MINIMAL GRAPHS OVER CONCAVE DOMAINS

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1 MINIMAL GRAPHS OVER CONCAVE DOMAINS ALLEN WEITSMAN Abstract. We consider minimal graphs u = u(x, y > 0 over concave unbounded domains D R 2 bounded by a Jordan arc γ on which u = 0. We prove that the sets u(x, y > c (c > 0 are then all concave. A consequence of this is that such such solutions u to the minimal surface equation are also superharmonic. Keywords: minimal surface, harmonic mapping, asymptotics MSC: 49Q05 1. INTRODUCTION Let D be a plane domain bounded by an unbounded C 2 Jordan arc γ. In this paper we consider the boundary value problem for the minimal surface equation u div = 0 and u > 0 in D ( u 2 u = 0 on γ Regarding the possible geometry of D, it follows from a theorem of Nitsche [4, p.256] that D cannot be convex unless D is a halfplane, since (1.1 cannot have nontrivial solutions over domains contained in a sector of opening less than π. On the other hand, amongst the examples given in [3], there is a continuum of graphs which do have concave domains; specifically those given parametrically in the right half plane H by (1.2 z(ζ = (ζ + 1 γ 1 γ(2 γ ( ζ γ (ζ H, 1 < γ < 2 together with the height function 2Re ζ. A concave domain D is taken to be one whose complement is an unbounded convex domain. The boundary of D is then a curve which bends away from the domain. There is a slight language irony here since then, by convention, we say here that a concave domain lies on the convex side of a convex curve. In 5 we will verify that the domains for the graphs of (1.2 are concave. In this note we shall prove the following Theorem 1. If u is a solution to (1.1 with D concave and bounded by a C 2 curve γ, then the sets {u > c} are concave for each c > 0. 1

2 2 ALLEN WEITSMAN This has the curious consequence Corollary. If u is as in Theorem 1 above, then u is also superharmonic in D. 2. PRELIMINARIES From the given hyyothesis, it follows that γ must have asymptotic angles in both directions as z. By a rotation we may assume that the asymptotic tangent vectors have directions ±α for some 0 α π/2.. For a solution u to the minimal surface equation over a simply connected domain D we shall slightly abuse notation by using u to also denote the solution to (1.1 when given in parametric form. We shall make use of the parametrization of the surface given by u in isothermal coordinates using Weierstrass functions (x(ζ, y(ζ, u(ζ with ζ in the right half plane H. Our notation will then be given by (2.1 f(ζ = x(ζ + iy(ζ ζ = σ + iτ H. Then f(ζ is univalent and harmonic, and since D is simply connected it can be written in the form (2.2 f(ζ = h(ζ + g(ζ where h(ζ and g(ζ are analytic in H, (2.3 h (ζ > g (ζ, and (2.4 u(ζ = 2Re i h (ζg (ζ dζ. (cf. [1, 10.2]. Now, u(ζ is harmonic and positive in H and vanishes on H. Thus, (cf. [5, p. 151], (2.5 u(ζ = K Re ζ, where K is a positive constant. This with (2.4 gives g (ζ = C h (ζ where C is a positive constant. By a rescaling of the surface we may assume that C = 1 and K = 2. Then from (2.3 we have, in particular, that (2.6 h (ζ = 1/ g (ζ > 1. It follows from (2.5 that the level curves of u can be parametrized by f(σ 0 + iτ for < τ < and fixed values σ 0. Then the curvature κ corresponding to height σ 0 is given for ϕ = arctan(y τ /x τ

3 by MINIMAL GRAPHS 3 (2.7 κ = κ(σ 0, τ = dϕ ds = 1 (x 2 τ + y 2 τ 3/2 (x τy ττ y τ x ττ. To compute (2.7 we use (2.1 and (2.4 to write (2.8 x τ = τ Re(h + g = Re i(h 1/h = Im(h 1/h = ( h 2 + 1Im 1 h (2.9 x ττ = τ Im(h 1/h = Re(h + h /h 2 (2.10 y τ = τ Im(h + g = Im i(h + 1/h = Re(h + 1/h = ( h 2 + 1Re 1 h (2.11 y ττ = τ Re(h + 1/h = Im(h h /h 2 Substituing (2.8-(2.11 into (2.7 we get κ = h h 3 which simplifies down to ( ( 1 h 1 h (h h h 2 h + h h 2 + ( 1 h + 1 h (h + h h + 2 h + h h 2 (2.12 κ = h h Re h h. The proof of Theorem 1 uses the comparison of κ in (2.12 with the corresponding curvature κ 1 of the image of the line σ 0 + iτ ( < τ < under h. Since arg h = Im log h, the formula (2.7 gives (2.13 κ 1 = 1 h Reh h. 3. PROOF OF THEOREM 1 From the concavity of D and the assumption that the asymptotic tangents to γ have angles ±α, it follows that y τ 0 for σ = 0. Thus, from (2.10 it follows that for σ = 0, Re 1/h 0, and hence Re 1/h 0. Since 1/h is bounded in H this means that Re 1/h 0 thoughout H. This in turn gives (3.1 Re h (ζ 0 ζ H. Let ψ(τ = arg h (iτ. By (2.12 and (2.13 it must be that (3.2 dψ dτ = τ Im(log h = Re h h > 0.

4 4 ALLEN WEITSMAN By (3.1 (3.3 π/2 ψ(τ π/2. Now, π/2 < Im(log h < π/2 in H, and in particular is a bounded harmonic function in H. So for ζ = σ + iτ H, Then Im log h (ζ = σ π Re h (ζ h (ζ = τ Im log h (ζ = τ An integration by parts yields Re h = σ ( ψ(t h π σ 2 + (t τ 2 ψ(tdt σ 2 + (t τ 2. ( σ ψ(tdt = 2σ π σ 2 + (t τ 2 π + ψ (tdt. σ 2 + (t τ 2 (t τψ(tdt (σ 2 + (t τ 2 2. By (3.3 it follows that the first term on the right vanishes, and by (3.2 the second term is positive. Thus κ 1 in (2.13 and hence κ in (2.12 are positive in H. 4. PROOF OF THE COROLLARY We may write the minimal surface equation for u as u + F u 3 = 0 where F = F (u, x, y = u 2 yu xx + u 2 xu yy 2u x u y u xy. Now, for a given function v(x, y > 0 the curvature of the level set v(x, y = 0 is given by F (v, x, y/ v 3 [2, p. 72] which is positive when the curve bends away from the interior of the domain. Since Theorem 1 shows that the level sets u = c which bound the sets u > c each have positive curvature, then applying this to F (u c, x, y we find that u < 0 and hence u is superharmonic in D For the examples (1.2 of 1, 5. Concluding Remark. Re h h = Re γ(γ 1(ζ + 1 γ 2 > 0 for 1 < γ < 2 so that by (2.12 these have concave domains.

5 MINIMAL GRAPHS 5 References 1. P. Duren, Harmonic mappings in the plane, Cambridge Tracts in Mathematics, A. Gray, Modern differential geometry of curves and surfaces, Studies in Advanced Mathematics, E. Lundberg, A. Weitsman, On the growth of solutions to the minimal surface equation over domains containing a half plane, Calc Var. Partial Differential Equations 54 ( J.C.C. Nitsche,On new results in the theory of minimal surfaces, Bull. Amer. Mat. Soc. 71 (1965, M. Tsuji, Potential Theory in Modern Function Theory, Maruzen Co., Ltd., Tokyo (1959. Department of Mathematics, Purdue University, West Lafayette, IN weitsman@purdue.edu