAT FLOW EQUATION OF SURFACES OF CONSTANT MEAN CURVATURE IN HIGHER DIMENSIONS∗

)

Tan Zhong (




Wu Guochun (

)†

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China E-mail: [email protected]; [email protected]

Abstract In this paper, we consider the heat flow for the H-system with constant mean curvature in higher dimensions. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solution to the heat flow, that converges to zero 2 in W 1,n with the decay rate t 2−n as time goes to infinity. Key words heat equation; mean curvature; higher dimensions 2000 MR Subject Classification

1

35Q99; 58J35

Introduction

Let Ω ⊂ Rn be a bounded set with smooth boundary. We call u a weak solution of the H-system if u ∈ W 1,n (Ω, Rn+1 ) satisfies div(|∇u|n−2 ∇u) = H(u)ux1 ∧ · · · ∧ uxn ,

(1.1)

here ux1 ∧ · · · ∧ uxn is the cross product of the partial derivatives ux1 , · · ·, uxn , which can be described as follows. For any vector v ∈ Rn+1 ,     1 2  v v · · · v n+1      1 2  ux1 ux1 · · · un+1 x1  .  v · u x1 ∧ · · · ∧ u xn =  . ... . . . ...   ..     1 2  uxn uxn · · · un+1 xn  For a given η ∈ W 1,n (Ω, Rn+1 ) and a constant c, we consider the minimization problem  min |∇u|n , u = η on ∂Ω, V (u) = c, (1.2) Ω

∗ Received

June 28, 2010. Supported by NSFC (10976026). author: Wu Guochun.

† Corresponding

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 1 where V (u) = n+1 u · ux1 ∧ · · · ∧ uxn is a well defined volume for u ∈ W 1,n (Ω, Rn+1 ) (see Ω L.Mou-P.Yang [12]). A critical point of (1.2) is a solution of (1.1). When n = 2, lots of effort were devoted since the pioneering work of Wente [11] in 1969, to the study of the existence of the equation of H-surface (1.1) if H is a constant, Hildebrant [7] considered the Plateau problem for surfaces of constant mean curvature, Brezis-Coron [1] considered the multiple solutions problem of H-surfaces, and Struwe [10] obtained the existence of surfaces of constant mean curvature H with free boundaries. For variable H, there were recent works by Caldiroli-Musina [2]. For n ≥ 3, Mou-Yang [12] proved the existence of multiple solutions of the H-systems with a given boundary data if H is constant. Duzzer-Grotowski [19] studied the existence of solutions to the H-systems for non-constant functions H in higher dimensional compact Riemannian manifolds without boundary. In this paper, we consider the heat flow for H-systems of the following form: ⎧ n−2 ⎪ ∇u) − nHux1 ∧ · · · ∧ uxn , (x, t) ∈ Ω × (0, T ), ⎪ ⎨ ut = div(|∇u| ⎪ ⎪ ⎩

u(x, t) = χ,

(x, t) ∈ ∂Ω × (0, T ),

u(x, 0) = u0 (x),

x ∈ Ω,

(1.3)

3

where H is a constant, n ≥ 2. In case n = 2, if χ ∈ H 2 (∂Ω), Struwe [10] obtained the existence of surfaces with free boundaries under the condition |H|||u0 ||L∞ < 1; Rey [9] extended the main result to variable H under the assumption ||H||L∞ ||u0 ||L∞ < 1. Chang and Liu [3–6] studied the evolution of minimal surfaces with Plateau boundary condition, and boundary flow for the minimal surfaces with Plateau boundary condition. If the boundary value χ is zero, Huang-Tan-Wang [8] showed the existence and asymptotic estimates of global solutions and blow-up of (1.3) by applying the idea of stable set or potential well. Tan-Wu [16] considered the non-constant boundary. In this paper, we also consider zero-boundary in higher dimensions. In other words, we consider the following equation ⎧ n−2 ⎪ ∇u) − nHux1 ∧ · · · ∧ uxn , (x, t) ∈ Ω × (0, T ), ⎪ ⎨ ut = div(|∇u| (1.4) u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ), ⎪ ⎪ ⎩ u(x, 0) = u0 (x), x ∈ Ω, we prove that the heat flow develops finite time singularity if the initial value is in an unstable set and exists global solution with the initial value in stable set, and estimate exactly the decay rate. In order to describe our results, we need the following notations. Define  Q(u) = (n + 1)V (u) = u · u x1 ∧ · · · ∧ u xn , Ω

 n H u · u x1 ∧ · · · ∧ u xn , QH (u) = nHV (u) = n+1 Ω  1 |∇u|n , E(u) = D(u) + QH (u). D(u) = n Ω Lemma 1.1 For any u ∈ W01,n (Ω, Rn+1 ), there holds n

