2004-Fez conference on Differential Equations and Mechanics Electronic Journal of Differential Equations, Conference 11, 2004, pp. 81–93. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

AN INFINITE-HARMONIC ANALOGUE OF A LELONG THEOREM AND INFINITE-HARMONICITY CELLS

MOHAMMED BOUTALEB

Abstract. We consider the problem of finding a function f in the set of ∞-harmonic functions, satisfying

lim_{w→ζ} |f̃(w)| = ∞, w ∈ H(D), ζ ∈ ∂H(D)

and being a solution to the quasi-linear parabolic equation

u_{x}^{2}u_{xx} + 2u_{x}u_{y}u_{xy} + u_{y}^{2}u_{yy} = 0 in D ⊂ ℝ^{2} , where D is a simply connected plane domain, H(D) ⊂ ℂ^{2} is the harmonicity cell of D, and f̃ is the holomorphic extension of f. As an application, we show a p-harmonic behaviour of the modulus of the velocity of an arbitrary stationary plane flow near an extreme point of the profile.

1. Introduction

The complexification problems for partial differential equations in a domain Ω ⊂ ℝ^{n} include the introduction of a common domain Ω̃ ⊃ Ω in ℂ^{n} to which all the solutions of a specified p.d.e. extend holomorphically. The complex domains in question are the so-called harmonicity cells H(Ω), in [4], for the following set of 2m-order elliptic operators:

Δ^{m}u = ∑_{|α|=m} \frac{m!}{α!} \frac{∂^{2|α|}u}{∂x_{1}^{2α_{1}} … ∂x_{n}^{2α_{n}}} = 0, m = 1, 2, 3 …  (1.1)

They often describe properties of physical processes which are governed by such a p.d.e [19]. The operator Δ^{2} has been widely studied in the literature, frequently in the contexts of biharmonic functions [3].

Motivation. Our objective is to introduce the complex domain D̃, and the adequate solution f = f_{ζ} in the space of ∞-harmonic functions H_{∞}(D) , for equation (1.5), below. In view of Theorem 2.5, part 2, we assign a domain D̃ ⊂ ℂ^{2}, denoted by H_{∞}(D), to the class H_{∞}(D). The definition of H_{∞}(D), is similar to the definition of H(D), although less explicit. Equation (1.5) is actually the formal limit, as p → +∞, of the p-harmonic equation in D ⊂ ℝ^{2}

2000 Mathematics Subject Classification. 31A30, 31B30, 35J30.

Key words and phrases. Infinite-harmonic functions; holomorphic extension; harmonicity cells; p-Laplace equation; stationary plane flow.

\begin{matrix}c \\ ◯ \end{matrix}2004 Texas State University - San Marcos.

Published October 15, 2004.

Δ_{p}u = div(|∇u|^{p−2}Δu) = 0, 1 < p < +∞,  (1.2)

For every finite real p > 1, the hodograph method transforms Δ_{p}u = 0 into a linear elliptic p.d.e. in the hodograph plane. Due to [5], the pull-back operation is possible from ℝ^{2}(u_{x}, u_{y}) to the physical plane. Although linear, the obtained equation is not easily computed since its limit conditions become more complicated.

Preliminaries. Let Ω be a domain in ℝ^{n}, n ≥ 2, Ω ≠ ∅, ∂Ω ≠ ∅. In 1935, Aronszajn [3] introduced the notion of harmonicity cells in order to study the singularities of m-polyharmonic functions. These functions, used in elasticity calculus of plates, are C^{∞}-solutions in Ω of (1.1). Recall that H(Ω) is the domain of ℂ^{n}, whose trace Tr H(Ω) on ℝ^{n} is Ω, and represented by the connected component containing Ω of the open set ℂ^{n}−∪_{t∈∂Ω}Γ(t), where Γ(t) = {z ∈ ℂ^{n} : (z_{1}−t_{1})^{2}+⋯+(z_{n}−t_{n})^{2} = 0} is the isotropic cone of ℂ^{n}, with vertex t ∈ ℝ^{n}. Lelong [16] proved that H(Ω) coincides with the set of points z ∈ ℂ^{n} such that there exists a path γ satisfying: γ(0) = z, γ(1) ∈ Ω and T [γ(τ)] ⊂ Ω for every τ in [0, 1], where T is the Lelong transformation, mapping points z = x + iy ∈ ℂ^{n} to Euclidean (n − 2)-spheres S^{n−2}(x, ∥y∥) of the hyperplane of ℝ^{n} defined by: ⟨t − x, y⟩ = 0. If Ω is starshaped at a_{0} ∈ Ω, H(Ω) = {z ∈ ℂ^{n}; T (z) ⊂ Ω} is also starshaped at a_{0}. Furthermore, for bounded convex domains Ω of ℝ^{n}, we get

H(Ω) = {z = x + iy ∈ ℂ^{n} : max_{t∈T (iy)} max[ max_{ξ∈S^{n−1}} (⟨x + t, ξ⟩ − max_{s∈Ω}⟨ξ, s⟩)] < 0} (1.3)

where S^{n−1} is the Euclidean unit sphere of ℝ^{n} [4, 6]. The harmonicity cell of the Euclidean unit ball B_{n} of ℝ^{n} gives a central example, since H(B_{n}) coincides with the Lie ball LB = \{z ∈ ℂ^{n}; L(z) = [∥z∥^{2} + \sqrt{∥z∥^{4} − |z_{1}^{2} + ⋯ + z_{n}^{2}|^{2}}]^{1/2} < 1\}, where ∥z∥ = (|z_{1}|^{2} + ⋯ + |z_{n}|^{2})^{1/2}. Besides, representing also the fourth type of symmetric bounded homogenous irreducible domains of ℂ^{n}, H(B_{n}) has been studied (specially in dimension n = 4) by theoretical physicits interested in a variety of different topics: particle physics, quantum field theory, quantum mechanics, statistical mechanics, geometric quantization, accelerated observers, general relativity and even harmony and sound analysis (For more details, see [11, 18, 19, 20].

From the point of view of complex analysis, Jarnicki [14] proved that if D_{1} and D_{2} are two analytically homeomorphic plane domains of ℂ ≃ ℝ^{2} then their harmonicity cells H(D_{1}) and H(D_{2}) are also analytically homeomorphic in ℂ^{2}. A generalization in ℂ^{n}, n ≥ 2, of this Jarnicki Theorem is established by the author [8], as well as a characterization of polyhedric harmonicity cells in ℂ^{2} [10]. Furthermore, recall that if A(Ω) and Ha(Ω) denote the spaces of all real analytic and harmonic functions (respectively) in Ω, then H(Ω) is characterized by the following feature

