Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 225–231

ON FOLDING SOLUTIONS OF THE BELTRAMI EQUATION

Uri Srebro and Eduard Yakubov

Technion – Israel Institute of Technology Haifa 32000, Israel; srebro@math.technion.ac.il Holon Academic Instite of Technology 52 Golomb St., Holon, Israel; yakubov@barley.cteh.ac.il

Abstract. A sufficient condition for the existence of a local folding solution to an alternating Beltrami equation is presented, and the uniformization of local folding solutions is studied.

1. Introduction

1.1. Consider the Beltrami equation

(B)  wz̄ = μ(z)w_{z}

in the unit disk D = \{z ∈ C : |z| < 1\}, where μ: D → \overline{C} is measurable, μ is locally bounded in D_{1} = {z ∈ D : Im z > 0}, i.e. ∥μ|K∥_{∞} < 1 for every compact set K in D_{1} , and 1/μ is locally bounded in D_{2} = {z ∈ D : Im z < 0}.

Equation (B) can be rewritten in the symmetric form

(B′)  A(z)w_{z} + B(z)wz̄ = 0

where A(z) and B(z) are complex-valued measurable functions such that μ = −A/B in D_{1} and 1/μ = −B/A in D_{2} . Let E = D ∩ R.

1.2. Let μ be as in 1.1. A function f: D → C is said to be a solution of (B), if f is continuous, f | (D \ R) is in W_{loc}^{1,2} , and its weak derivatives satisfy (B′) a.e. in D. A solution f in D is a folding solution if it is injective in D_{1} , in D_{2} and in E. If, in addition, f(D_{1}) = f(D_{2}) f is a proper folding solution.

In Section 2 below, we provide a sufficient condition for the existence of a local folding solution, i.e. a local solution which is a folding, and in Section 3, we check to what extend local folding solutions can be uniformized to a global folding solution.

2000 Mathematics Subject Classification: Primary 30C62, 35J70,30B40.

2. Existence of local folding solutions

2.1. Theorem. Let D, D_{1} , D_{2} , E and μ be as in 1.1. If there exist r ∈ (0, 1), a non-negative integer m, a real-valued real analytic function θ(z), |z| < r with |θ(z)| < \frac{1}{2}π and a complex-valued real analytic function M(x, y) in |z| < r, z = x + iy, with

(2.2)  Re M(0, 0) > 0

and such that for |z| < r,

(2.3)  μ(z) = e^{2iθ(z)}[1 − y^{2m+1}M(x, y)],

then (B) has a folding solution in some neighborhood of 0.

Proof. In [SY2, Theorem 1.1] we considered a Beltrami equation (B) with

(M)  μ(z) = e^{2iθ(z)}[1 − ϱ(y)M(z)]

where θ and ϱ are real-valued functions, and M is a complex-valued function which satisfy certain conditions. We then applied a process, which we called a deformation of the complex dilatation, and reduced (B) to a Beltrami equation in the (ξ, η)-plane with complex dilatation

μ̃(ξ, η) = 1 − ϱ(η)M̃(ξ, η)

where ϱ is the same as in (M), and M̃ satisfies a similar condition as M, see [SY2, pp. 475–477]. The same method can be applied here. Thus we will assume that θ(z) ≡ 0. We will also assume that m = 0 and that M is a polynomial. The general case is proved in a similar way.

In view of (2.2), M has the form

(2.4)  M(x, y) = 2a + 2ib + ∑_{n+k=1}^{N} q_{nk}z^{n}z̄^{k}

where N ≥ 1, a and b are real and a > 0.

As in [L, pp. 23–24] we set in (B)

(2.5)  w = f(z) = z + c_{01}z̄ + ∑_{n+k=2}^{∞} c_{nk}z^{n}z̄^{k}

with c_{n0} = 0 for all n > 1. With application of (2.3), one obtains all c_{nk} (uniquely). Again, exactly as in [L], one can show that the series in (2.5) converges uniformly in U_{0} = {z : |z| < r_{0}} for some r_{0} ∈ (0, r).

It remains to show that f is a folding in some neighborhood U of 0 which is contained in U_{0} . Let f = u + iv, then by (2.5)

(2.6)  u = 2x − b(x^{2} + y^{2}) + ∑_{n+k=3}^{∞} α_{nk}x^{n}y^{k}

and

(2.7)  v = a(x^{2} + y^{2}) + ∑_{n+k=3}^{∞} β_{nk}x^{n}y^{k}

for some α_{nk} and β_{nk} which depend on the c_{nk} ’s.

