INVERTIBLE HARMONIC MAPPINGS, BEYOND KNESER

Theorem 1.1 (H. Kneser). If D is convex, then U is a homeomorphism of B onto D. We recall that this Theorem had a remarkable impact in the development of the theory of minimal surfaces, see for instance [17]. Its influence appears also in other areas of mathematics, let us mention here homogenization and effective properties of materials [5, 2, 3], inverse boundary value problems [10, 1, 11] and, quite recently, variational problems for maps of finite distortion [4]. See also, as general references

Theorem 1.2 (G. Choquet). For every Jordan domain D which is not convex, there exists a homeomorphism Φ : ∂B →∂D such that the solution U to (1.1) is not a homeomorphism. A proof for this Theorem is due to Choquet [7, §3]. In Section 6 we present a new proof aimed at having a more explicit description of the homeomorphism Φ. In the final part of this Introduction, when presenting the content of Section 6, we shall illustrate the advantages of this new proof with more details.

Theorem 1.2 shows that, given a non–convex domain D and its boundary γ, one can find some parameterization of the latter which give rise to a non–invertible solution to (1.1). On the other hand, by the Riemann Mapping Theorem, see for instance [15, Theorem 3.4], for any such γ one can also find other parameterizations for which the corresponding solution to (1.1) is a homeomorphism and, in fact, a conformal mapping. Thus the question arises, for a given simply connected target domain D, possibly n

Theorem 1.3. Let Φ : ∂B →γ ⊂R2 be an orientation preserving diffeomorphism of class C1 onto a simple closed curve γ. Let D be the bounded domain such that ∂D = γ. Let U ∈C2(B; R2) ∩C(B; R2) be the solution to (1.1) and assume, in addition, that U ∈C1(B; R2). The mapping U is a diffeomorphism of B onto D if and only if (1.2) det DU > 0 everywhere on ∂B.

Theorem 1.3, by requiring (1.2) on a suitable proper subset of ∂B. This is the content of Theorem 5.2. Furthermore, it may be worth stressing that (1.2) is, in fact, a constraint on the boundary mapping Φ only. Indeed in Theorem 5.4, by means of the Hilbert transform, we shall express the Jacobian bound det DU > 0 on ∂B as an explicit, although nonlocal, constraint on the components of Φ.

Theorem 1.7. Let Φ : ∂B →γ ⊂R2 be a homeomorphism onto a simple closed curve γ. Let D be the bounded domain such that ∂D = γ. Let U ∈W 1,2 loc (B; R2) ∩ C(B; R2) be the solution to (1.1). The mapping U is a homeomorphism of B onto D if and only if, for every P ∈∂B, the mapping U is a local homeomorphism at P.

Theorem 1.3 . In this case an analogous result could be stated when the disk B is replaced by any simply connected domain Ω, provided the boundary of Ωis smooth enough to guarantee that the map ω mapping conformally Ωonto B, extends to a C1 diffeomorphism of Ωonto B. The paper is organized as follows. In Section 2 we recall two classical results of global invertibility, Theorems 2.1, 2.2, and a fundamental result by H. Lewy [13], about invertible harmonic mappings,

Theorem 2.3. Section 3 collects a sequence of results which are useful for the proofs of The- orems 1.3, 1.7. In view of Theorem 2.2 on the inversion of C1 mappings, our guiding light towards Theorem 1.3 is to obtain that det DU > 0 everywhere in B. This is equivalent to show the absence of critical points for any linear combination uα = cos(α) u + sin(α) v of the components u, v of U. This goal will be achieved through a number of steps. In Proposition 3.2 we show that, assuming (1.2), the numb

Theorem 2.1 (Monodromy). Let U ∈C(B; R2) be such that a) Φ = U

Theorem 2.2. Let U ∈C1(B; R2) be such that a′) Φ = U

Theorem 2.3 (H. Lewy). Let U : B →R2 be harmonic. If U is a sense preserving homeomorphism, then det DU > 0 everywhere in B.

