AA SHARP BOUND FOR THE GROWTH OF MINIMAL GRAPHS
ALLEN WEITSMAN
Abstract.
We consider minimal graphs u = u ( x, y ) > D ⊂ R bounded by a Jordan arc γ on which u = 0. We prove a sort of reversePhragm´en-Lindel¨of theorem by showing that if D contains a sector S λ = { ( r, θ ) := {− λ/ < θ < λ/ } ( π ≤ λ ≤ π ) , then the order of growth is at most r π/λ . Keywords: minimal surface, harmonic mapping, asymptotics
MSC: Introduction
Let D be an unbounded plane domain. In this paper we consider the boundary valueproblem for the minimal surface equation(1.1) div ∇ u (cid:112) |∇ u | = 0 and u > Du = 0 on ∂D We shall study the constraints on growth of nontrivial solutions to (1.1) as determinedby the maximum M ( r ) = max u ( x, y ) , where the max is taken over the values r = (cid:112) x + y and ( x, y ) ∈ D .The methods of this paper extend the results of [9], where the following is proved. Theorem A.
Let D be a simply connected domain whose boundary is a Jordan arc,and D contains a sector S λ := { z : | arg z | ≤ λ/ } , with λ > π . With M ( r ) definedas above, if u satisfies (1.1) in D , then there exist positive constants K and R suchthat (1.2) M ( r ) ≤ Kr, | z | > R. As in Theorem A above, throughout this paper we shall use complex notation forconvenience. a r X i v : . [ m a t h . DG ] A ug ALLEN WEITSMAN
The study of upper and lower bounds for the growth of solutions to (1.1) are ratherscarce and fragmented. To begin with, the first relevant theorem in this directionwas proved by Nitsche [12, p. 256] who observed that if D is contained in a sector ofopening strictly less than π , then there are no nontrivial solutions.However, for domains contained in a half plane, but not contained in any such sector,a continuum of solutions to (1.1) with differing growth rates were constructed in [9].For angles λ ≥ π , in terms of the order ρ of u defined by ρ = lim r →∞ sup log M ( r )log r , there is a ”Phragm´en-Lindel¨of” type result proved in [15]. By this it is meant thatlower bounds on the size of D translate into lower bounds on the growth of u . In thecurrent setting, this implies that if D omits a sector of opening 2 π − α , ( π ≤ α ≤ π ),the omitted set in the case α = 2 π being a line, then the order ρ of any nontrivialsolution to (1.1) is at least π/α , More precisely, the results in [15] are phrased interms of the asymptotic angle β defined as follows.Let Θ( r ) be the angular measure of the set D ∩ {| z | = r } , and Θ ∗ ( r ) = Θ( r ) if D does not contain the circle | z | = r , and + ∞ otherwise. Then β = lim sup r →∞ Θ ∗ ( r ) . With this definition, the lower bound is given by
Theorem B.
Let D be an unbounded domain whose boundary ∂D is a piecewisedifferentiable arc, and u satisfies (1.1). If β ≥ π , then ρ ≥ π/β . Regarding upper bounds, it has been conjectured [16] that solutions to (1.1) in generalhave at most exponential growth, and this is achieved by the horizontal catenoid. In[16] the following is proved.
Theorem C. If D is an unbounded domain contained in a half plane and boundedby a Jordan arc, then Cr ≤ M ( r ) ≤ e Cr ( r > r ) for some positive constants C and r . The main result of this paper is the following upper bound for the order ρ of solutionswhen D contains a sector. It is a sort of ”reverse Phragm´en-Lindel¨of” phenomenon. Theorem 1.1.
Let D be a simply connected domain whose boundary is a Jordan arc,and D contains a sector S λ := { z : | arg z | < λ/ } , with π < λ ≤ π . If u satisfies(1.1) in D , then ρ ≤ π/λ . INIMAL GRAPHS 3
The examples given in [9] show that the theorem is sharp.2.