Next, define φ: U_{0} → C by letting

(2.8)  φ(x, y) = (u(x, y), y)

with u as in (2.6). Then φ has a non-zero Jacobian at 0, and hence it is a diffeomorphism in some neighborhood U_{1} ⊂ U_{0} of 0. Consequently, there exists a mapping F: φ(U_{1}) → C such that f = F ∘ φ. Since φ | U_{1} is injective, it sufficies to show that F is a folding. Now, (φ | U_{1})^{−1} has the form

(2.9)  φ^{−1}(u, y) = (x(u, y), y),

hence,

(2.10)  F (u, y) = (u, ṽ(u, y)),

where, by (2.7),

(2.11)  ṽ(u, y) = v(x(u, y), y) = a[x^{2}(u, y) + y^{2}] + ∑_{n+k=3}^{∞} β_{nk}x^{n}(u, y)y^{k},

or

(2.12)  ṽ(u, y) = A_{0}(u) + A_{1}(u)y + A_{2}(u)y^{2} + ∑_{k=3}^{∞} A_{k}(u)y^{k}

for some functions A_{k}(u).

We now show that A_{1}(u) ≡ 0 and A_{2}(u) ≠ 0 in some neighborhood of 0. This will imply that F is a folding in some neighborhood of 0. Indeed, in view of (2.3), the Jacobian of f vanishes at all points (x, 0). Hence for y = 0,

A_{1}(u) = \frac{∂ṽ(u, y)}{∂y} |_{y=0}= J_{F} |_{y=0} = J_{f∘φ^{−}1}|_{y=0} = 0.

By (2.11) and (2.12)

2A_{2}(0) = \frac{∂^{2}ṽ(0, 0)}{∂y^{2}} = 2a > 0,

and by continuity, A_{2}(u) > 0 in |u| < δ for some δ > 0.

Finally, since A_{2}(u) > 0 and A_{1}(u) ≡ 0 for |u| < δ we can get from (2.12)

F (u, y) = (u, ṽ(u, y)) = (u, A_{0}(u) + w^{2}(u, y)),

where

w(u, y) = A_{2}^{1/2}(u)(y + ∑_{k=2}^{∞} b_{k}(u)y^{k})

for some functions b_{k}(u). This shows that F is a local folding in some neighborhood of 0, and hence so is f.

3. Uniformization

3.1. Let D, D_{1} , D_{2} , E and μ be as in 1.1. The main question addressed in this section is whether the existence of local folding solutions at every point of E implies the existence of a global folding solution in D.

In the proof of the following theorem, we will use the fact that if f: U → C is a folding solution such that U ∩E is connected, then U ∩E has a simply connected neighborhood V , V ⊂ U such that V ∩ E is connected, and f | V is a proper folding solution, see [SY1, Lemma 2.2].

3.2. Theorem. Let D, D_{1} , D_{2} , E and μ be as in 1.1, and suppose that every point x in E has a neighborhood U_{x}, U_{x} ⊂ D, where (B) has a folding solution. Then E has a neighborhood V , V ⊂ ⋃_{x∈E} U_{x}, such that

(i) (B) has a proper folding solution in V . (ii) (B) has a folding solution g in D_{1} ∪ V and a folding solution h in D_{2} ∪ V .

Proof. We first prove (ii). Let x ∈ E. Then x has a simply connected neighborhood V_{x}, V_{x} ⊂ U_{x} such that V_{x} ∩ E is connected, and such that (B) has a proper folding solution f_{x}: V_{x} → C. Let V = ⋃_{x∈E} V_{x}. Since μ is locally bounded in D_{1} , (B) has a homeomorphic solution g_{1} in D_{1} , cf. [B]. We may assume that g_{1}(D_{1}) = D_{1} . We will show that g_{1} has a homeomorphic extension, denoted again by g_{1} , on D_{1} ∪ E. Indeed, given x ∈ E, f_{x} maps V_{x} \ D_{2} homeomorphically onto f_{x}(V_{x}), and thus, every subarc of f_{x}(V_{x} ∩ E) is a free boundary arc of f_{x}(V_{x} ∩ D_{1}). Let f̂_{x} = f | V_{x} ∩ D_{1} . Then, g_{1} ∘ f̂_{x}^{−1} is conformal, and hence has a homeomorphic extension on f_{x}(V_{x} \D_{2}). Therefore, g_{1} has a homeomorphic extension on D_{1} ∪ (E ∩ V_{x}). Consequently, g_{1} has a homeomorphic extension on D_{1} ∪ E. We may assume that g_{1}(E) = E.