S|Q(u)| n+1 < nD(u),

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n

1

where S = n 2 wnn+1 and wn is the area of the unit n-sphere S n . In [12], we can find the details of it proof. Now, we introduce the stable set and the unstable set. Set M1 = {u ∈ W01,n (Ω, Rn+1 ) : E(u) ≤ 0} ∪ u ∈ W01,n (Ω, Rn+1 ) : S n+1 0 < E(u) < , |Q(u)| > (n + 1)nn+1 |H|n M2 =

u∈

W01,n (Ω, Rn+1 )



S n|H|

n+1 ,

S n+1 : 0 < E(u) < , |Q(u)| < (n + 1)nn+1 |H|n



S n|H|

n+1 .

Suppose n ≥ 3, then we have Theorem 1.1 If 0 =

u0 ∈ M1 , then the local regular solution to (1.4) must blow up at a finite time. Theorem 1.2 If u0 ∈ M2 , then there exists a unique global regular solution to (1.4). Moreover, 2 ||u(t)||W 1,n (Ω) = O(t 2−n ) as t → ∞. (1.5) 0

2

Proof of Theorem 1.1 This section is devoted to the proof of Theorem 1.1.

S n+1 Lemma 2.1 If u ∈ W01,n (Ω, Rn+1 ) satisfies |Q(u)| = n|H| , then E(u) ≥

S n+1 . (n + 1)nn+1 |H|n

Proof Applying Lemma 1.1 we obtain D(u) ≥

n S n+1 n |Q(u)|

=

S n+1 nn+1 |H|n .

Hence

n+1 S S n+1 n|H| S n+1 − = . E(u) ≥ D(u) − |QH (u)| ≥ n+1 n |H|n n + 1 n|H| (n + 1)nn+1 |H|n This completes the proof. We also need the following energy inequality for regular solutions to (1.4). Lemma 2.2 For any 0 ≤ T ≤ ∞, suppose that u(x, t) is a regular solution to (1.4), then it holds   t2

t1

Ω

|ut |2 + E(u(t2 )) = E(u(t1 )).

(2.1)

Proof Multiplying (1.4)1 by ut , integrating over (t1 , t2 ) × Ω, and applying integration by parts, we obtain Lemma 2.2. Proof of Theorem 1.1 Similar to Tan [15, 17], suppose that tmax = ∞ and denote  t f (t) = |u(s)|2 , t > 0. 0

Ω

Multiplying (1.4)1 by u and integrating over (0, t) × Ω, we have    t |u(t)|2 − |u0 |2 = −2n (D(u) + HQ(u)). Ω

Ω

0

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By the definition of f (t), we have f  (t) = f  (t) =

 Ω



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|u(t)|2 , thus we get

Ω



|u0 |2 − 2n

t

0

(D(u) + HQ(u))

(2.2)

and f  (t) = −2n(D(u(t)) + HQ(u(t))).

(2.3)

nHQ(u(t)) = (n + 1)(E(u(t)) − D(u(t))),

(2.4)

Since

(2.1), (2.3) and (2.4) give 

f (t) = 2D(u(t)) − 2(n + 1)E(u0 ) + 2(n + 1)

 t 0

Ω

|ut |2 .

(2.5)

Now we claim D(u(t)) − (n + 1)E(u0 ) ≥ 0, ∀ t > 0. Assume this claim for the moment. Then we obtain  t f  (t) ≥ 2(n + 1) |ut |2 . 0

Now we need to show

1 0

Ω

(2.6)

(2.7)

Ω

|ut |2 > 0. For, otherwise, u0 solves

div(|∇u0 |n−2 ∇u0 ) = nHu0x1 ∧ · · · ∧ u0xn . Multiplying the equation by u0 and integrating over Ω, we have D(u0 ) + HQ(u0 ) = 0, which implies 1 E(u0 ) = D(u0 ) > 0. (2.8) n+1

S n+1 . By Lemma 1.1 and (2.8), we have Since u0 ∈ M1 , we have |Q(u0 )| = n|H| E(u0 ) ≥ which contradicts to E(u0 ) < t ≥ 1, i.e.,

S n+1 , (n + 1)nn+1 |H|n

S n+1 (n+1)nn+1 |H|n .



f (t) ≥ 2(n + 1)

From (2.7), we obtain f (t) is strictly convex for

 t Ω

0

|ut |2 > 0, ∀ t > 1.