PRELIMINARIES
Let u be a solution to the minimal surface equation over a simply connected domain D . We shall make use of the parametrization of the surface given by u in isothermalcoordinates using Weierstrass functions ( x ( ζ ) , y ( ζ ) , U ( ζ )) with ζ in the right half plane H , U ( ζ ) = u ( x ( ζ ) , y ( ζ )). Our notation will then be given by(2.1) f ( ζ ) = x ( ζ ) + iy ( ζ ) ζ ∈ H. Then f ( ζ ) is univalent and harmonic, and since D is simply connected it can bewritten in the form(2.2) f ( ζ ) = h ( ζ ) + g ( ζ )where h ( ζ ) and g ( ζ ) are analytic in H ,(2.3) | h (cid:48) ( ζ ) | > | g (cid:48) ( ζ ) | , and(2.4) U ( ζ ) = 2 (cid:60) e i (cid:90) (cid:112) h (cid:48) ( ζ ) g (cid:48) ( ζ ) dζ. (cf. [3]).Now, U ( ζ ) is harmonic and positive in H and vanishes on ∂H . Thus, (cf. [14, p.151]),(2.5) U ( ζ ) = K (cid:60) e ζ, where K is a positive constant. This with (2.4) gives(2.6) g (cid:48) ( ζ ) = − Ch (cid:48) ( ζ )where C is a positive constant.In order to estimate the function f ( ζ ) in (2.2), we shall use the following lemma onquasiconformal mappings from [2] (see [1, Lemma 5.8]). Lemma A.
Let ϕ be quasiconformal in the plane such that ϕ (0) = 0 , ϕ (1) =1 , ϕ ( ∞ ) = ∞ , and the dilatation µ ( z ) = ϕ z ( z ) /ϕ z ( z ) satisfies (cid:90) π | µ ( re iθ ) | dθ → r → ∞ . ALLEN WEITSMAN
Then, in any fixed annulus A R = { R − ≤ | z | ≤ R } ( R > , ϕ ( tz ) ϕ ( t ) → z uniformly in A ( R ) as < t → ∞ . In particular, | ϕ ( z ) | = | z | o (1)) z → ∞ . At the last stage we shall need a barrier argument based on the following [4, p.827].
Lemma B.
Let u ( z ) be a solution to the minimal surface equation over a domain Ωof the form S λ \ E with λ < π and u ( z ) ≤ ax m + b (0 < m < , a, b ≥
0) for z ∈ ∂S λ and u ( z ) = 0 on ∂E . Then u ( z ) ≤ ax m + b in Ω. Proof.
Let T = S λ ∩ { z : (cid:60) e z < } . Then, there exists [7, p.322] a solution v ( z ) tothe minimal surface equation over T with values ax + b on ∂S λ , and v ( z ) → + ∞ if (cid:60) e z → | arg z | ≤ λ (cid:48) < λ in T . The dilations v R ( z ) = Rv ( z/R ) havecorresponding properties for T R = S λ ∩ { z : (cid:60) e z < R } . Now { v R } is a decreasingsequence on compact subsets of S λ , so V R → v on S λ , where v is a solution to theminimal surface equation with planar boundary values in a sector of opening less than π . Thus [12, p.256] v ( z ) ≡ ax + b on S λ . If Ω is as in the hypothesis, and U ( z ) isa solution to the minimal surface equation over Ω with U ( z ) ≤ ax + b on ∂S λ and U ( z ) = 0 on ∂E , then for any R > v R ( z ) dominates U ( z ) inside Ω ∩{ z : (cid:60) e z < R } ..Thus letting R → ∞ , it follows that U ( z ) ≤ ax + b in Ω.To apply this to u ( z ) as in the statement of the lemma, take x > u ( z ) ≤ a (cid:0) x m + mx m − ( x − x ) (cid:1) + b z ∈ ∂S λ . By the above linear case, it follows that the inequality in (2.7) holds in Ω. Applyingthis in particular a point z = x + iy in Ω, we get u ( x + iy ) ≤ ax m + b . Since x wasarbitrary, the lemma is proved. (cid:3) Proof of Theorem 1.1 for λ > π
We shall need the following qualitative growth estimate.
Lemma 3.1.
Let u ( z ) be a solution to (1.1) over a domain D containing a sector S λ with λ > π , and f ( ζ ) , h ( ζ ) , g ( ζ ) , and U ( ζ ) be as in (2.2), (2.4), and (2.6)corresponding to u ( z ) . Then, for any proper subsector S λ (cid:48) with π < λ (cid:48) < λ and D λ (cid:48) = f − ( S λ (cid:48) ) , h (cid:48) ( ζ ) → ∞ as ζ → ∞ uniformly for ζ ∈ D λ (cid:48) . INIMAL GRAPHS 5
Proof.
