Journal of Differential Equations 171, 54�62 (2001)

Constant Mean Curvature Graphs on UnboundedConvex Domains

Rafael Lo� pez1

Departamento de Geometr@� a y Topolog@� a, Universidad de Granada,18071 Granada, Spain

E-mail: rcamino�ugr.es

Received July 26, 1999; revised January 31, 2000

In this paper, we consider the Dirichlet problem for the constant mean curvatureequation on an unbounded convex planar domain 0. Let H>0. We prove thatthere exists a graph with constant mean curvature H and with boundary �0 if andonly if 0 is included in an infinite strip of width 1

H . We also establish an existenceresult for convex bounded domains contained in a strip. � 2001 Academic Press

1. INTRODUCTION AND STATEMENT OF RESULTS

Consider a smooth surface M in three-dimensional Euclidean space,having a non-parametric representation z=u(x, y) and consider theupwards orientation. Then M has mean curvature H provided that thefunction u=u(x, y) satisfies the partial differential equation

LH(u)#div{u

- 1+|{u| 2&2H=0, (1)

where div and { are the Euclidean divergence and gradient operators.A solution of (1) is called an H-graph. The Dirichlet problem for Eq. (1)corresponding to a fixed (smooth) domain 0 in the (x, y)-plane and H agiven constant consists of finding a solution of LH(u)=0 in 0 taking onassigned continuous values on the boundary �0 of 0. Geometrically, thishas the meaning that the resulting surface z=u(x, y) is defined over 0 andspans a given space curve with single valued projection on the (x, y)-plane.

The first general existence theorem for H-graphs is due to Serrin. Heshowed in [9] that when the curvature of �0 satisfies

}�2H>0, (2)

doi:10.1006�jdeq.2000.3830, available online at http:��www.idealibrary.com on

540022-0396�01 �35.00Copyright � 2001 by Academic PressAll rights of reproduction in any form reserved.

1 Research partially supported by DGICYT Grant PB97-0785.

then there exists a solution of (1) for arbitrary prescribed boundary values.His result requires, in particular, that 0 be bounded and convex. In thiscontext, Serrin also showed that the lower bound (2) is necessary for theexistence of solutions with arbitrary boundary values. Serrin's necessary andsufficient condition was a natural extension of that given by Finn in [3]who showed that minimal graphs (H=0) exist over a bounded domain forarbitrary boundary values if and only if the domain is convex.

Necessary Condition. In the same paper, Finn derived a necessary condi-tion for the existence of a solution on the unbounded domain (strip) betweentwo parallel lines: If u is a solution of (1) on a strip, then necessarily the widthh of the strip satisfies h� 1

H . Our first result generalizes this to unbounded con-vex domains and shows that a similarly strict condition is imposed on thegeometry of 0 by the existence of a solution.

Theorem 1.1. Let H>0 and let 0 be an unbounded convex planar domain.If Eq. (1) has a solution in 0, then 0 is included in a strip of width 1

H .

Sufficient Conditions. It is important to notice that the results of Finnand Serrin give sufficient conditions under which a domain admits solu-tions with arbitrary boundary values. We now consider the particular caseof zero boundary data. In this particular case, there is a unique value H0>0depending on 0 for which solutions of (1) exist (as graphs over 0)precisely for |H |�H0 (a discussion of this point may be found in [7]). In[6] the author in collaboration with Montiel gives estimates on the valueof H0 in terms of the geometry of 0. Further related results are alsoobtained by the author in [5]

Theorem 1.2. Let H>0 and let 0 be an unbounded convex planardomain included in a strip of width 1

H . Then there exists an H-graph on 0with boundary �0.

As a consequence of Theorems 1.1 and 1.2, we obtain:

Corollary 1.3. Let H>0 and let 0 be an unbounded convex planardomain. Then there exists an H-graph in 0 with boundary �0 if and only if0 is included in a strip of width 1

H .

Finally, we establish an existence result of H-graphs for bounded convexdomains included in a strip.

Theorem 1.4. Let H>0 and let 0 be a bounded convex domain includedin a strip of width h>0. If H< 1

h , there exists an H-graph on 0 whoseboundary is �0.

55CONSTANT MEAN CURVATURE GRAPHS

Theorems 1.2 and 1.4 assert us that the Dirichlet problem with zeroboundary data associated to the constant mean curvature equation (1) canbe solved for convex domains included in a strip of width 1

H . When 0 is notbounded, the result is optimum. However, when 0 is a bounded convexdomain included in a strip, it is to be wished that the assumption H< 1

h onthe width of the strip can be relaxed in some sense, as it occurs when 0 isa disk.

2. PROOF OF RESULTS

Proof of Theorem 1.1. Let u be a solution of (1) in 0 and let G denotethe graph of u. Next result follows the same ideas as in [3].

Lemma 2.1. For each p # 0, the disk D( p, 1H) centred at p and with radius

1H is not included in 0.

Proof. We proceed by contradiction. Let 7 be a sphere of radius 1H

whose center lies on the straight-line through p that is orthogonal to the(x, y)-plane. Lift 7 vertically upwards until 7 is completely above G. Then,let us descend 7 until to find a first point of contact of 7 and G. SinceD( p, 1

H)/0, the contact point is interior and the maximum principle givesus a contradiction. K

We continue with the proof of Theorem 1.1. From Lemma 2.1 and theconvexity of 0, it follows that 0 is included in a strip B$ of width 2

H . Bymeans of a rigid motion, we can assume that the strip is given by

B$={(x, y); &1H

< y<1H= . (3)

Since 0 is a convex domain, then we have two possibilities: if the x-coor-dinate is not bounded from above or below in 0, then 0 is a strip includedin B$; in other case, 0 is asymptotic to a strip included in B$. In the firstcase, it is known that the width of 0 is less than 1

H . Consider the secondone. Without loss of generality, we suppose that 0/[x>0] and 0� & [x=0]{<. Let B$h /B$ be the strip of width h>0 such that 0 is asymptotic tothe boundary of B$h . For each n>0, let [ y1(n), y2(n)]=0 & [x=n], withy1(n)< y2(n). For each m>n>0, consider 0n, m=0 & [(x, y); n<x<m,y1(n)< y< y2(n)]. An integration of (1) over 0n, m yields

56 RAFAEL LO� PEZ

2H(m&n)( y2(n)& y1(n))=2H area(0n, m)

= }|�0n, m

({u, &)

- 1+|{u| 2 }�length(�0n, m)

=2(m&n+ y2(n)& y1(n))�2 \m&n+2H+ ,

where & denotes the exterior directed unit normal along �0n, m . Lettingm � �, we have y2(n)& y1(n)� 1

H . Since the two branches of �0 areasymptotic to �B$h , in the limit case (n � �), we conclude h� 1

H , and thetheorem is proved.

Proof of Theorem 1.2. Without loss of generality, we suppose that thestrip B in the statement of the theorem is given by B=[(x, y); & 1

2H<y< 1

2H]. Solutions of Eq. (1) on unbounded convex planar domains andwith zero boundary data on the boundary satisfy a priori height estimates.Let u be a solution of (1) such that u=0 on �0, being 0 an unboundeddomain. First, since u is zero along �0, u is bounded in 0 (see [8, Lemma 2.4]).Suppose now that 0 is convex. Then a standard barrier argument [1] byusing halfcylinders of radius 1

2H of type z(x, y)=&- 1�4H 2& y2 and theplane z=0 gives us

z(x, y)�u(x, y)�0, (x, y) # 0. (4)

Now we are in position to prove Theorem 2. There exists two types ofunbounded convex domains in B: either an infinite strip parallel to thex-axis, or a domain with the x-coordinate is lower (or upper) bounded andasymptotic to a strip included in B.