We now define g in D_{1} ∪ V as follows. If z ∈ D_{1} ∪ E, we set g(z) = g_{1}(z). Given x in E, we define g in V_{x} ∩ D_{2} by letting g(z) = g_{1} ∘ f̂_{x}^{−1} ∘ f_{x}(z). The mapping g is well defined in V ∩ D_{2} , since by the uniqueness theorem for proper folding solutions [SY1, Theorem 3.2], once two points have the same image by a proper folding solution, they will have the same image under any other folding solution. Clearly, g is a folding solution in D_{1} ∪ V .

The mapping h for (ii) is constructed in the same way. As for (i), note that g | V , with g and V as above, is a proper folding solution of (B) in V , as needed.

In spite of Theorem 3.2 and contrary to the uniformization theorem, we have the following two theorems.

3.3. Theorem. Let D, D_{1} , D_{2} , E be as in 1.1. Given a point z_{0} ∈ D \ E, there exists μ as in 1.1 such that

(i) (B) has a local folding solution at every point of E. (ii) (B) has a global solution in D. (iii) Every global solution of (B) in D branches at z_{0} , and in particular, (B) has

no (global) folding solution in D.

Proof. Let T (z) = i(z − \frac{1}{4})(1 − \frac{1}{4}z) and ψ(z) = z^{2} . Let E′ be the connected component of ψ^{−1}(T^{−1}(E)), which contains the point \frac{1}{2} . Let φ be a diffeomorphism of D onto itself such that φ(z_{0}) = 0 and φ(E) = E′, and let F (x, y) = (x, |y|). Set f = F ∘ T ∘ ψ ∘ φ and μ = μ_{f}. Then μ satisfies the conditions of 1.1, f is a solution of (B) in D, and every point x of E has a neighborhood U_{x} where f | U_{x} is a local proper folding solution of (B). Note that f branches at z_{0} with local index i(z_{0}, f) = 2.

We now show that every other global solution branches at z_{0} . Suppose that g is a solution in D. We may assume that Im z_{0} < 0. The case Im z_{0} > 0 is similar. Let G_{1} = D_{1} , let G_{2} be the subdomain of D which lies between E and φ^{−1}(L) where L is the intersection of D with the imaginary axes, and let G_{3} = D_{2} \ \overline{G_{2}} . Next for i = 1, 2, 3, let f_{i} = f | G_{i} and h_{i} = g ∘ f_{i}^{−1} . Then h_{1} and h_{3} are defined in D_{1} , and h_{2} is defined in D_{1} \ F ∘ T ∘ ψ(L). Since g and all f_{i} are W_{loc}^{1,2} solutions of (B) off E it follows that each h_{i} is analytic. Also,

(3.4)  g = h_{i} ∘ f_{i}.

The mappings f_{1} and f_{2} extend homeomorphically on G_{1}∪E and G_{2}∪E, respectively, and they coincide on E. Therefore, h_{1} and h_{2} have the same boundary values on E, and hence h_{1} = h_{2} in D_{1} \ L′, where L′ = T ∘ ψ(L). However, h_{1} is analytic in D_{1} , therefore h_{2} can be extended analytically to D_{1} , and h_{1} = h_{2} in D_{1} .

Finally, f_{2} and f_{3} coincide on φ^{−1}(L), therefore h_{2} and h_{3} have the same boundary values on L′ and therefore h_{2} = h_{3} in D_{1} \ L′, and hence in D_{1} , when continued analytically across L′.

We obtain that h_{1} = h_{2} = h_{3} := h is analytic in D_{1} , and has a continuous extension on D_{1} ∪ E, and in view of (3.4), satisfy g = h ∘ f. Therefore g branches at z_{0} and consequently there is no global folding solution in D.

In view of the last theorem, one may ask: Suppose that (B) has a global solution in D which folds along E, and which is locally injective off E. Does it follow that (B) has a global folding solution in D? The following example answers this question negatively.