Let f ( ζ ), U ( ζ ) be as above. So, u ( f ( ζ )) = U ( ζ ) = (cid:60) e ζ .Let P α := { ζ : (cid:60) e e iα f ( ζ ) > } ( | α | < λ/ − π/
2) and introduce a new variable ˜ ζ and let ψ ( ˜ ζ ) be a conformal map from the right half ˜ ζ -plane H := { ˜ ζ : (cid:60) e ˜ ζ > } onto P .Define(3.1) ˜ f ( ˜ ζ ) := f ( ψ ( ˜ ζ ))˜ g ( ˜ ζ ) := g ( ψ ( ˜ ζ ))˜ h ( ˜ ζ ) := h ( ψ ( ˜ ζ ))Then ˜ f is a harmonic map, and ˜ f ( ˜ ζ ) = ˜ h ( ˜ ζ ) + ˜ g ( ˜ ζ ) . Let ˜ F ( ˜ ζ ) = ˜ h ( ˜ ζ ) + ˜ g ( ˜ ζ ) be the analytic function with the same real part as ˜ f . Then (cid:60) e ˜ F is positive in H and vanishes on ∂H , and therefore, after renormalizing we maywrite (see [14, p. 151])(3.2) ˜ F ( ˜ ζ ) = ˜ ζ = ⇒ ˜ F (cid:48) ( ˜ ζ ) = 1 . In particular we have (cid:60) e { ˜ h ( ˜ ζ ) + ˜ g ( ˜ ζ ) } = (cid:60) e { ˜ h ( ˜ ζ ) + ˜ g ( ˜ ζ ) } = (cid:60) e ˜ ζ. Now,(3.3) ˜ h (cid:48) ( ˜ ζ ) = h (cid:48) ( ψ ( ˜ ζ )) · ψ (cid:48) ( ˜ ζ ) , and(3.4) ˜ g (cid:48) ( ˜ ζ ) = − ψ (cid:48) ( ˜ ζ ) h (cid:48) ( ψ ( ˜ ζ )) = − ψ (cid:48) ( ˜ ζ ) ˜ h (cid:48) ( ˜ ζ ) . Combining this with (3.2)) we have1 = ˜ F (cid:48) ( ˜ ζ ) = ˜ h (cid:48) ( ˜ ζ ) − ψ (cid:48) ( ˜ ζ ) ˜ h (cid:48) ( ˜ ζ )which implies ˜ h (cid:48) ( ˜ ζ ) − ˜ h (cid:48) ( ˜ ζ ) − ψ (cid:48) ( ˜ ζ ) = 0 . Thus,(3.5) ˜ h (cid:48) ( ˜ ζ ) = 1 + (cid:113) ψ (cid:48) ( ˜ ζ ) . Since ψ ( ˜ ζ ) is a conformal map with (cid:60) e ψ ( ˜ ζ ) > H , there exists a real constant0 ≤ c < ∞ such that in any sector S β := { ˜ ζ : | arg ˜ ζ | ≤ β < π/ } the limit ψ (cid:48) ( ˜ ζ ) → c exists as ˜ ζ → ∞ in S β (see [14, p. 152]). ALLEN WEITSMAN
Suppose first that c > ζ = ψ ( ˜ ζ ) = ˜ ζ (1 + o (1)) as ˜ ζ → ∞ in S β . This meansthat P is asymptotically the half plane H , that is, for any sector S β (cid:48) (0 < β (cid:48) < π/ S β (cid:48) ∩ {| ζ | > R } ⊆ P for some R = R ( β (cid:48) ).Furthermore, by (3.4) and (3.5),(3.6) ˜ h (cid:48) ( ˜ ζ ) → √ c , ˜ g (cid:48) ( ˜ ζ ) → − c √ c , ˜ ζ/ζ → c. which implies that˜ h ( ˜ ζ ) + ˜ g ( ˜ ζ ) = (cid:20) (cid:60) e ˜ ζ + i (cid:18) c √ c (cid:19) (cid:61) m ˜ ζ (cid:21) (1 + o (1))as ˜ ζ → ∞ uniformly in S β . From this it follows that(3.7) h ( ζ ) + g ( ζ ) = (cid:20) (cid:60) e ζ/c + i (cid:18) c √ c (cid:19) (cid:61) m ζ/c (cid:21) (1 + o (1)))uniformly as ζ → ∞ in proper subsectors of H .By (2.1) and (2.5), the graph of the minimal surface is given parametrically by( (cid:60) e f ( ζ ) , (cid:61) m f ( ζ ) , K (cid:60) e ζ ). Using (3.7) we then have that the suface is asymptotic toa plane, that is, its parametrization has the form(3.8) (cid:18) (1 + o (1)) (cid:60) e ζ, (1 + o (1))(1 + 4 c √ c ) (cid:61) m ζ, (cid:60) e ζ (cid:19) as ζ → ∞ in proper subsectors of H .Now, if we consider P α and P − α for α = ( λ − π ) / P , then f ( P α ) , f ( P − α ),and f ( P ) are an overlapping cover of S λ . The mapping e iα f ( ζ ) = e iα