(1) In the case that 0 is an infinite strip, by means of using a rigidmotion, we can suppose that 0=[(x, y); &h< y<h] with h� 1

2H . Thenthe piece of the cylinder of radius 1

2H given by

u(x, y)=� 14H 2&

1h2&� 1

4H2& y2 , (x, y) # 0,

is a solution of (1) with u=0 on �0.

(2) Now consider that 0 is not a strip. In order to find a solution uof the corresponding Dirichlet problem, we shall use of the classical Perronmethod of superfunctions for (1) (see [4]; also [1] for an example in thesame context). This will be possible provided that for sufficiently smalldisks there exist solutions of (1) with arbitrary continuous boundaryvalues. Let v # C0(0� ) be a continuous function in 0� and let D be a closeddisk in 0. We denote by v� the unique solution of the Dirichlet problem

57CONSTANT MEAN CURVATURE GRAPHS

LH(v� )=0 in D satisfying the condition v� =v on �D. The existence of thefunction v� is assured by the Serrin result stated in Introduction: as D/B,the radius of the disk D is less than 1

2H and so, the curvature of �D isgreater than 2H. We define the uniquely continuous function MD(v) in 0as

MD(v)( p)={v� ( p),v( p),

p # Dp # 0&D.

Following [4], we call a function v # C0(0� ) supersolution in 0 if it satisfiesv�MD(v) for every disk D in 0. On the other hand, a continuous functionin 0� will be called a superfunction relative to 0 if v is a supersolution in0 and if, on �D, v�0.

Let F denote the class all superfunctions relative to 0. The set F is notempty since 0 # F: for any disk D/0, 0�MD(0) in D by the maximumprinciple of Eq. (1). The next lemma gives some properties of F that arenecessary in the Perron technique.

Lemma 2.2. (1) If [v1 , ..., vn]/F, then min[v1 , ..., vn] also belongs to F.

(2) If v # F and if D is a disk in 0, then MD(v) # F.

Proof. (1) Let D be a disk in 0 and denote v=min[v1 , ..., vn]. On�D, we have vi�v and the maximum principle gives MD(v i)�MD(v) foreach i. Then min[MD(v1), ..., MD(vn)]�MD(v). Furthermore, as the functionsvi are supersolutions, vi�MD(vi) on D and then v�min[MD(v1), ..., MD(vn)].As a consequence, we obtain the desired inequality v�MD(v). This provesthat v is a supersolution. On the other hand, vi�0 on �0. Thus v�0 on�0, and we arrive that v is a superfunction in 0 relative to 0.

(2) First, we must show that

MD(v)�MD$(MD(v))

for any disk D$/0. This clearly holds if D$ lies either completely inside orcompletely outside of D. Thus, we only consider the case where D$ is partlyinside and partly outside of D. As v # F, it follows that v�MD(v) and themaximum principle yields v� �MD(v) on D$. The fact that v�MD$(v) andthe maximum principle leads again

v�MD$(v)�MD$(MD(v)). (5)

We now observe that by (5), we have MD(v)=v�MD$(MD(v)) in D$&D.We study the situation in D$ & D.

58 RAFAEL LO� PEZ

(a) If p # �(D$ & D) & �D, MD(v)( p)=v( p) and by (5), MD(v)( p)�MD$(MD(v))( p).

(b) If p # �(D$ & D) & �D$, MD$(MD(v))( p)=MD(v)( p).

As a consequence, it follows

MD(v)�MD$(MD(v)) on �(D$ & D)

and the maximum principles yields the desired inequality MD(v)�MD$(MD(v))in D$ & D. Finally, since �0/0� &D, MD(v)=v�0 on �0. Hence, weconclude that MD(v) is a superfunction in 0 relative to 0. K

Let z be the function defined in B given by z(x, y)=&- 1�4H2& y2. LetZ=z | 0 be the restriction of z to the domain 0. Consider

F*=[v # F; Z�v�0 in 0].