Proposition 3.2. Let U ∈C1(B; R2) be harmonic in B. If det DU > 0 on ∂B, then for every α ∈[0, 2π], the number Mα is finite and we have Mα = M for every α ∈[0, 2π].

Corollary 3.3. Let U be as in Proposition 3.2. We have det DU > 0 everywhere in B if and only if there exists α ∈[0, 2π], such that ∇uα ̸= 0 everywhere in B.

Proposition 3.6. Let u ∈C1(B) be harmonic in B. If ∇u ̸= 0 on ∂B, then M = WN(f(∂B)) −1, with M as in Definition 3.1.

Lemma 3.8. Let U = (u, v) ∈C1(B; R2) be harmonic in B. If det DU > 0, on ∂B, then ∂˜u ∂v > 0, on ∂B, where ˜u is the harmonic conjugate of u.

Theorem 3.9. Let U ∈C1(B; R2) be harmonic in B and let Φ = U

Proposition 3.10. Given a C1 curve parameterized by Φ = (φ, ψ) : [a, b] →R2 and such that φ′ ̸= 0 in (a, b) and φ′(a) = φ′(b) = 0 and given a C1 function g : ψ([a, b]) →R with g′ > 0 in ψ((a, b)), consider the curve eΦ : [a, b] →R2 given by eΦ = (φ, g(ψ)). We have (3.3) Z b a d arg(eΦ′) = Z b a d arg(Φ′).

Lemma 3.11. Under the same assumptions of Theorem 3.9, assuming in addition that ∂u ∂θ

Lemma 4.1. Assume Φ : ∂B →γ ⊂R2 is a homeomorphism onto a simple closed curve γ. Let U ∈C2(B; R2)∩C(B; R2) be the solution to (1.1). If, in addition, for every P ∈∂B the mapping U is a local homeomorphism near P, then there exists ρ ∈(0, 1) such that U is a diffeomorphism of B \ Bρ(0) onto U  B \ Bρ(0)  .

Theorem 5.2. Under the same assumptions as in Theorem 1.3, the mapping U is a diffeomorphism of B onto D if and only if (5.1) det DU > 0 everywhere on Φ−1(γnc), where γnc is the set introduced in Definition 5.1 above. First we prove the following Lemma.

Lemma 5.3. Under the assumptions of Theorem 1.3, we always have (5.2) det DU > 0 everywhere on Φ−1(γc).

Theorem 5.4. Under the same assumptions as in Theorem 1.3, U is a diffeomor- phism of B onto D if and only if the components φ and ψ of Φ satisfy (5.4) ∂φ ∂θ H ∂ψ ∂θ  −∂ψ ∂θ H ∂φ ∂θ  > 0 everywhere on Φ−1(γnc).

Theorem 1.7 is. In fact, the conclusion of Theorem 1.7 does not hold if the condition for every P ∈∂B, the mapping U is a local homeomorphism at P is relaxed to for every P ∈∂B, except possibly at two points, the mapping U is a local homeomorphism at P.

Theorem 7.2 (Clunie and Sheil-Small). Let U be a harmonic mapping on B with canonical representation as in (7.4), let f be defined by (7.5) and assume that (7.6) det DU > 0 in B .

Theorem 7.3. Let U ∈C2(B; R2) ∩C1(B; R2) be harmonic on B with canonical representation as in (7.4), let f be defined by (7.5) and assume that (7.9) det DU > 0 on ∂B , then the following conditions are equivalent (7.10) U

Theorem 7.3 we can remove this kind of requirement.

Corollary 7.4. Let f, ω be holomorphic functions in B such that f extends to a C1 invertible mapping on B, ω extends continuously to B and it satisfies |ω| < 1 , in B . Then, given G, H the holomorphic solutions to (7.15), the harmonic mapping U = G + H is a diffeomorphism on B, it satisfies ReU = Ref and its dilatation equals ω in B.