[∩_{f∈Ha(Ω)}Ω^{f}]^{0} = H(Ω),  (1.4)

while [∩Ω^{f}]^{0} = ∅, when f runs through A(Ω), where Ω^{f} is the greatest domain of ℂ^{n} to which f extends holomorphically. We emphasize that in (1.4), Ω is actually required to be star-shaped at some point a_{0}, or a C-domain (that is, Ω contains the convex hull Ch(S^{n−2}) of any (n − 2)-Euclidean sphere S^{n−2} included in Ω) or Ω ⊂ ℝ^{2p} with 2p ≥ 4, or Ω is a simply connected domain in ℝ^{2} (cf. [4]). The technique of holomorphic extension, used for harmonic functions in [22], has been generalized for solutions of partial differential equations with constant coefficients by Kiselman [15]. In a recent paper, Ebenfelt [12] considers the holomorphic extension to the so-called kernel N H(Ω) of Ω’s harmonicity cell, for solutions in simply connected domains Ω in ℝ^{n}, of linear elliptic partial differential equations of type: Δ^{k}u + ∑_{|α|<2k} a_{α}(x)D^{α}u = g, where N H(Ω) = {z ∈ H(Ω); Ch[T (z)] ⊂ Ω}. It can be observed that one of the central results in the theory of harmonicity cells is the following Lelong theorem (stated here in the harmonic case)

Theorem 1.1. Let Ω be a non empty domain in ℝ^{n}, n ≥ 2, with non empty boundary and H(Ω) its harmonicity cell in ℂ^{n}. For every ζ ∈ ∂H(Ω) there exists f = f_{ζ}, a harmonic function in Ω, which is the restriction to Ω = H(Ω) ∩ ℝ^{n} of a (unique) holomorphic function f̃_{ζ} defined in H(Ω) such that f̃_{ζ} can not be extended holomorphically in any open neighborhood of ζ.

Statement of the problem. In this paper we consider the simpler case of a nonempty plane domain D (with ∂D ≠ ∅) which we set to be simply connected and look for a suitable ∞-harmonic function f_{ζ} in D. We state the problem as follows:

Let ζ be a boundary point of H(D) and put T (ζ) = {ζ_{1} + iζ_{2}, ζ̄_{1} + iζ̄_{2}}. We will assume first that ζ belongs to Γ(ζ_{1} + iζ_{2}). The problem is to find a solution f_{ζ} in the classical sense, i.e. f_{ζ} ∈ C^{2}(D) and f_{ζ} a.e. continuous on ∂D of the quasi-elliptic system:

u_{x_{1}}^{2}u_{x_{1}x_{1}} + 2u_{x_{1}}u_{x_{2}}u_{x_{1}x_{2}} + u_{x_{2}}^{2}u_{x_{2}x_{2}} = 0 inD  (1.5)
\frac{∂}{∂w̄_{j}} ũ = 0 j = 1, 2  in H(D)  (1.6)
lim_{w→ζ, w∈H(D)} |ũ(w)| = ∞ .  (1.7)

This problem has already been considered in [16] in the harmonic case, and in [7] in the p-polyharmonic case. It has also been solved in the (non linear) p-harmonic case with 1 < p < +∞ and n = 2 [9]. We used in [9] radial p-harmonic functions and their stream functions, centered at points of ∂D; but this approach limited our results to finite real p (with p > 1) and to real valued p-harmonic functions. Our main result in the present paper consists of introducing infinite-harmonicity cells and proving an existence theorem for the ∞-Laplace equation. In Theorem 2.5, we prove that to ζ ∈ ∂H(D) corresponds a f_{ζ} ∈ H_{∞}(D) such that f̃_{ζ} is holomorphic in H(D) and satisfies |f̃_{ζ}(w)| → ∞, when w → ζ with w inside H(D).

2. Infinite-harmonicity cells

The next four propositions are used in this work and their proofs are found in the references as cited.

Proposition 2.1 ([16]). Let Ω be a domain in ℝ^{n}, n ≥ 2, Ω ≠ ∅, ∂Ω ≠ ∅, and H(Ω) ⊂ ℂ^{n} be its harmonicity cell. For every point ζ ∈ ∂H(Ω), the topological boundary of H(Ω), one can associate a point t ∈ ∂Ω, the topological boundary of Ω, such that ζ ∈ Γ(t), the isotropic cone of ℂ^{n} with vertex t.

Proposition 2.2 ([2, 17]). A classical solution u = u(x_{1}, x_{2}) ∈ C^{2} of the partial differential equation

Δ_{∞}u = u_{x_{1}}^{2}u_{x_{1}x_{1}} + 2u_{x_{1}}u_{x_{2}}u_{x_{1}x_{2}} + u_{x_{2}}^{2}u_{x_{2}x_{2}} = 0,

in every non-empty domain D ⊂ ℝ^{2}, is real analytic in D, and cannot have a stationary point without being constant Proposition 2.3 ([4]). To every couple (Ω, f), where Ω is an open set of ℝ^{n} = {x + iy ∈ ℂ^{n}; y = 0} (equipped with the induced topology from ℂ^{n}), f is a real analytic function on D, one can associate a couple (Ω̃, f̃) such that Ω̃ is an open set of ℂ^{n} whose trace Ω̃∩ℝ^{n} with ℝ^{n} is the starting domain Ω, and f̃ is a holomorphic function in Ω̃ whose restriction f̃|Ω to Ω coincides with f. Furthermore, (i) if Ω is connected, so is Ω̃; (ii) Among all the Ω̃’s above, there exists a unique domain, denoted Ω^{f}, which is maximal in the inclusion meaning.

Proposition 2.4 ([13]). Let A ⊂ ℂ^{n} be a connected open set, f and g be two holomorphic functions in A with values in a complex Banach space E. If there exists an open subset U of A such that f(z) = g(z) for every z in U ∩ ℝ^{n}, then f(z) = g(z) for every z in A.

Theorem 2.5. Let D be a simply connected domain of ℝ^{2} ≃ ℂ, with D ≠ ∅, and ∂D ≠ ∅. Let H(D) = {z ∈ ℂ^{2}; z_{1} + iz_{2} ∈ D and z̄_{1} + iz̄_{2} ∈ D} be the harmonicity cell of D. Then

(1) For every ζ ∈ ∂H(D), and every open neighbourhood V_{ζ} of ζ in ℂ^{2}, there exists a classical (∈ C^{2}) ∞-harmonic function f_{ζ} on D, whose complex extension is holomorphic in H(D), but cannot be analytically continued through V_{ζ}.

(2) For the given domain D, let us denote by H_{∞}(D) the interior in ℂ^{2} of ∩{D^{u}; u ∈ H_{∞}(D)}. The set H_{∞}(D) which may be called the infinite-harmonicity cell of D, satisfies:

(a) The trace of H_{∞}(D) with ℝ^{2} is D, under the hypothesis that H_{∞}(D) ≠ ∅ (b) H_{∞}(D) is a connected open of ℂ^{2}

(c) The inclusion H_{∞}(D) ⊂ H(D) always holds

(d) If D is such that every u ∈ H_{∞}(D) extends holomorphically to H(D) then H_{∞}(D) ≠ ∅, and both the cells H(D) and H_{∞}(D) coincide.

(e) Suppose D is bounded and covered by a finite union of open rectangles P_{2}^{r}(a_{j}; ρ_{j1}, ρ_{j2}), centered at a_{j} ∈ D, j = 1, …, m, such that for every u ∈ H_{∞}(D)

lim sup_{n_{k}→+∞} [\frac{1}{(n_{k})!}|\frac{∂^{n_{k}}u}{∂x_{k}^{n_{k}}}(a_{j})|]^{1/n_{k}} ≤ \frac{1}{ρ_{jk}} , k = 1, 2, 1 ≤ j ≤ m .