h ( ζ ) + e − iα g ( ζ )maps P α onto the right half plane, and an analysis corresponding to the above for P α gives a conformal mapping which we again denote by ψ ( ˜ ζ ) mapping H onto P α , andas before ψ (cid:48) ( ˜ ζ ) tends to a limit in proper subsectors. We claim that if this limit for P was not 0, then the same must be true for P α . In fact, in P , it follows from (3.3) and(3.6) that h (cid:48) ( ζ ) remains bounded as ζ → ∞ in proper subsectors of P . However, for P α if the corresponding ψ (cid:48) ( ˜ ζ ) →
0, then by (3.3) and (3.5), h (cid:48) ( ζ ) becomes unbounded.This creates a contradiction in the overlapping regions. The same analysis applies to P − λ .Thus, the graphs of the minmal surface over proper subsectors of P − λ , P , and P λ are all asymptotic to planes. However, since these subsectors overlap, they mustbe asymptotic to the same plane over compact subsets of S λ , But since λ > π thiscontradicts the fact that u ( z ) > D .Since this shows that in D λ (cid:48) the above conformal maps ψ ( ˜ ζ ) all satisfy ψ (cid:48) ( ˜ ζ ) → h (cid:48) ( ζ ) → ∞ . (cid:3) INIMAL GRAPHS 7
Proof.
We now complete the proof in the case λ > π .For fixed λ , let f ( ζ ) denote the function in (2.1) corresponding to a solution to(1.1) over a domain D containing S λ . Then for λ (cid:48) such that π < λ (cid:48) < λ we define f ( ζ ) = ζ λ (cid:48) /π +1. Let ˜ S λ (cid:48) = f ( H ) and ˜ H = f − ( ˜ S λ (cid:48) ). Then if ψ ( ζ ) is a 1 − H onto ˜ H with ψ ( ∞ ) = ∞ , it follows that f ( ψ ( H )) = f ( H ) and thereexists an orientation preserving homeomorphism ϕ : H → H with ϕ ( ∞ ) = ∞ suchthat(3.9) f ( ψ ( ζ )) = f ( ϕ ( ζ )) ζ ∈ H. Differentiating (3.9) with respect to ζ and ζ , and using (2.6) we obtain(3.10) ψ (cid:48) ( ζ ) h (cid:48) ( ψ ( ζ )) = ϕ ζ ( ζ ) f (cid:48) ( ϕ ( ζ ))and(3.11) − C ψ (cid:48) ( ζ ) h (cid:48) ( ψ ( ζ )) = ϕ ζ ( ζ ) f (cid:48) ( ϕ ( ζ ))Dividing (3.11) by (3.10) we have(3.12) C | h (cid:48) ( ψ ( ζ )) | = (cid:12)(cid:12)(cid:12)(cid:12) ϕ ζ ( ζ ) ϕ ζ ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) . Now, ψ ( ζ ) → ∞ as ζ → ∞ in H , so by Lemma 3.1 it follows that the left side of(3.12) tends to 0.It therefore follows from (3.12) and the fact that ϕ is a sense preserving differentiablehomeomorphism, that ϕ is quasiconformal in H and that the dilatation of ϕ satisfies(3.13) (cid:12)(cid:12)(cid:12)(cid:12) ϕ ζ ( ζ ) ϕ ζ ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) → ζ → ∞ ζ ∈ H. The mapping ϕ can then be extended by reflection to a quasiconformal mapping ofthe complex plane onto the complex plane with (3.13) still in force. Thus, Lemma Acan be applied to ϕ . In fact, since ϕ maps the vertical axis to itself, the conclusionin Lemma A can be improved to ϕ ( re iθ ) = r (1+ o (1)) e i ( θ + o (1)) so that f ( ψ ( re iθ )) = f ( ϕ ( re iθ )) = r ( λ (cid:48) /π + o (1)) e i ( λ (cid:48) θ/π + o (1)) ζ = re iθ → ∞ ζ ∈ H. From this we see that, given any λ (cid:48)(cid:48) such that π < λ (cid:48)(cid:48) < λ (cid:48) , there is a proper sectorΣ λ (cid:48)(cid:48) in H such that f ( ψ (Σ λ (cid:48)(cid:48)(cid:48) )) covers S λ (cid:48)(cid:48) . But ψ ( ζ ) is a conformal mapping of H into H , so ψ (cid:48) ( ζ ) → C as ζ → ∞ in Σ λ (cid:48)(cid:48) for some C ≥ z ,(3.14) u ( z ) < | z | ( π/λ (cid:48) + o (1)) z ∈ S λ (cid:48)(cid:48) . ALLEN WEITSMAN
The boundary of the sector Σ λ (cid:48)(cid:48) on which (3.14) holds forms an angle in the left halfplane of opening less than π . On the remaining portion of the boundary of D in theleft half plane u ( z ) = 0. Therefore Lemma B implies that (3.14) holds in the lefthalf plane as well. Thus (3.14) holds throughout D . Since λ (cid:48) can be taken arbitrarilyclose to λ in (3.14), the proof is complete. (cid:3) References
1. D .Drasin,
On Nevanlinna’s inverse problem , Complex Variables (1999), 123-1432. D. Drasin, A. Weitsman, Meromorphic functions with large sums of deficiencies , Advances inMath. (1974), 93-126.3. P. Duren, Harmonic mappings in the plane , Cambridge Tracts in Mathematics, 2004.4. J-F Hwang,
Phragm´en Lindel¨of theorem for the minimal surface equation , Proc. Amer. Math.Soc. (1988), 825-828.5. J-F Hwang,
Catenoid-like solutions for the minimal surface equation , Pacific Jour. Math. (1998), 91-102.6. J-F Hwang,
How many theorems can be derived from a vector function - on uniqueness theoremsfor the minimal surface equation , Taiwanese Jour. Math. (2003), 513-539.7. H. Jenkins, j. Serrin, Variational problems of minimal surface type II. Boundary value problemsfor the minimal surface equation , Arch. Rat. Mech. Anal., (1965/66), 321-342.8. R. Langevin, G. Levitt, H. Rosenberg, Complete minimal surfaces with long line boundaries , DukeMath. Jour. (1987), 985-995.9. E. Lundberg, A. Weitsman, On the growth of solutions to the minimal surface equation overdomains containing a half plane , preprint, arXiv:1311.3334.10. V. Miklyukov,
Some singularities in the behavior of solutions of equations of minimal surfacetype in unbounded domains , Math. USSR Sbornik (1983), 61-73.11. V. Miklyukov, Conformal maps of nonsmooth surfaces and their applications , Exlibris Corp.(2008).12. J.C.C. Nitsche,
On new results in the theory of minimal surfaces , Bull. Amer. Mat. Soc. (1965), 195-270.13. R. Osserman, A survey of minimal surfaces . Dover Publications Inc. (1986).14. M. Tsuji,
Potential Theory in Modern Function Theory , Maruzen Co., Ltd., Tokyo (1959).15. A. Weitsman,
On the growth of minimal graphs , Indiana Univ. Math. J. (2005), 617-625.16. A. Weitsman, Growth of solutions to the minimal surface equation over domains in a half plane ,Communications in Analysis and Geometry (2005), 1077-1087.(2005), 1077-1087.