3.5. Theorem. Given D, D_{1} , D_{2} , E as in 1.1, there exists μ satisfying the conditions of 1.1 such that

(i) (B) has a global solution in D, which folds along E and is locally injective off E.

(ii) (B) has no global folding solution in D.

Proof. For k = 0, 1, …, 8, let z_{k} = e^{ikπ/8} , and let I_{k} denote the subarc of ∂D_{1} having end points z_{k−1} and z_{k}. Let Q′ denote the square having vertices at the points \frac{1}{4}i, \frac{3}{4}i, \frac{3}{4}i − \frac{1}{2} and \frac{1}{4}i − \frac{1}{2} .

Let f be a mapping of \overline{D} onto \overline{D_{1}} which has the following properties:

(a) f(z) = z̄ for Im z ≤ 0.

(b) f(z_{0}) = 1, f(z_{1}) = i, f(z_{2}) = \frac{3}{4}i, f(z_{3}) = \frac{3}{4}i − \frac{1}{2} , f(z_{4}) = \frac{1}{4}i − \frac{1}{2} , f(z_{5}) = \frac{1}{4}i, f(z_{6}) = \frac{1}{8}i, f(z_{7}) = i and f(z_{8}) = −1.

(c) f is a local diffeomorphism in D_{1} , injective on each arc I_{k}, and maps I_{1} and I_{8} into |z| = 1 and each other arc I_{k} into a line segment.

(d) f(D_{1}) = D_{1} \ ([\frac{1}{8}i, \frac{1}{4}i] ∪ [\frac{1}{2}i, i]), where f | D_{1} covers every point of Q′ twice, and every point in D_{1} \ ([\frac{1}{8}i, i] ∪ Q′) once.

Then μ = μ_{f} satisfies the conditions of 1.1, and f is a solution of (B) which folds along E and is a local homeomorphism in D \ E. Let Q denote the connected component of f^{−1}(Q′) whose boundary meets I_{4} , and let D_{0} = D \ (Q ∪ [−i, −\frac{1}{8}i]). Then f_{0} = f | D_{0} is a proper folding solution in D_{0} which maps D_{0} onto E ∪ D_{1} \ [\frac{1}{8}i, i].

Suppose now that (B) has a global folding solution in D. Then, by the uniqueness theorem for proper folding solutions [SY1, Theorem 3.2], g | D_{0} = h ∘ f_{0} , for some mapping h which is conformal in int f(D_{0}) = D_{1} \ [\frac{1}{8}i, i] := D′_{1} . However, g is homeomorphic in D_{2} , and g = h ∘ f, hence h has a homeomorphic extension on D_{1} , denoted again by h.

Now f^{−1}(\frac{1}{2}i) consists of three points b_{1} ∈ D_{1} , b_{2} ∈ D_{2} and b_{3} ∈ ∂D_{1} , or more precisely b_{3} ∈ I_{7} , and thus a small neighborhood V of b_{3} relative D_{1} is mapped by f into Q′. It follows that g^{−1}(w) has two pre-image points in D_{1} , whenever w ∈ g(V) = h(f(V)), contradicting the assumption that g is a folding.

3.6. Remark. In [SY3] we showed that the existence of local folding solution at every point of E does not imply the existence of a (global) folding solution in D. There, the proof was based on the fact that |a_{2}| ≤ 2 for every schlicht function f(z) = z + a_{2}z^{2} + ⋯ in D.

Acknowledgment. The research of the first author was partially supported by the Fund for the Promotion of Research at the Technion. The authors acknowledge

On folding solutions of the Beltrami equation 231

the hospitality of the Mathematics Department at the University of Helsinki.

References

[B] Bers, L.: Uniformization by Beltrami equation. - Comm. Pure Appl. Math. 14, 1961, 215–228.

[SY1] Srebro, U., and E. Yakubov: Branched folded maps and alternating Beltrami equations. - J. Anal. Math. 70, 1996, 65–90.

[SY2] Srebro, U., and E. Yakubov: μ-homeomorphisms. - Contemp. Math. 211, 1997, 473– 479.

[SY3] Srebro, U., and E. Yakubov: Uniformization of maps with folds. - Israel Math. Conf. Proc. 11, 1997, 229–232.

Received 16 August 1999