Then H_{∞}(D) ⊃ ∪_{j=1}^{m}P_{2}^{c}(a_{j}, ρ_{j}), and therefore H_{∞}(D) ≠ ∅.

In the proof of Theorem 2.5, we will use the following two lemmas.

Lemma 2.6. In every sector −π < θ < π, the ∞-Laplace equation Δ_{∞}u = 0 has a solution in the form u = \frac{v(θ)}{ρ}, where θ = Arg z, ρ = |z|, and v satisfies the ordinary differential equation (not containing θ)

(v′)^{2}v” + 3v(v′)^{2} + 2v^{3} = 0  (2.1)

Proof. It is clear that we have to use polar coordinates. With x_{1} = ρ cos θ, x_{2} = ρ sin θ in (1.5), we get by a simple calculation: u_{x_{1}} = u_{ρ} cos θ − \frac{1}{ρ}u_{θ} sin θ, u_{x_{2}} = u_{ρ} sin θ + \frac{1}{ρ}u_{θ} cos θ, u_{x_{1}x_{1}} = u_{ρρ} cos^{2} θ + \frac{1}{ρ^{2}} u_{θθ} sin^{2} θ − \frac{1}{ρ}u_{θρ} sin 2θ + \frac{1}{ρ}u_{ρ} sin θ + \frac{1}{ρ^{2}} u_{θ} sin 2θ, u_{x_{2}x_{2}} = u_{ρρ} sin^{2} θ + \frac{1}{ρ^{2}} u_{θθ} cos^{2} θ + \frac{1}{ρ}u_{θρ} sin 2θ + \frac{1}{ρ}u_{ρ} cos^{2} θ − \frac{1}{ρ^{2}} u_{θ} sin 2θ, u_{x_{1}x_{2}} = \frac{1}{2}u_{ρρ} sin 2θ − \frac{1}{2ρ^{2}} u_{θθ} sin 2θ + \frac{1}{ρ}u_{θρ} cos 2θ − \frac{1}{2ρ}u_{ρ} sin 2θ − \frac{1}{ρ^{2}} u_{θ} cos 2θ. Finally, after expanding the terms and rearranging, the ∞-Laplace equation (1.5) takes the form (in polar coordinates)

Δ_{∞}u = u_{ρ}^{2}u_{ρρ} + \frac{2u_{ρ}u_{θ}u_{ρθ}}{ρ^{2}} + \frac{u_{θ}^{2}u_{θθ}}{ρ^{4}} − \frac{u_{ρ}u_{θ}^{2}}{ρ^{3}} = 0  (2.2)

Putting u = \frac{v(θ)}{ρ} in (2.2) we find that v satisfies the non-linear o.d.e. (2.1). □

Lemma 2.7. Let D be a simply connected domain in ℂ, D ≠ ∅, ∂D ≠ ∅. For every t ∈ ∂D, there exists a complex valued ∞-harmonic function in D which cannot be extended continuously in any given open neighborhood of t.

Proof. Let us look for a solution of (1.5) in D in the form u(z) = \frac{v(θ)}{|z−t|} , where the argument θ is the unique angle in ] − π, π[ satisfying z − t = e^{iθ}|z − t|, v is assumed to be C^{2} in ] − π, π[. Note here that the simple connexity of D guarantees that u is uniform in D. As it can be shown that the ∞−Laplacien operator: Δ_{∞}u = u_{x_{1}}^{2}u_{x_{1}x_{1}} + 2u_{x_{1}}u_{x_{2}}u_{x_{x}x_{2}} + u_{x_{2}}^{2}u_{x_{2}x_{2}} is invariant under translations τ_{a} of ℂ ≃ ℝ^{2}, z = x_{1} + ix_{2}, a = a_{1} + ia_{2} - that is Δ_{∞}(u ∘ τ_{a}) = (Δ_{∞}u) ∘ τ_{a} - we may assume without loss of generality that t = 0. Insertion of v = e^{γθ}, where γ ∈ ℂ is a constant, in (2.1) gives: γ^{4} + 3γ^{2} + 2 = 0 or (γ^{2} + 1)(γ^{2} + 2) = 0. Take γ = i and consider the ∞-harmonic function in D defined by: u(z) = \frac{e^{iθ}}{|z−t|} , or more explicitly:

u(z) = \begin{cases}\frac{1}{|z−t|} exp(i arcsin \frac{x_{2}−t_{2}}{|z−t|}) & if x_{1} ≥ t_{1} \\ \frac{π}{|z−t|} − \frac{1}{|z−t|} exp(i arcsin \frac{x_{2}−t_{2}}{|z−t|}) & if x_{1} < t_{1} and x_{2} > t_{2} \\ \frac{−π}{|z−t|} − \frac{1}{|z−t|} exp(i arcsin \frac{x_{2}−t_{2}}{|z−t|}) & if x_{1} < t_{1} and x_{2} < t_{2} \end{cases}

The result follows immediately by taking the principal argument z ↦ Arg z ∈ ] − π, π[; here for δ ∈ [−1, 1], arcsin δ signifies the unique number β in [−\frac{π}{2} , \frac{π}{2} ] satisfying sin β = δ. □

Proof of Theorem 2.5. For ζ ∈ ∂H(D) there exists, due to Proposition 2.1, a boundary point t of D, such that ζ ∈ Γ(t) (which is equivalent to t ∈ T (ζ)). As T (w) reduces in the two-dimensional case to the pair {w_{1} + iw_{2}, w̄_{1} + iw̄_{2}}, we have t = ζ_{1} + iζ_{2} or t = ζ̄_{1} + iζ̄_{2}. 1.a. Suppose at first that t = ζ_{1} + iζ_{2}. By Lemma 2.6, we deduce that (1.5) has a solution u(z) in D in the form |z − t|^{−1}e^{iθ}. We conclude then as the solutions of (1.5) are in particular real analytic in D (Proposition 2.2), that the so-defined function u(z) (given by Lemma 2.7) has a holomorphic extension ũ to a maximal domain A_{1} = D^{u} in ℂ^{2} (Proposition 2.3). Since H(D) is the connected component containing D of the open ℂ^{2} − ∪_{t′∈∂D}{w ∈ ℂ^{2}; (w_{1} − t′_{1})^{2} + (w_{2} − t′_{2})^{2}}, we have A_{1} ⊃ H(D). Substituting in u(z) complex variables w_{1}, w_{2} to real ones and putting h(w) = \sqrt{(w_{1} − t_{1})^{2} + (w_{2} − t_{2})^{2}}, we obtain

ũ(w) = \begin{cases}\frac{1}{h(w)} exp(i arcsin \frac{w_{2}−t_{2}}{h(w)} ) & if Re w_{1} ≥ t_{1} \\ \frac{π}{h(w)} − \frac{1}{h(w)} exp(i arcsin \frac{w_{2}−t_{2}}{h(w)} ( & if Re w_{1} < t_{1} and Re w_{2} > t_{2} \\ \frac{−π}{h(w)} − \frac{1}{h(w)} exp(i arcsin \frac{w_{2}−t_{2}}{h(w)} ) & if Re w_{1} < t_{1} and Re w_{2} < t_{2}, \end{cases}

where the branches are taken such that the square root is positive when it is restricted to D, and for arcsin the branch is chosen such that its values are real (in ] − π, π[) whenever z belongs to D. To see that ũ(w) is holomorphic in H(D), we consider