Remark that the functions of F* are uniformly bounded. Also, the functionsof F* satisfy the properties (1) and (2) of Lemma 2.2. For the property (1),Z�vi�0 and the same occurs with the function min[v1 , ..., vn]. In relationto the second property, let v # F*. As Z�v�0, the maximum principleimplies MD(Z )�MD(v)�MD(0) on D. But MD(Z )=Z and MD(0)�0.

Define

u=inf [v; v # F*]=inf [MD(v); v # F*, D/0].

Perron's method shows that the function u is a solution of equation LH(u)=0in 0. We want to prove that u is continuous in 0� and assumes 0 as boundaryvalues. The fact that u is continuous in 0 is a consequence of Harnack'sprinciple: if D is a disk in 0, the functions MD(v) are uniformly boundedand this implies uniform boundedness of their first derivatives (see [9]).Thus the family [MD(v); v # F*] is uniformly continuous in D and thesame occurs for the function u.

It remains to show that the Perron solution actually takes the value 0 onthe boundary. The study of the boundary behaviour of the Perron solutionis determined by the geometric properties of the boundary of 0. This canbe accomplished by the technique of barriers. Indeed, let p # �0. Consideran (infinite) quarter-cylinder Cp of radius 1

2H whose axis is parallel to thetangent line l to �0 at the point p. Now let us place Cp in such a way thatthe axis and one component of Cp lie in the (x, y)-plane. Also, assume thatCp /[z�0], the concave side of Cp is looking towards p and �Cp &[z=0] lies in the component of [z=0]&l that does not contain 0. Bythe convexity of �0, we get Cp & G=<, where G is the graph of the func-tion u. Move Cp towards 0 by horizontal translations until to touch G. By

59CONSTANT MEAN CURVATURE GRAPHS

taking account of maximum principle, the convexity of 0 and the heightestimate (4), the first point of contact between Cp and G occurs at p.Denote by vp the function whose graph is Cp . Consequently, the functionsvp and w=0 are a modulus of continuity in a neighbourhood of p. Sincevp( p)=w( p)=0, we obtain u( p)=0. This completes the proof of Theorem 2.

Remark 1. The solution obtained in Theorem 1.2 is unique at infinity [2].In the paper [3] mentioned above, Finn also conjectured that the half-cylinder of radius 1�2H was the only H-graph in a strip of width 1�H.A counterexample can be obtained from the 1990 paper of Collin [1] whosucceeded in proving the existence of H-graphs over a strip of width 1�Hwith boundary formed by two convex functions on �0.

Proof of Theorem 1.4. Let 0 be a convex bounded domain included ina strip of width h. If H< 1

h , the solvability of the Dirichlet problem

{div{u

- 1+|{u|2=2H

u=0

in 0,

on �0,

(6)

can be achieved using the continuity method. Following the usual Leray�Schauder approach (see [4, Theorem 13.8]), solvability of (6) follows ifthere is a constant M>0 such that an apriori bound

|u|C 1 (0)=sup0

|u|+sup0

|{u|<M

holds for any solution u of (6) with 0�H�c, where c is a fixed realpositive number such that c< 1

h . The constant M is required to be inde-pendent of u and H. Standard theory of quasilinear equations assures thatthe estimate of |{u| in 0 is obtained if we have bounds of |{u| along �0.

For H=0, we have the trivial solution u=0 and from the implicit func-tion theorem applied to Eq. (1), there exists a range of solutions for (6)with 0�H�c0 , for a small number c0 . In the interval [0, c0], there existsuniform C0-bounds depending only on 0. For every solution u of (6) withc0�H�c the height estimate for graphs due to Serrin implies that |u|�1�|H |�1�c0 ([9]; see also [8]). Thus we obtain an apriori estimationof sup0 |u|.