Thus, we have lim f (t) = lim f  (t) = ∞.

t→∞

t→∞

However, we have, for t ≥ 1,   t  

 t 2 2 |u| |ut | f (t)f (t) ≥ 2(n + 1) 

0

Ω

0

Ω

 t 2 n+1  ≥ 2(n + 1) (f (t) − f  (0))2 . uut = 2 Ω 0

(2.9)

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By (2.9), there exists a positive κ and t1 , such that for all t ≥ t1 , f (t)f  (t) ≥ (1 + κ)(f  (t))2 . Hence f (t)−κ is strictly concave on [t1 , ∞), contradict to the fact that f (t)−κ > 0,

lim f (t)−κ = 0.

t→∞

Thus, the short time regular solution to (1.4) must blow up at finite time. Now, we return to the proof of (2.6). Case 1 If E(u0 ) ≤ 0, (2.6) obviously holds. S n+1 S n+1 Case 2 0 < E(u0 ) < (n+1)n . By Lemma 2.2, we obtain n+1 |H|n , |Q(u0 )| > ( n|H| ) E(u(t)) <

S n+1 , ∀ t ≥ 0. (n + 1)nn+1 |H|n

Now we claim

|Q(u(t))| >

S n|H|

n+1

, ∀ t ≥ 0.

(2.10)

(2.11)

Notice (2.11) holds for t = 0. If (2.11) were false, then there would exist t0 > 0 such that

|Q(u(t0 ))| =

S n|H|

n+1 , ∀ t ≥ 0.

(2.12)

But (2.10) and (2.12) contradict to Lemma 2.1, hence (2.11) holds. S n+1 It follows from (2.11) and Lemma 1.1 that D(u(t)) ≥ nn+1 |H|n , ∀ t ≥ 0. Hence D(u(t)) − (n + 1)E(u0 ) ≥

S n+1 nn+1 |H|n

− (n + 1)

S n+1 = 0. (n + 1)nn+1 |H|n

This proves (2.6). Hence the proof of Theorem 1.1 is completed.

3

Proof of Theorem 1.2

We divide the proof of Theorem 1.2, into several steps. Step 1 Local existence We apply the same arguments as in [14], and for the sake of completeness, we give the proof here. As in [14], we consider the regularized nonlinear parabolic problem ⎧ n−2 2 ⎪ 2 ∇u) − nHu x1 ∧ · · · ∧ uxn , (x, t) ∈ Ω × (0, T ), ⎪ ⎨ ut = div(|∇u| + ) u(x, t) = χ, (x, t) ∈ ∂Ω × (0, T ), ⎪ ⎪ ⎩ u(x, 0) = u0 (x), x ∈ Ω.

(3.1)

By using semigroup theory in Banach space, Lunardi’s Theorem (see [18]) yields: Lemma 3.1 For any  > 0, problem (3.1) has a unique solution u in the sense of Lunardi [12] on a time interval [0, T ], and u ∈ C 1 ([0, T ], L2 (Ω)) ∩ C 0 ([0, T ], W 2,2 (Ω) ∩ C 1,α (Ω)). For the proof of Lemma 3.1 we refer to Section 4 of [13].

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Definition 3.1 For a map u : [t1 , t2 ] × Ω → Rn+1 , u ∈ L∞ ([t1 , t2 ], W 1,n (Ω)) and  > 0, P ∗ (, u) = sup P :

let

 sup (x,t)∈[t1 ,t2 ]×Ω

BP (x)

|∇u|n <  .

