height estimates for graph parametrized surfaces

Bull. Sci. math. 136 (2012) 848–856www.elsevier.com/locate/bulsci

Height estimates for graph parametrized surfaces

Marcelo Montenegro a,∗, Sebastián Lorca b

a Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651,CEP 13083-859, Campinas, SP, Brazil

b Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7 D, Arica, Chile

Received 7 June 2012

Available online 13 July 2012

Abstract

We show a priori bounds for positive solutions of the equation −div( ∇u√1+|∇u|2 ) = f (x,u) on a general

bounded domain Ω with u = 0 on ∂Ω . The results can be interpreted as height estimates for parametrizedsurfaces as graphs. We also show existence and regularity of solutions.© 2012 Elsevier Masson SAS. All rights reserved.

MSC: 53A10; 35J65; 35Q35; 76B45

Keywords: Dirichlet problem; Prescribed mean curvature; A priori bounds; Height estimates; Regularity

1. Introduction

In this paper we prove an a priori bound for positive C2(Ω) ∩ C1(Ω) solutions of the pre-scribed mean curvature equation⎧⎨

⎩−div

( ∇u√1 + |∇u|2

)= f (x,u) in Ω,

u = 0 in ∂Ω,

(1)

where Ω ⊂ RN , N � 2, is a bounded domain with boundary ∂Ω of class C2. No convexity

assumption is assumed on Ω . The solutions u of (1) are surfaces parameterized as graphs inR

N+1 over Ω . We assume that

* Corresponding author.E-mail addresses: [email protected] (M. Montenegro), [email protected] (S. Lorca).

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M. Montenegro, S. Lorca / Bull. Sci. math. 136 (2012) 848–856 849

f : Ω ×R→ [0,∞) is C1, (2)

lims→∞

f (x, s)

sq= 0 uniformly for x ∈ Ω for some 0 < q <

1

N − 1, (3)

either

lim sups→∞

sf (x, s) − τ [F(x, s) + 1N

x∇xF (x,u)]sf (x, s)

1N

� 0

uniformly for x ∈ Ω for some 0 � τ <N

N − 1, (4)

or

lim infs→∞

sf (x, s) − τ [F(x, s) + 1N

x∇xF (x,u)]sf (x, s)

1N

� 0

uniformly for x ∈ Ω for some τ >N

N − 1, (5)

where F(x, s) = ∫ s

0 f (x, ξ) dξ .If f (x,u) = uq with 0 < q < 1/(N − 1), assumptions (3), (4) and (5) are immediately sat-

isfied. If f ≡ H > 0 is constant or f = f (x) depending only on x, then (4) is satisfied with1 < τ < N/(N − 1). The Dirichlet problem (1) has been addressed in [2,8,16–19]. A surveydealing with constant mean curvature surfaces is [15]. For compact graphs in R

3 with planarboundary, it was obtained in [9] that the optimal height estimate of u from the plane is 1/H , andit is attained by the hemisphere of radius 1/H .

There are many unbounded surfaces with constant mean curvature, e.g., the cylinder, catenoid,helicoid and Delunay surfaces, see [12] for an account. Constant mean curvature surfaces overcircular domains with zero Dirichlet boundary data have been constructed in [11]. Notice thatsuch CMC surfaces need not to be C1 up to the boundary of Ω . A simple example is a hemisphereu over a disk B . If ∂B is an equator, than ∇u(x) blows up as x ∈ B approaches the boundary ∂B .A classical assumption yielding C2,α(Ω) CMC surfaces over convex sets is N

N−1 |H | � K , whereK is the mean curvature of the boundary ∂Ω and f ≡ NH is constant, see [22]. More generalassumptions over convex sets have been devised in [19]. Estimates of solutions on non-convexdomains have been treated in [23].

The a priori bound stated in Theorem 1.1 is related to height estimates for surfaces. There isa well developed theory for tridimensional surfaces with positive constant mean curvature H .Similar results to [9] were obtained in the hyperbolic 3-dimensional space by [13]. The sameproblem was studied in the product space of a Riemannian surface and R by [10], and optimalestimates have been obtained in [1]. We refer to [3,5,14,20] and [21] for more results of thisnature and applications of such estimates to other issues.