F (w) = \begin{cases}\frac{1}{g(w)} exp(i arcsin \frac{w_{2}−Im(ζ_{1}+iζ_{2})}{g(w)}) & if Re w_{1} ≥ Re(ζ_{1} + iζ_{2}) \\ \frac{π}{g(w)} − \frac{1}{g(w)} exp(i arcsin \frac{w_{2}−Im(ζ_{1}+iζ_{2})}{g(w)}) & if Re w_{1} < Re(ζ_{1} + iζ_{2}), \\ & Re w_{2} > Im(ζ_{1} + iζ_{2}) \\ \frac{−π}{g(w)} − \frac{1}{g(w)} exp(i arcsin \frac{w_{2}−Im(ζ_{1}+iζ_{2})}{g(w)}) & if Re w_{1} < Re(ζ_{1} + iζ_{2}), \\ & Re w_{2} < Im(ζ_{1} + iζ_{2}), \end{cases}

where g(w) = \sqrt{[(w_{1} + iw_{2}) − (ζ_{1} + iζ_{2})][(w̄_{1} + iw̄_{2}) − (ζ_{1} + iζ_{2})]}, and the branches are chosen as in ũ(w). Seeing that by [16], H(D) = {w ∈ ℂ^{2}; T (w) ⊂ D}, and noting that g(w) = 0 if and only if w ∈ Γ(t) with t ∈ ∂D, the function F (w) is well defined in some open A_{2} ⊃ H(D). Observe that ũ and F are both holomorphic in A = A_{1} ∩ A_{2} - since \frac{∂ũ}{∂w̄_{j}} = \frac{∂F}{∂w̄_{j}} = 0 in A ⊃ H(D), w_{j} = x_{j} + iy_{j} , j = 1, 2- having the same restriction on D = U ∩ℝ^{2}, with U = H(D): ũ|_{D}(z) = F |_{D}(z) = u(z), with z = x_{1} + ix_{2}. By Proposition 2.4, we deduce that ũ = F in H(D). Furthermore, since ζ and t satisfies (ζ_{1} − t_{1})^{2} + (ζ_{2} − t_{2})^{2} = 0, one has by letting w ∈ H(D) tend to ζ: |ũ(w)| = |h(w)^{−1}| → ∞; consequently the function ũ(w) cannot be extended holomorphically across ζ ∈ ∂H(D). 1.b If t = ζ̄_{1} + iζ̄_{2}, the function G(w) defined in the same way by substituting ζ̄_{1} + iζ̄_{2} to ζ_{1} + iζ_{2} in F (w) (with similar branches) satisfies: (i) G(w) exists for every w ∈ H(D), (ii) G(w) is holomorphic in H(D), (iii) G(w) cannot be extended holomorphically to any open neighborhood of ζ in ℂ^{2} (since |G(w)| → ∞ when w ∈ H(D) → ζ), (iv) The restriction of G(w) on D is an infinite-harmonic function on D.

2) It might happen that the set ∩{D^{u}; u ∈ H_{∞}(D)} reduces to only the starting domain D, we would obtain thus an empty ∞-harmonicity cell, and consequently (b), (c) are held to be true if this eventual case occur.

(a) Suppose the above intersection is of non empty interior in ℂ^{2}. Since D is considered as a relative domain in ℝ^{2}, with respect to the induced topology from ℂ^{2}, and since D^{u} ∩ℝ^{2} = D for every u ∈ H_{∞}(D), we have: ∩{D^{u}; u ∈ H_{∞}(D)}∩ℝ^{2} = ∩{D^{u} ∩ ℝ^{2}; u ∈ H_{∞}(D)} = D; so Tr H_{∞}(D) = H_{∞}(D) ∩ ℝ^{2} ⊂ D. On the other hand, since D ⊂ D^{u} for every u ∈ H_{∞}(D), we have D ⊂ (∩_{u∈H_{∞}(D)}D^{u}) ∩ ℝ^{2}. Moreover, from the real analyticity of a function u ∈ H_{∞}(D) in D, we deduce that for every point a ∈ D, there exist radius ρ_{j}^{u} = ρ_{j}^{u}(a) > 0, j = 1, 2, small enough such that u(z) = ∑_{α∈ℕ^{2}} a_{α}(z − a)^{α}, for all z in the rectangle P_{2}^{r}(a, ρ_{j}^{u}(a)) = {x ∈ ℝ^{2}; |x_{j} − a_{j}| < ρ_{j}^{u}(a), j = 1, 2} ⊂ D, where (z − a)^{α} = (x_{1} − a_{1})^{α_{1}}(x_{2} − a_{2})^{α_{2}}. Substituting w ∈ ℂ^{2} to z, we obtain ũ(w) = ∑_{α∈ℕ^{2}} a_{α}(w − a)^{α} which is of course holomorphic in the complex bidisk P_{2}^{c}(a, ρ_{j}^{u}(a)) = {w ∈ ℂ^{2}; |w_{j} − a_{j}| < ρ_{j}^{u}(a), j = 1, 2} ⊂ ℂ^{2}, where (w − a)^{α} = (w_{1} − a_{1})^{α_{1}}(w_{2} − a_{2})^{α_{2}}, the chosen branch being such that the restriction of (w − a)^{α} to ℝ^{2} is > 0. Thus the domain of holomorphic extension of u is nothing else but the union of all the P_{2}^{c}(a, ρ_{j}^{u}(a)) ’s with a running through D. The above construction involves D ⊂ [∩{D^{u}; u ∈ H_{∞}(D)}]^{0} ∩ ℝ^{2}; so one has Tr H_{∞}(D) = D.

(b) Let w, w′ be two arbitrary points in B = ∩{D^{u}; u ∈ H_{∞}(D)}. By (a), B = ∩_{u∈H_{∞}(D)} ∪_{a∈D} P_{2}^{c}(a, ρ_{j}^{u}(a)), where ρ_{j}^{u}(a), j = 1, 2, are the greatest radius corresponding to the power series expansion of u at a. Note that the set B is connected in case the above intersection reduces to D. Suppose then B ≠ D and take w, w′ in B. Since w, w′ are in D^{u} for every u ∈ H_{∞}(D), there exist, by construction of D^{u}, a, a′ ∈ D, such that w ∈ P_{2}^{c}(a, ρ_{j}^{u}(a)), and w′ ∈ P_{2}^{c}(a′, ρ_{j}^{u}(a′)). Putting ρ_{j}(a) = inf{ρ_{j}^{u}(a); u ∈ H_{∞}(D)}, ρ_{j}(a′) = inf{ρ_{j}^{u}(a′); u ∈ H_{∞}(D)}, we obtain w ∈ P_{2}^{c}(a, ρ_{j}(a)), w′ ∈ P_{2}^{c}(a′, ρ_{j}(a′)), with ρ_{j}(a) ≥ 0 and ρ_{j}(a′) ≥ 0. Let then β denote a path in D joining Re w ∈ P_{2}^{r}(a, ρ_{j}(a)) ⊂ D to Re w′ ∈ P_{2}^{r}(a′, ρ_{j}(a′)) ⊂ D. The path γ, constituted successively with the paths [w, Re w], β, and [Re w′, w′] joins w to w′ and is included into the union P_{2}^{c}(a, ρ_{j}(a)) ∪ D ∪ P_{2}^{c}(a′, ρ_{j}(a′)) ⊂ D^{u}. We conclude that γ ⊂ B, B is connected and therefore so is H_{∞}(D) = B^{0}.