Theorem 3.1 There exists a constant 0 > 0 depending on Ω with following property. For arbitrary u0 ∈ C 1,α (Ω, Rn+1 ) ∩ W 2,2 (Ω, Rn+1 ) there exists a time T0 > 0, only depending on D(u0 ), P ∗ (0 , u0 ) and the geometry of Ω, such that for every  ∈ (0, 0 ), there exists a solution u ∈ C 0 ([0, T0 ], W 2,2 (Ω) ∩ C 1,α1 (Ω)) ∩ C 1 ([0, T0 ], L2 (Ω)) of (3.1) with initial value u0 . Moreover, there hold the following -independent estimates: ||ut ||L2 ([0,T0 ]×Ω) ≤ C(u0 ), ||∇u||L∞ ([0, T0 ] × Ω) ≤ C(||∇u0 ||L∞ (Ω) ), ||∇u||C 0,α1 ([0, T0 ] × Ω) ≤ C(||∇u0 ||C 0,α (Ω ), where the constants also depend on n and Ω, and the constant α1 ≤ α depends on α and ||∇u0 ||L∞ . The proof of Theorem 3.1 is almost parallel to the case of n-harmonic map flow as in [13]: in this case the right-hand term does not cause any trouble in obtaining high regularity estimates. We refer to [13] Section 4. By Theorem 3.1, now we can prove local existence of solution for problem (3.1) by passing to the limit as  → 0. Theorem 3.2 There exists a constant 0 > 0, depending on Ω with the following property. For arbitrary u0 ∈ C 1,α (Ω, Rn+1 ) ∩ W 2,2 (Ω, Rn+1 ), there exists a time T0 > 0 only depending on D(u0 ), P ∗ (0 , u0 ) and the geometry of Ω, such that there exists a weak solution u : [0, T0 ] × Ω → Rn+1 of (1.3). Furthermore, u satisfies the inequalities as in Theorem 3.1. Proof Let u be the solution of (3.1) corresponding to a given  on the time interval [0, T0 ], where T0 is constructed in Theorem 3.1. The aim is to pass the limit on the distributional form of (3.1) on [0, T0 ]. From the -independent estimates in Theorem 3.1 we know that at least {ut } is bounded in L2 ([0, T0 ] × Ω). Thus, we may choose a sequence k → 0 such that ut k  u weakly in L2 ([0, T0 ] × Ω). And from Theorem 3.1, we have ||∇uk ||C 0,α1 ([0, T0 ] × Ω) ≤ C(||∇u0 ||C 0,α (Ω) ). Without loss of generality, we obtain that ∇uk → ∇u

strongly in C 0,

α1 2

([0, T0 ] × Ω).

Now, for any C0∞ test function ϕ we pass to the limit k → 0 in  T0   T0  n−2 (∂t u ϕ) = − (( + |∇u |2 ) 2 ∇u ∇ϕ + nHϕux1 ∧ · · · ∧ uxn , 0

Ω

0

Ω

which implies that u is a weak solution of (1.3). To prove that u satisfies the bounds as in Theorem 3.1, we notice that the estimates in Theorem 3.1 is independent on . By standard argument of approximation, we can prove the local existence result of (1.3) in a more general case of initial value u0 ∈ W 1,n (Ω) and u ∈ C 0 ([0, T0 ], W 1,n (Ω)) Step 2 Global existence By Lemma 1.1 and a few simple calculations we obtain that, if u ∈ M2 , then E(u) > 0. Now we prove that u(x, t) ∈ M2 for any t > 0. Indeed, if

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S S ∗ n+1 there exists a t∗ ∈ ∂M2 , then we have E(u(x, t∗ )) = (n+1)n , n+1 |H|n or |Q(u(x, t ))| = ( n|H| ) which contradicts to Lemma 2.1 and Lemma 2.2. Therefore,  t S n+1 |ut |2 < (n + 1)nn+1 |H|n Ω 0

and D(u(t)) ≤ E(u(t)) + |QH (u(t))| ≤

S n+1 nn+1 |H|n

for any t > 0. Thus, u(x, t) is a global solution of equation (1.4). Step 3 Proof of (1.5) Set F (u) = nD(u) + nHQ(u). By Lemma 1.1, we have D(u) ≥

S n|Q(u)|

1 n+1

|Q(u)| ≥

Hence E(u) ≥ D(u) − |QH (u)| = D(u) −

S

= |H||Q(u)|.

(3.2)

1 n|H| |Q(u)| ≥ D(u). n+1 n+1

(3.3)

nS n|H|

Lemma 1.1 also implies that

  n+1  n1

n  n 1 n |H|n n n |H||Q(u)| ≤ |H| |∇u| = |∇u| D(u) S Ω S n+1 Ω   n1  n1

 (n + 1)nn+1 |H|n  (n + 1)nn+1 |H|n  E(u) D(u) ≤  E(u0 ) D(u). ≤ n+1 n+1 S S Since 0 < E(u0 ) < 0 < δ < 1, s.t.

n+1

S n+1 (n+1)nn+1 |H|n ,

n

|H| we have | (n+1)n E(u0 )| n < 1, therefore, there exists δ, S n+1

|H||Q(u(t))| ≤ (1 − δ)D(u(t)),

1

∀ t > 0.