We state now our main result.

Theorem 1.1. Let u ∈ C2(Ω)∩C1(Ω) be a positive solution of (1). Assume (2), (3), (4), (5) andthat Ω is a bounded domain with ∂Ω of class C2. Then supΩ u � C, where the constant C doesnot depend on u.

To obtain the a priori bound of Theorem 1.1, we first prove that the existing positive solutionu of (1) belongs to Lp for every p. For that matter we use conditions (3), (4) and (5), which areinspired in those from [4]. We also need the so-called Pohozaev identity for the mean curvature

850 M. Montenegro, S. Lorca / Bull. Sci. math. 136 (2012) 848–856

equation of Lemma 2.1. We cannot simply let p → ∞ to conclude boundedness of u, since theinvolved constants in the Lp estimate depend on p. We bypass this step with a careful iterationscheme, starting from a reversed Hölder inequality leading to supΩ u� const.

In Section 2 we prove Theorem 1.1. Section 3 is devoted to some remarks about regularity,existence and nonexistence of solutions.

2. Proof of Theorem 1.1

Throughout the paper C,Cε,Cα,Cp denote various positive constants which are independentof u and that may vary from place to place.

We will need the following Pohozaev type identity which follows by a multiplication of Eq. (1)by x∇u and an integration, we omit the proof.

Lemma 2.1. Let u ∈ C2(Ω) ∩ C1(Ω) be a solution of (1). Then∫Ω

{N

(1 + |∇u|2)1/2 − uf (x,u) − NF(x,u) − x∇xF (x,u)

}dx

=∫

∂Ω

1

(1 + |∇u|2)1/2(x.ν) ds, (6)

where ν denotes the unit outward normal vector to ∂Ω .

Proof of Theorem 1.1. Step 1. First we show that u ∈ Lp(Ω) for every p > 1.We denote by ν the unit outward normal vector to ∂Ω . An integration furnishes∫

Ω

f (x,u) =∫Ω

−div

( ∇u√1 + |∇u|2

)= −

∫∂Ω

∇u√1 + |∇u|2 ν.

Thus, ∫Ω

f (x,u) �∫

∂Ω

|ν| � |∂Ω|. (7)

Multiplying Eq. (1) by u we obtain∫Ω

|∇u| �∫Ω

(1 + |∇u|2)1/2 =

∫Ω

1

(1 + |∇u|2)1/2+ uf (x,u) � |Ω| +

∫Ω

uf (x,u) (8)

and ∫Ω

uf (x,u) =∫Ω

|∇u|2(1 + |∇u|2)1/2

�∫Ω

|∇u|. (9)

Recall the Sobolev imbedding (see [7])

N1/2(∫

|u| NN−1

)N−1N

�∫

|∇u|. (10)

Ω Ω


By estimate (7), (8), (10) and Hölder inequality one obtains

∫Ω

uf (x,u)1N �

(∫Ω

uN

N−1

)N−1N

(∫Ω

f (x,u)

) 1N

� |∂Ω|1/N

(∫Ω

uN

N−1

)N−1N

� N−1/2|∂Ω|1/N

(|Ω| +

∫Ω

uf (x,u)

). (11)

The boundary integral of the Pohozaev identity of Lemma 2.1 is bounded, since∣∣∣∣∫

∂Ω

1

(1 + |∇u|2)1/2(x.ν) ds

∣∣∣∣ �∫

∂Ω

∣∣(x.ν)∣∣ds � C

then we have from (6),∫Ω

|∇u| �∫Ω

F(x,u) + 1

N

∫Ω

uf (x,u) + 1

N

∫Ω

x∇xF (x,u) + C (12)

and inserting (9) into (6) we obtain∫Ω

F(x,u) + 1

N

∫Ω

x∇xF (x,u) � N − 1

N

∫Ω

uf (x,u) + C (13)

where the constant C does not depend on u nor on ∇u. Estimates (11) and (13) together withcondition (4) imply∫