On the other hand, C1-estimates of u along the boundary �0 are obtainedprovided that we are able to establish apriori estimates of the slope ofthe H-graph G of u along its boundary �G=�0. This estimate will beaccomplished with quarter-cylinders as geometrical barrier surfaces as in

60 RAFAEL LO� PEZ

[6, Corollary 4]. This is summarized as follows. Without loss of generality,consider that 0 is contained in the strip

B*={(x, y); &c1

2< y<

c1

2 = ,

with c1<h�2. We need first a slight improvement on the height estimatesfor u. By comparing with halfcylinders of radius h

2 as in Theorem 1.2, themaximum principle gets immediately &h

2<u�0 (remark that H�c< 1h).

Let K be the halfcylinder

K={(x, y, z); &h2

< y<h2

, z=&12

- h2&4y2= ,

whose mean curvature is 1h . Consider K$ the piece of K that projects ortho-

gonally onto B*. Then the maximum principle asserts that we can moveupwards K$ until to touch the first time the plane z=0 and without con-tacting with G. Therefore,

c2=: &12 (h&- h2&4c2

1)<u(x, y)�0, (x, y) # 0. (7)

In order to estimate the slope of the graph G, let p # �G=�0 be anarbitrary boundary point. Consider Kp a quarter-cylinder of radius h

2 whoseaxis is parallel to the tangent line l to �0 at the point p and with the samecharacteristics as Cp in Theorem 1.2. Applying the estimate (7), we moveup Kp slightly without touching G until to obtain a piece K$p of Kp suchthat its height is c2 with respect to the plane z=0. After a rigid motion, thesurface K$p can be viewed as the half portion of K$. By moving K$p towardsp and by the convexity of 0, the same reasoning as in Theorem 1.2 provesthat the slope of G at the point p is less than the slope of K$ along itsboundary. This estimate depends only on h and c2 , or in others words, itis independent on H. This completes the proof of the theorem.

Remark 2. It is worth pointing out that the bounded domain 0 inTheorem 1.4 is not necessarily strictly convex, in contrast to the Serrin resultin [9]. As a consequence, the boundary of our domain 0 can contain piecesof straight-lines or, in other situation, the size of 0 can be too big insidethe strip.

ACKNOWLEDGMENTS

The author thanks both Antonio Ros for useful conversations on the subject and thereferee, who contributed to improve the final presentation of the manuscript.

61CONSTANT MEAN CURVATURE GRAPHS

REFERENCES

1. P. Collin, Deux examples de graphes de courbure moyenne constante sur une bande de R2,C.R. Acad. Sci. Paris Se� r. I 311 (1990), 539�542.

2. P. Collin and R. Krust, Le proble� me de Dirichlet pour l'e� quation des surfaces minimalessur des domaines non borne� s, Bull. Soc. Math. France 119 (1991), 443�462.

3. R. Finn, Remarks relevant to minimal surfaces and to surfaces of prescribed meancurvature, J. Anal. Math. 14 (1965), 139�160.

4. D. Gilbarg and N. S. Trudinger, ``Elliptic Partial Differential Equations of Second Order,''2nd ed., Springer-Verlag, Berlin, 1983.

5. R. Lo� pez, Constant mean curvature surfaces with boundary in Euclidean three-space,Tsukuba J. Math. 23 (1999), 27�36.

6. R. Lo� pez and S. Montiel, Constant mean curvature surfaces with planar boundary, DukeMath. J. 85 (1996), 583�604.

7. J. McCuan, Continua of H-graphs: convexity and isoperimetric stability, Calc. Var. 9(1999), 297�325.

8. W. H. Meeks III, The topology and geometry of embedded surfaces of constant meancurvature, J. Differential Geom. 27 (1988), 539�552.

9. J. Serrin, The problem of Dirichlet for quasilinear elliptic equations with many independentvariables, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 413�496.

62 RAFAEL LO� PEZ