(c) By contradiction, suppose that H(D) does not contain H_{∞}(D). Take w_{0} ∈ H_{∞}(D) with w_{0} ∉ H(D). Since H_{∞}(D) is connected and D ⊂ H_{∞}(D), there would exist a continuous path γ_{w_{0},a} joining w_{0} to some point a ∈ D, with γ_{w_{0},a} ⊂ H_{∞}(D). Next, due to the inclusion D ⊂ H(D), we ensure the existence of a point ζ_{0} belonging to γ_{w_{0},a}∩∂H(D). Due to Part 1 above, to the boundary point ζ_{0} of H(D) corresponds some function f_{ζ_{0}} which is ∞-harmonic in D and whose extension f̃_{ζ_{0}} in ℂ^{2} is a holomorphic function in H(D) which can not be holomorphically continued beyond ζ_{0}. Now, the ∞-harmonicity cell H_{∞}(D) is characterized by: (i) Every u ∈ H_{∞}(D) is the restriction on D of a holomorphic function ũ : H_{∞}(D) → ℂ; (ii) H_{∞}(D) is the maximal domain of ℂ^{2}, in the inclusion sense, whose trace on ℝ^{2} is D, and satisfying (i). Then f̃_{ζ_{0}} is not holomorphic at ζ_{0} with ζ_{0} inside H_{∞}(D), which contradicts the property (i). Consequently, the inclusion H_{∞}(D) ⊂ H(D) always holds.

(d) By Proposition 2.3, given u ∈ H_{∞}(D), there exists a maximal domain D^{u} ⊂ ℂ^{2} to which u extends holomorphically. The domain D^{u} is then a domain of holomorphy of ũ (also called domain of holomorphy of u). Suppose that every u ∈ H_{∞}(D) extends holomorphically to H(D). One has then H(D) ⊂ D^{u}, for every u ∈ H_{∞}(D); therefore, H(D) = H(D)^{0} ⊂ [∩_{u∈H_{∞}(D)}D^{u}]^{0} = H_{∞}(D). The result follows by (c).

(e) Due to Proposition 2.2, every ∞-harmonic function u in D is in particular real analytic in D, and thereby partially real analytic in D. Since D ⊂ ∪_{j=1}^{m}P_{2}^{r}(a_{j}, ρ_{j}), there exist open rectangles P_{2}^{r}(a_{j}, ρ_{j}^{u}) ⊂ D in which u writes as the sum of a power series in (x_{1} − a_{j1})(x_{2} − a_{j2}). More, the convergence radius ρ_{j1}^{u}, ρ_{j2}^{u} corresponding to the development of x_{1} ↦ u(x_{1}, a_{j2}) and x_{2} ↦ u(a_{j1}, x_{2}) at a_{j1} and a_{j2} (respectively) are given by ρ_{jk}^{u} = \{lim sup_{n_{k}→+∞}[\frac{1}{(n_{k})!}|\frac{∂^{n_{k}} u}{∂x_{k}^{n_{k}}}(a_{j})|]^{1/n_{k}}\}^{−1} k = 1, 2, 1 ≤ j ≤ m. By assumption, the given covering of D satisfies inf_{u∈H_{∞}(D)} ρ_{jk}^{u} ≥ ρ_{jk}, that is for every x ∈ P_{2}^{r}(a_{j}, ρ_{j}):

u(x) = ∑_{n_{1}∈ℕ} ∑_{n_{2}∈ℕ} \frac{1}{n_{1}! n_{2}!} \frac{∂^{n_{1}+n_{2}}u}{∂x_{1}^{n_{1}} ∂x_{2}^{n_{2}}} (a_{j})(x_{1} − a_{j1})^{n_{1}}(x_{2} − a_{j2})^{n_{2}},

where x = (x_{1}, x_{2}), a_{j} = (a_{j1}, a_{j2}) and ρ_{j} = (ρ_{j1,}ρ_{j2}). It is clear that the complex series obtained by substituting w_{1}, w_{2} ∈ ℂ to x_{1}, x_{2} ∈ ℝ is convergent on every complex bidisk P_{2}^{c}(a_{j}, ρ_{j}) = {w ∈ ℂ^{2}; |w_{1} − a_{j1}| < ρ_{j1} and |w_{2} − a_{j2}| < ρ_{j2}}. Due to the maximality of D^{u}, we have ∪_{j=1}^{m}P_{2}^{c}(a_{j}, ρ_{j}) ⊂ D^{u} for every u ∈ H_{∞}(D), and thereby ∪_{j=1}^{m}P_{2}^{c}(a_{j}, ρ_{j}) ⊂ ∩{D^{u}; u ∈ H_{∞}(D)}. The last union being an open set, one deduces that H_{∞}(D) ⊃ ∪_{j=1}^{m}P_{2}^{c}(a_{j}, ρ_{j}); this mean in particular that H_{∞}(D) ≠ ∅. □

Remark 2.8. The significant fact of the inclusion H_{∞}(D) ⊂ H(D) is that the common complex domain D̃, denoted H_{∞}(D), for the whole class H_{∞}(D), cannot pass beyond H(D). Nevertheless, given a specified ∞−harmonic function u in D, we may have: D^{u} ⊃ H(D) with D^{u} ≠ H(D) .

Example 2.9. Consider D = {(x_{1}, x_{2}) ∈ ℝ^{2}; x_{1} > 0, x_{2} > 0}, and look for a C^{2} solution u in D of Δ_{∞}u = 0 in the form u = Ax_{1}^{α} + Bx_{2}^{β} (where A, B, α, β are constant). Since Δ_{∞}u = A^{3}α^{3}(α − 1)x_{1}^{3α−4} + B^{3}β^{3}(β − 1)x_{2}^{3β−4}, we deduce that u = x1^{\frac{4}{3}} − x2^{\frac{4}{3}} is a classical ∞-harmonic function u in D. Putting w_{j} = x_{j} + iy_{j},

j = 1, 2 and ũ(w_{1}, w_{2}) = w1^{\frac{4}{3}} − w2^{\frac{4}{3}} , where the branch is chosen such that the restriction of ũ to D ⊂ ℝ^{2} is a real valued function, we observe that ũ is holomorphic in ℂ^{2} − (L_{1} ∪ L_{2}), where L_{1} = ℂ × {0}, L_{2} = {0} × ℂ, and ũ|D = u. Since ℂ^{2} − (L_{1} ∪ L_{2}) = ℂ^{∗} × ℂ^{∗} is a domain (connected open) in ℂ^{2}, we deduce that D^{u} = ℂ^{∗} ×ℂ^{∗}. The harmonicity cell of D is given explicitly by the set of all w ∈ ℂ^{2} satisfying: w_{1}+iw_{2} = x_{1}−y_{2}+i(x_{2}+y_{1}) ∈ D and w̄_{1}+iw̄_{2} = x_{1}+y_{2}+i(x_{2}−y_{1}) ∈ D (here ℝ^{2} ≃ ℂ ). Thus H(D) = {w ∈ ℂ^{2}; x_{1} > |y_{2}| and x_{2} > |y_{1}|} ⊂ D^{u}, and H(D) ≠ D^{u}.