Hence F (u(t)) ≥ nδD(u(t)),

∀ t > 0.

(3.4)

Direct calculation implies that d dt

 Ω

|u(t)|2 = −2F (u(t)),

∀ t > 0.

Integrating over [t, +∞) yields 

+∞ t

1 2

F (u(s)) ≤

 Ω

|u(t)|2 ≤ C

 Ω

2

|∇u(t)|2 ≤ C(D(u(t))) n ,

(3.5)

where we have used the Poincar´e inequality and H¨older inequality. Combining (3.4) and (3.5),  +∞ 2 we obtain t D(u(s)) ≤ C(D(u(t))) n . Hence

 t

Set G(t) =

 +∞ t

+∞

 n2 D(u(s)) ≤ C(D(u(t))).

(3.6)

n

D(u(s)), then (3.6) becomes CG (t) + (G(t)) 2 ≤ 0, ∀ t > 0. Hence 2

G(t) ≤ C(t + 1) 2−n ,

∀ t > 0.

(3.7)

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Since there implies E(u(t)) ≤ D(u(t)) + (3.7) gives

 +∞ t

n 2n + 1 D(u(t)) = D(u(t)), n+1 n+1

2

E(u(s)) ≤ C(t + 1) 2−n . Since E(u(t)) is monotone decreasing, it implies  E(u(t + 1)) ≤

t

t+1

2

E(u(s)) ≤ C(t + 1) 2−n .

(3.8)

(3.3), (3.8) and Poincar´e inequality, imply (1.5). The proof of Theorem 1.2 is now completed. References [1] Brezis H, Coron J M. Multiple Solutions of H-systems and Rellich’s conjecture. Comm Pure Appl Math, 1984, 37(2): 149–187 [2] Caldiroli P, Musina R. The Dirichlet problem for H-systems with small boundary data: blow-up phenomena and nonexistence results. Arch Rat Mech Anal, 2006, 181: 142–183 [3] Chang K C, Liu J Q. Heat flow for the minimal surface with Plateau boundary condition. Acta Mathematica Sinica (English Series), 2003, 19: 1–28 [4] Chang K C, Liu J Q. Another approach to the heat flow for Plateau problem. J Differ Equ, 2003, 189: 46–70 [5] Chang K C, Liu J Q. An evolution of minimal surfaces with Plateau condition. Calc Var, 2004, 19: 117–163 [6] Chang K C, Liu J Q. Boundary flow for the minimal surfaces in Rn with Plateau boundary condition. Proc Royal Soc Edinb, 2005, 135A: 537–562 [7] Hildebrant S. On the Plateau problem for surfaces of constant mean curvature. Comm Pure Appl Math, 1970, 23: 97–114 [8] Huang T, Tan Z, Wang C Y. On the heat flow of equation of surfaces of constant mean curvature. Manuscripta Math, 2001, 134(1/2): 259–271 [9] Rey O. Heat flow for the equation of surfaces with prescribed mean curvature. Math Ann, 1991, 297: 123–146 [10] Struwe M. The existence of surfaces of constant mean curvature with free boundaries. Acta Math, 1988, 1960(1/2): 19–64 [11] Wente H C. An existence theorem for surfaces of constant mean curvature. J Math Anal Appl, 1969, 26: 318–344 [12] Mou L, Yang P. Multiple solutions and regularity of H-systems. Indiana Univ Math J, 1996, 14: 1193–1222 [13] Hungerb¨ uhler N. m-harmonic flow. Ann Scuola Norm Sup Pisa Cl Sci, 1997, 24(4): 593–631 [14] Hong M C, Hsu D L. Global existence of the heat flow for H-system in higher dimensions. Preprint. [15] Tan Z. Asymptotic behavior and blowup of some degenerate parabolic equation with critical Sobolev exponent. Commun Appl Anal, 2004, 8(1): 67–85 [16] Tan Z, Wu G C. On the heat flow of equation of surfaces of constant mean curvature with non-zero boundary. Preprint. [17] Tan Z. Non-Newton filtration equation with special mediam v = id. Acta Math Sci, 2004, 24B(1): 118–128 [18] Lunardi A. On the local dynamical system associated to a fully nonlinear abstract parabolic equation//Laksmikantham V, ed. Nonlinear Analysis and Applications. Marcel Decker Inc, 1987: 319–326 [19] Duzaar F, Grotowski J F. Existence and regularity for higher-dimensional H-systems. Duke Math J, 2000, 101(3): 459–485