Ω

uf (x,u) � τ

∫Ω

F(x,u) + τ

N

∫Ω

x∇xF (x,u) + ε

∫Ω

uf (x,u)1N + Cε

�(

N − 1

N

)τ

∫Ω

f (x,u)u + εC

∫Ω

f (x,u)u + Cε,

where ε > 0 is small enough and Cε > 0 is a large constant depending only on ε. Then∫Ω

uf (x,u) � C,

hence (8) implies∫Ω

|∇u| � C. (14)

Assume now (5) holds, then

−∫

uf (x,u) + τ

∫F(x,u) + τ

N

∫x∇xF (x,u) � ε

∫uf (x,u)

1N + Cε (15)

Ω Ω Ω Ω


where ε > 0 can be assumed to be very small and Cε > 0 is a large constant depending only on ε.By (12),

τ

∫Ω

|∇u| − τ

N

∫Ω

uf (x,u) −∫Ω

uf (x,u) � ε

∫Ω

uf (x,u)1N + Cε. (16)

Using (9) and (11),((N − 1)τ − N

N

)∫Ω

uf (x,u) − εC

∫Ω

uf (x,u) � Cε. (17)

Again we obtain∫Ω

uf (x,u) � C,

by choosing ε > 0 small enough. Hence (8) implies (14).Multiply now Eq. (1) by uα with α > 1 to be chosen later. We obtain∫

Ω

∣∣∇(uα

)∣∣ � ∫Ω

αuα−1(1 + |∇u|2)1/2

=∫Ω

uαf (x,u) + αuα−1

(1 + |∇u|2)1/2

�∫Ω

uαf (x,u) + αuα−1. (18)

Using (3) one sees that f (x,u) � εuq for u � Cε where ε > 0 can be assumed to be very smalland C := Cε > 0 is a large constant depending only on ε. Hence∫

Ω

uαf (x,u) �∫

{u<C}uαf (x,u) +

∫{u>C}

uαf (x,u) � C

∫Ω

uα−1 + ε

∫Ω

uα+q . (19)

Estimating the left-hand side of (18) by Sobolev imbedding (10) and using (19) yields

N1/2(∫

Ω

us

)N−1N

� ε

(∫Ω

us

)N−1N

(∫Ω

uqN

) 1N + (α + C)

∫Ω

uα−1

where s = αNN−1 . We take a constant C0 > 0 such that α + C < C0α. Again, Sobolev inequality

(10) and the bound (14) provides us with(∫Ω

us

)N−1N

� εC

(∫Ω

us

)N−1N + αC

∫Ω

uα−1. (20)

Using Hölder inequality(∫us

)N−1N

� εC

(∫us

)N−1N + αC

(∫us

)(N−1)(α−1)/αN

. (21)

Ω Ω Ω


Thus∫Ω

us � Cα,

when ε is taken to be small enough. Notice that constant Cα increases as α increases. A recursivescheme using (20) and the fact that s > α − 1 gives∫

Ω

up � Cp for every p > 1, (22)

hence by (3),∫Ω

f (x,u)p � Cp for every p > 1.

The constant Cp depends on p and increases as p increases.Step 2. Our aim now is to prove that u is a priori bounded. This fact can be resorted from an

iterative scheme.Since α − 1 < αN

N−1 , we also obtain from (20) a reversed Hölder inequality

(∫Ω

uαN

N−1

)N−1N

� Cα

∫Ω

uα−1. (23)

We proceed to iterate (23). Let αj be the sequence defined by α1 −1 = q +1 and αj −1 = αj−1N

N−1 .Hence(∫

Ω

uαj N

N−1

)N−1N

� Cαj

∫Ω

uαj −1

and

αj =j−2∑i=0

(N

N − 1

)i

+ α1

(N

N − 1

)k−1

.

Denoting r = NN−1 we compute

αj = rj−1 − 1

r − 1+ α1r

j−1.

By a recursive scheme we obtain

∫Ω

uαj N

N−1 � C∑j−1

i=0 rij∏

i=1

(αi)rj−i

(∫Ω

uq+1)rj−1

.