Remark 2.10. The inclusion H_{∞}(D) ⊂ H(D) can be strengthened. Indeed, let D ⊂ ℂ be a simply connected domain, with smooth boundary, and let H_{qr}(D) denote the sub-class of all ∞-harmonic functions which are quasi-radial with respect to some boundary point of D. A function u ∈ H_{qr}(D) if there exists t ∈ ∂D such that u(z) = ρ^{m}f(θ), where z = t + ρe^{iθ} ∈ D, f is a real or complex-valued C^{2} function in ] − π, π[, and m is a constant (no restriction on m also). Note that by Aronsson [2], H_{qr}(D) is not empty. For instance, for m > 1, one can find functions Z = f(θ) in parametric representation: Z = \frac{C}{m}(1 − \frac{1}{m} cos^{2} τ)^{\frac{m−1}{2}} cos τ, θ = θ_{0} + ∫_{τ_{0}}^{τ} \frac{sin^{2} τ′}{m−cos^{2} τ′} dτ′, τ_{1} < τ < τ_{2} (C, θ_{0}, τ_{0}, τ_{1}, τ_{2} are constants). Similarly, let H_{qr}(D) denote the complex domain D̃ corresponding to H_{qr}(D). Since H_{qr}(D) = [∩_{u∈Hqr(D)}D^{u}]^{0} , H_{qr}(D) ⊂ H_{∞}(D), and the constructed function f_{ζ} in the proof of Theorem 2.5 is quasi-radial, we have: H_{∞}(D) ⊂ H_{qr}(D) ⊂ H(D).

Remark 2.11. To see that the property: lim_{w→ζ} |f̃(w)| = ∞, (w ∈ H(D), ζ ∈ ∂H(D)) may fail, we give the following example.

Example 2.12. Let D be an arbitrary simply connected plane domain, D ≠ ∅, ∂D ≠ ∅. For a fixed ζ ∈ ∂H(D), take t = ζ_{1} + iζ_{2} ∈ T (ζ) and consider

F (w) = \sqrt{(w_{1} − t_{1})^{2} + (w_{2} − t_{2})^{2}} exp(\frac{1}{2} arctan \frac{w_{2} − t_{2}}{w_{1} − t_{1}} ),

where the branches are taken such that their restriction to D ⊂ ℝ^{2} is positive for the square root and in ] − \frac{π}{2} , \frac{π}{2} [ for arctg). This function verifies: F (w) is well defined and holomorphic on H(D), its restriction f to D is ∞-harmonic in D since f(z) = \sqrt{ρ}e^{θ/2} where z − t = ρe^{iθ}; nevertheless lim_{w→ζ} |F (w)| = 0. Indeed, if ζ is assumed in ∂H(D)−∂D, one has w_{1}+iw_{2} → ζ_{1}+iζ_{2}, so that (w_{1}−t_{1})^{2}+(w_{2}−t_{2})^{2} = [(w_{1} + iw_{2}) − (ζ_{1} + iζ_{2})][(w̄_{1} + iw̄_{2}) − (ζ_{1} + iζ_{2})] → 0; on the other hand, by definition of T (ζ), (ζ_{1} − t_{1})^{2} + (ζ_{2} − t_{2})^{2} = 0, thus arctan \frac{w_{2}−t_{2}}{w_{1}−t_{1}} → arctan \frac{ζ_{2}−t_{2}}{ζ_{1}−t_{1}} = arctan ±i = ±i∞, and | exp(\frac{1}{2}arctg \frac{w_{2}−t_{2}}{w_{1}−t_{1}} )| → 1. Otherwise, the result is immediate if ζ ∈ ∂D ⊂ ∂H(D). sectionHolomorphic extension in Fluids dynamic In this section, we consider two general examples where the above techniques, of

complexification and analytic continuation to ℂ^{n}, are used for the study of some physical problems. The main application we are interested in is the problem of the behaviour of a flow near an extreme point. In the following, H_{p}(D) denotes the class of all p-harmonic functions on D.

Proposition 2.13. Let D ⊂ ℂ be an arbitrary profile limited by a connected closed curve C, and consider a stationary plane flow round D defined by the data of a vanishing point and its velocity V_{∞} at the infinite. Suppose that C contains two straight segments [a, z_{1}], [a, z_{2}] originated at a = a_{1} + ia_{2} and forming an angle νπ, 0 < ν < 1. Then there exist a suitable real p > 1 and an open simply connected neighborhood U of a, such that the quasi-linear p.d.e: Δ_{p}u = |∇u|^{2}Δu + (p − 2)Δ_{∞}u = 0, has a radial (with respect to a) positive solution φ in U, which approximates the modulus of the velocity V (z) of the fluid. More precisely:

(i) |V (z)| ∼ φ(z) as z → a, (z ∈ U).

(ii) φ ∈ H_{(3ν−4)/(2ν−2)}(U).

(iii) Let C > 0 be a constant, and put δ = (\frac{2−ν}{ν} C)^{(ν−2)/(2ν−2)}; then a stream function φ_{c} associated with a function φ of the form C|z−a|^{ν/(2−ν)} is given by

φ_{c}(x_{1} + ix_{2}) = \begin{cases}δ arcsin \frac{x_{2}−a_{2}}{|z−a|} & if x_{1} ≥ a_{1} \\ δπ − δ arcsin \frac{x_{2}−a_{2}}{|z−a|} & if x_{1} < a_{1} and x_{2} > a_{2} \\ −δπ − δ arcsin \frac{x_{2}−a_{2}}{|z−a|} & if x_{1} < a_{1} and x_{2} < a_{2} \end{cases}

which is q-harmonic in U, with \frac{1}{p} + \frac{1}{q} = 1,; that is, φ_{c} ∈ H_{(3ν−4)/(ν−2)}(U).

Proof. Since D is connected and simply connected, there exists an injective holomorphic transformation f_{1} sending the closed lower half-plane

P^{−} − {−i} = {β ∈ ℂ : Im β ≤ 0, β ≠ −i},

sharpened at −i, onto the the exterior of D, with a = f_{1}(β_{0}) and Im β_{0} = 0. The composed map γ = g_{1}(β) = [f_{1}(β)−a]^{1/(2−ν)} is holomorphic, injective, and sends an open simply connected neighborhood V(β_{0}) ⊂ P^{−}−{−i} of β_{0} onto a neighborhood V(0) ⊂ P_{1}, where P_{1} ⊂ ℂ[γ] is one of the half-planes limited by the straight line passing through γ_{1} = (z_{1} −a)^{1/(2−ν)} and γ_{2} = (z_{2} −a)^{1/(2−ν)}. Due to the symmetry principle of Schwartz [13], the function g_{1} extends as a holomorphic function g̃_{1} in some open simply connected neighborhood Ṽ(β_{0}) ⊂ ℂ − {−i}, Ṽ(β_{0}) ⊃ V(β_{0}). Thus, for every β ∈ Ṽ(β_{0}), one has the absolutely convergent expansion for g̃_{1} : g̃_{1}(β) = ∑_{j=1}^{+∞} A_{j}(β − β_{0})^{j}. Moreover, seeing that g̃_{1} is holomorphic and injective in a neighborhood of β_{0}, the first coefficient A_{1} = [g̃_{1}]′(β_{0}) = g′_{1}(β_{0}) is ≠ 0. This implies in particular: f_{1}(β) = a + (β − β_{0})^{2−ν}f_{2}(β), for β ∈ V(β_{0}), where the function f_{2}(β) = [∑_{j=1}^{+∞} A_{j}(β − β_{0})^{j−1}]^{2−ν} is a holomorphic function in V(β_{0}) and may be taken uniform (seeing that A_{1} is ≠ 0) and holomorphic in a certain open neighborhood Ṽ_{1}(β_{0}). Thus, for β ∈ V_{1}(β_{0}), one has f_{1}(β) = a + (β − β_{0})^{2−ν} ∑_{j=0}^{+∞} B_{j}(β − β_{0})^{j}, and

f_{1}(β) − a ∼ B_{0}(β − β_{0})^{2−ν} as β → β_{0}

where B_{0} = A_{1}^{2−ν}. Recall that the flow is supposed to be held round a profile D with an angular point a. Due to the well known Chaplygine condition, the vanishing point of the current moves under the effect of viscosity and the formation of whirlpools, to the extreme point a of D̄. As a simple calculus reveals, the complex potential of the flow round D is given by:

w = f(z) = Re^{−iθ}g(z) + \frac{Re^{iθ}r^{2}}{g(z)} − [2irR sin(ψ − θ)] ln(z),

where μ = g(z) is the bijective holomorphic function from D^{c}, the exterior of D, onto the domain |μ| > r. The values of r and ψ are such that lim_{z→∞} g(z) = ∞, lim_{z→∞} g′(z) = 1, μ_{0} = g(a) = re^{iψ} and V_{∞} = Re^{iθ} is the velocity at the infinite. The holomorphic bijection f_{3} = f_{1}^{−1} ∘ g^{−1} maps {|μ| ≥ r} onto P^{−} − {−i}. Thus (2) gives

f_{1} ∘ f_{3}(μ) − f_{1} ∘ f_{3}(μ_{0}) ∼ B_{0}[f_{3}(μ) − f_{3}(μ_{0})]^{2−ν} as μ → μ_{0} .  (2.3)

Since f′_{3}(μ_{0}) ≠ 0 one has f_{3}(μ) − f_{3}(μ_{0}) ∼ f′_{3}(μ_{0})(μ − μ_{0}) as μ → μ_{0}, so that (2.3) implies g^{−1}(μ) − g^{−1}(μ_{0}) ∼ C_{0}(μ − μ_{0})^{2−ν}, where

C_{0} = B_{0}f′_{3}(μ_{0})^{2−ν} = [\frac{g′_{1}(β_{0}).g′(a)}{f′_{1}(β_{0})}]^{2−ν};

that is, g(z) − g(a) ∼ C_{0}^{1/(ν−2)}(z − a)^{1/(2−ν)} as z → a. Consequently, near the vanishing point a of the flow, the derivative of g satisfies

g′(z) ∼ \frac{g(z) − g(a)}{z − a} ∼ C_{0}^{1/(ν−2)}(z −a)^{1/(2−ν)−1} = C_{0}^{1/(ν−2)}(z −a)^{(ν−1)/(2−ν)} (2.4)

as z → a. On the other hand, putting μ = g(z), we obtain

\frac{df}{dμ} = R e^{−iθ} − R e^{iθ} \frac{r^{2}}{μ^{2}} − \frac{2irR}{μ} sin(ψ − θ)  (2.5)

Since the velocity satisfies V (z) = f̄′(z), for z ∈ D^{c}, Equality (2.5) at μ_{0} gives

Re^{−iθ} − Re^{iθ} \frac{r^{2}}{μ_{0}^{2}} − \frac{2irR}{μ_{0}} sin(ψ − θ) = 0

From the above equation and (2.5), we get for |μ| ≥ r : \frac{df}{dμ}(μ) − \frac{df}{dμ}(μ_{0}) = (μ − μ_{0})h(μ), with h(μ) = r^{2}R e^{iθ} \frac{μ+μ_{0}}{μ^{2}μ_{0}^{2}} + \frac{2irR}{μμ_{0}} sin(ψ − θ). By a simple calcu-

lus, lim_{μ→μ_{0}} h(μ) = \frac{2R}{r}e^{−2iψ} cos(θ − ψ) ≠ 0, here we will have to suppose that V_{∞} is such that θ ≠ ψ ± \frac{π}{2} (otherwise, if θ = ψ ± \frac{π}{2} , a direct calculus will do). Hence,

\frac{df}{dμ} ∼ D_{0}(z − a)^{1/(2−ν)} as μ → μ_{0} ,  (2.6)

with D_{0} = 2C_{0}^{1/(ν−2)}R cos(θ−ψ)/(re^{2iψ}). Writing \frac{df}{dz} = \frac{df}{dμ}.\frac{dμ}{dz} and combining (2.4)

and (2.6), we obtain the equivalence \frac{df}{dz} ∼ C_{0}^{1/(ν−2)}D_{0}(z − a)^{1/(2−ν)}(z − a)^{\frac{ν−1}{2−ν}} as z → a. Consequently |V (z)| ∼ C|z − a|^{ν/(2−ν)}, where

C = \frac{2R| cos(θ − ψ)|}{r}|\frac{f′_{1}(β_{0})}{g′_{1}(β_{0}).g′a)}|.

Therefore, (i) and (ii) may be obtained by taking p = \frac{3ν−4}{2ν−2}, η = 0, ε = \frac{νC}{2−ν} in the following lemma. □

Lemma 2.14. For every real p > 1 and fixed complex point z_{0} ∈ ℂ, the p-Laplace equation (1.2) has radial solutions (with respect to the origin point z_{0}) defined in any sharpened disk X^{∗} at z_{0}: X^{∗} = {z ∈ ℂ; 0 < |z − z_{0}| < R_{0}}. All these functions may be given by: ε\frac{p−1}{p−2}|z − z_{0}|^{\frac{p−2}{p−1}} + η, if p ≠ 2, and ε ln |z − z_{0}| + η, if p = 2, where a, b are arbitrary in ℝ.