Hence

(∫u

αj N

N−1

)N−1Nαj � C

rj −1(r−1)αj

(j∏

i=1

(αi)rj−i

) 1αj

(∫uq+1

) rj−1αj

. (24)

Ω Ω


Notice that

rj − 1

(r − 1)αj

→ 1

1 + (r − 1)α1as j → ∞

and

rj−1

αj

→ r − 1

1 + (r − 1)α1as j → ∞.

Computing log of middle term in (24), yields

1

αj

j∑i=1

rj−i logαi

= 1

αj

j∑i=1

rj−i log

(ri−1

(1

r − 1+ α1

)− 1

r − 1

)

= rj

αj

j∑i=1

r−i log

(ri−1

(1

r − 1+ α1

))

= rj−1

αj

log

(1

r − 1+ α1

)+ rj

j∑i=2

r−i (i − 1) log

(r

(1

r − 1+ α1

) 1i−1

)

� rj−1

αj

log

(1

r − 1+ α1

)+ rj log

(r

(1

r − 1+ α1

)) j∑i=2

i − 1

ri→ L as j → ∞

where L is a constant depending only on α1 and r .In synthesis, letting j → ∞ in the expression (24) we obtain

supΩ

u � C1

1+(r−1)α1 exp(L)

(∫Ω

uq+1) r−1

1+(r−1)α1. �

3. Regularity, existence, nonexistence of solutions

Lipschitz regularity. We state the following Lipschitz regularity result which is an improve-ment of Theorem 2.1 of [6]. There the statement says that the Lipschitz constant M dependson u. By Theorem 1.1 one sees that the Lipschitz constant M does not depend on u, thus u isuniformly Lipschitz continuous on Ω .

Proposition 3.1. If u is the solution of (1) with f = f (x) being a Lipschitz continuous func-tion on Ω satisfying (25) and ∂Ω ∈ C3, then the solution u is Lipschitz continuous on Ω withLipschitz constant M independent of supΩ u.

Existence of solution. We state next an existence of solution result. It is an application of ourTheorem 1.1 joint with Theorem 16.9 of [7].

Proposition 3.2. Assume the hypotheses of Theorem 1.1. In addition suppose that ∂Ω ∈ C2,α

and that f ∈ C1(Ω) satisfies


N

N − 1f (y) � K(y)for every y ∈ ∂Ω, (25)

where K(y) is the mean curvature of the surface ∂Ω at y. Then there is a solution u ∈ C2,α(Ω),0 < α < 1, of (1).

Nonexistence of solution. If f ≡ NH > 0 is constant, notice that (2) are (3) are triviallysatisfied and (4) is satisfied for 1 � τ < N

N−1 . Then positive solutions u ∈ C2(Ω) ∩ C1(Ω) ofproblem⎧⎨

⎩−div

( ∇u√1 + |∇u|2

)= NH in Ω,

u = 0 in ∂Ω

(26)

are a priori bounded. But an a priori bound itself may not be useful to show existence of a solutionof (1). A sufficient condition yielding C2,α(Ω) CMC surfaces is (25), that is, H � (N − 1)K/N .As we shall see in the sequel, if H > 2λ1/N there is no solution of (26), where λ1 stands for thefirst eigenvalue of the Laplacian operator and ϕ1 its corresponding positive eigenfunction suchthat ϕ1 = 0 on ∂Ω .

Proposition 3.3. Let u ∈ C2(Ω) ∩ C1(Ω) be a positive solution to (26), then H � 2λ1/N .

Proof. Multiplying Eq. (1) by ϕ21 and integrating,

NH

∫Ω

ϕ21 =

∫Ω

2ϕ1∇ϕ1∇u√

1 + |∇u|2 � 2∫Ω

ϕ1|∇ϕ1| � 2

(∫Ω

ϕ21

)1/2(∫Ω

|∇ϕ1|2)1/2

,

NH � 2

(∫Ω

|∇ϕ1|2∫Ω

ϕ21

)1/2

= 2λ1,

then H � 2λ1/N . �Acknowledgements

S.L. was partially supported by FONDECYT and M.M. by CNPq and FAPESP.

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height estimates for graph parametrized surfaces

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