Proof of Lemma 2.14. Since Δ_{p}(u ∘ τ_{z_{0}}) = (Δ_{p}u) ∘ τ_{z_{0}}, we may assume that z_{0} = 0. Firstly, the case p = 2 is well known since a 2-harmonic function u in a domain Ω of ℝ^{n} is also harmonic outside the zeros of grad u. If p ≠ 2 and if x = ρ cos θ, y = ρ sin θ is used in the p-Laplace equation

Δ_{p}u = (p − 2)[u_{x}^{2}u_{xx} + 2u_{x}u_{y}u_{xy} + u_{y}^{2}u_{yy}] + (u_{x}^{2} + u_{y}^{2})(u_{xx} + u_{yy}) = 0 (2.7)

we observe, via a simple substitution of u_{x}, u_{y}, u_{xx}, u_{yy}, u_{xy}, expressed by means of the polar coordinates (ρ, θ), and taking into account that the usual Laplace operator Δ, and the gradient of u give in polar form: Δu = u_{ρρ} + ρ^{−1}u_{ρ} + ρ^{−2}u_{θθ} , |∇u|^{2} = u_{ρ}^{2} + ρ^{−2}u_{θ}^{2}, that (2.7) takes the form

Δ_{p}u = (p − 2)[u_{ρ}^{2}u_{ρρ} + \frac{2u_{ρ}u_{θ}u_{ρθ}}{ρ^{2}} + \frac{u_{θ}^{2}u_{θθ}}{ρ^{4}} − \frac{u_{ρ}u_{θ}^{2}}{ρ^{3}} ] + (u_{ρ}^{2} + \frac{u_{θ}^{2}}{ρ^{2}} )[u_{ρρ} + \frac{u_{ρ}}{ρ} + \frac{u_{θθ}}{ρ^{2}} ] = 0

(2.8)

To look for a radial solution u of (2.7), it suffices to put u(x + iy) = h(ρ) in (2.8). We obtain Δ_{p}u = (h′)^{2}[(p − 1)h′′ + \frac{1}{ρ}h′] = 0 which is computed without difficulty and gives the result stated in Lemma 2.14. (iii) Writing (2.7) in the divergence form, and seing that U is simply connected, one can associate to φ = C|z − a|^{ν/(2−ν)} a conjugate q-harmonic function φ_{c} in U, defined by (φ_{c})_{x_{1}} = −|∇φ|^{ν/(2−ν)}φ_{x_{2}} and (φ_{c})_{x_{2}} = |∇φ|^{ν/(2−ν)}φ_{x_{1}}. □

Remark 2.15. There is a physical interpretation of the p-Laplace equation (1.2) in terms of the laminar pipe flow of so-called power-law fluids [1]. Using the terminology of non-linear fluid mechanic, one is motivated to call the stream function v, corresponding to the potential u, the solution of Δ_{q}v = div(|∇v|^{q−2}Δv) = 0, 1 < q < +∞, where \frac{1}{p} + \frac{1}{q} = 1. In the language of Potential theory we say that u and v are conjugate functions.

Proposition 2.16. Under the same hypothesis than proposition above, suppose that a is a non angular point, V (a + iγ) ≠ 0 for γ real ≠ 0 sufficiently small, and \frac{∂^{r}V}{∂x_{2}^{r}} (a) ≠ 0 for some integer r ≥ 1. Then in some neighborhood U′ of a, the velocity of the fluid writes as

V (z) = [(x_{2} −a_{2})^{r} +(x_{2} −a_{2})^{r−1}h_{1}(x_{1})+⋯+h_{r}(x_{1})]h(z) = W (x_{2} −a_{2})h(z) (2.9)

where z = x_{1} +ix_{2}, a = a_{1} +ia_{2}, h is a real analytic function in some neighborhood U′ of a with h(z) ≠ 0 for every z ∈ U′, and h_{1}, …, h_{r}, appearing in the Weierstrass’ unitary polynomial in (x_{2} − a_{2}), are real analytic functions in some interval ]a_{1} − ε, a_{1} + ε[, ε > 0.

Proof. Due to Propositions 2.3 and 2.4 above, we can extend holomorphically in ℂ^{2} the velocity function V : Ω = (D̄)^{c} → ℂ, (x_{1}, x_{2}) ↦ V (x_{1} + ix_{2}), which is real analytic (in fact even antiholomorphic) in Ω. Using the same technique above, and putting: w = (w_{1}, w_{2}) = (x_{1} +iy_{1}, x_{2} +iy_{2}) ∈ ℂ^{2}, we find a maximal domain Ω^{V} in ℂ^{2} whose trace with ℝ^{2} is Ω, and to which V extends holomorphically. Let then Ṽ denote the unique complexified function of V with Ṽ|_{Ω} = V and Ṽ is holomorphic in Ω^{V}. Since Ṽ : Ω^{V} → ℂ satisfies also Ṽ(a) = V (a) = 0, Ṽ(a_{1}, a_{2}+γ) = V (a_{1}, a_{2}+γ) is ≠ 0 for some (a_{1}, a_{2} + γ) ∈ Ω ⊂ Ω^{V} with γ ≠ 0, and \frac{∂^{r} Ṽ}{∂w_{2}^{r}} (a) ≠ 0 -seing

that \frac{∂^{r} Ṽ}{∂w_{2}^{r}} (a) = \frac{∂^{r} Ṽ}{∂w_{2}^{r}} |_{Ω}(a) = \frac{∂^{r}V}{∂x_{2}^{r}} (a) - there exist, owing to Weierstass’ preparation Theorem in ℂ^{n} [21, p.290] with n = 2, r functions H_{1}(w_{1}), …, H_{r}(w_{1}) which are holomorphic in some open neighborhood Ω̃_{1} of a_{1} in ℂ, and a function H(w) which is holomorphic in some open neighborhood Ω̃ ⊂ Ω^{V} of a in ℂ^{2} with H(w) ≠ 0 in Ω̃, such that

Ṽ(w) = [(w_{2} − a_{2})^{r} + (w_{2} − a_{2})^{r−1}H_{1}(w_{1}) + ⋯ + H_{r}(w_{1})]H(w)  (2.10)

for every w in some open neighborhood (Ω̃)′ of a in ℂ^{2} with (Ω̃)′ ⊂ Ω̃ ⊂ Ω^{V}. Taking now the restriction of Equality (2.10) to ℝ^{2}, and seeing that the restriction h_{1}, …, h_{r} of each holomorphic function H_{1}(w_{1}), …, H_{r}(w_{1}) is (real) analytic in Ω̃_{1} ∩ ℝ, we find the announced result (2.9) by putting H_{j}|_{ℝ^{2}} = h_{j}, H|_{ℝ^{2}} = h, and (Ω̃)′ ∩ ℝ^{2} = U ⊂ Ω. Note also that the restriction h is analytic in U. □

Some concrete examples and physical interpretations of the above results will be discussed in a further paper; nevertheless, the determination of the h_{j} ’s rests heavily upon an identification process and a residue formula. These functions stand for the analytic coefficients of what we will call the Weierstrass polynomial associated to the velocity of the flow in a neighborhood of a vanishing point.

Following Lelong’s method who introduced the transformation T in 1954 (which was useful for constructing the harmonicity cells defined by Aronszajn in 1936), it seems advisable now that an analogue T_{∞} of T must be precise in order to give explicitly some infinite-harmonicity cells.

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Dép. de Mathématiques. Fac de Sciences Fès D. M, B.P. 1796 Atlas Maroc

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