Laurent Baratchart

Laurent Bourgeois

Juliette Leblond

Uniqueness results for inverse Robin problems with bounded coefficient

Keywords: Robin inverse problem, holomorphic Hardy-Smirnov classes, elliptic regularity, unique continuation

In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain Ω ⊂ R n , with L ∞ Robin coefficient, L 2 Neumann data and conductivity of class W 1,r (Ω), r > n. We show that uniqueness of the Robin coefficient on a subpart of the boundary, given Cauchy data on the complementary part, does hold in dimension n = 2 but needs not hold in higher dimension. We also raise on open issue on harmonic gradients which is of interest in this context.

Introduction

This study deals with uniqueness issues for the classical Robin inverse boundary value problem. Mathematically speaking, the inverse Robin problem for Email addresses: laurent.baratchart@inria.fr (Laurent Baratchart), laurent.bourgeois@ensta.fr (Laurent Bourgeois), juliette.leblond@inria.fr (Juliette Leblond) Preprint submitted to Elsevier February 11, 2016 an elliptic partial differential equation on a domain consists in finding the ratio between the normal derivative and the trace of the solution (the socalled Robin coefficient) on a subset of the boundary, granted the Cauchy data (i.e. the normal derivative and the trace of the solution) on the complementary subset. In this paper, we deal primarily with L ∞ Robin coefficients and L 2 Neumann data, for isotropic conductivity equations of the type div (σ grad u) = 0 on Lipschitz domains Ω ⊂ R n , with Sobolev-smooth realvalued strictly elliptic conductivity σ of class W 1,r (Ω), r > n. An anisotropic analog to our uniqueness result is discussed in a separate section.

The Robin inverse problem arises for example when considering non-destructive testing of corrosion in an electrostatic conductor. In this case, data consist of surface measurements of both the current and the voltage on some (accessible) part of the boundary of the conductor, while the complementary (inaccessible) part of the boundary is subject to corrosion. Non-destructive testing consists in quantifying corrosion from the data. Robin boundary condition can be regarded as a simple model for corrosion [START_REF] Inglese | An inverse problem in corrosion detection[END_REF]. Indeed, as was proved in [START_REF] Buttazzo | Reinforcement by a thin layer with oscillating thickness[END_REF], such boundary conditions arise when considering a thin oscillating coating surrounding a homogeneous background medium such that the thickness of the layer and the wavelength of the oscillations tend simultaneously to 0. A mathematical framework for corrosion detection can then be described as follows. We consider a conductivity equation in an open domain Ω, as a generalization of Laplace equation to non-homogeneous media, the boundary of which is divided into two parts. The first part Γ is characterized by a homogeneous Robin condition with functional coefficient λ. A non vanishing flux is imposed on the second part Γ 0 of the boundary. This provides us with a well-posed forward problem, that is, there uniquely exists a solution in Ω meeting the prescribed boundary conditions. The inverse problem consists in recovering the unknown Robin coefficient λ on Γ from measurements of the trace of the solution on Γ 0 . Further motivation to solve the Robin problem are indicated in [START_REF] Lanzani | On the Robin boundary condition for Laplace's equation in Lipschitz domains[END_REF] and its bibliography. A basic question is uniqueness: is the coefficient λ on Γ uniquely defined by the available Cauchy data on Γ 0 as soon as the latter has positive measure? In other words, can we find two different Robin coefficients that produce the same measurements? The answer naturally depends on the smoothness assumed for λ.

On smooth domains, for the Laplace operator at least, uniqueness of the inverse Robin problem for (piecewise) continuous λ has been known for decades to hold in all dimensions. The proof is for example given in [START_REF] Inglese | An inverse problem in corrosion detection[END_REF], and in [START_REF] Colton | The determination of the surface impedance of an obstacle from measurements of the far field pattern[END_REF] for the Helmholtz equation. It relies on a strong unique continuation property (Holmgren's theorem), i.e. on the fact that a harmonic function in Ω, the trace and normal derivative of which both vanish on a non-empty open subset of the boundary ∂Ω, vanishes identically. This argument no longer works for functions λ that are merely bounded. In this case we meet the following weaker unique continuation problem: does a harmonic function, the trace and normal derivative of which both vanish on a subset of ∂Ω with positive measure, vanish identically? A famous counterexample in [START_REF] Bourgain | A remark on gradients of harmonic functions in dimension ≥ 3[END_REF] shows that such a unique continuation result is false in dimension 3 and higher. In dimension 2, a proof that such a unique continuation property holds for the Laplace equation can be found in [START_REF] Aleksandrov | Uniqueness and free interpolation for logarithmic potentials and the Cauchy problem for the Laplace equation in R 2[END_REF] when the solution is assumed to be C 1 up to the boundary and Ω is the unit disk.

In this work, we prove more generally that this unique continuation result still holds for a W 3/2,2 solution to a conductivity equation with W 1,rconductivity σ, r > 2, in a bounded simply connected Lipschitz domain Ω ⊂ R 2 . This enables us to conclude to uniqueness in the inverse Robin problem. Our proof relies on two devices: -A factorization result for the complex derivative of a solution to an isotropic conductivity equation, where one factor is holomorphic and the other is smoothly invertible. This factorization implicitly appears in [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF], but we shall have to work out its regularity on a Lipschitz domain. The holomorphic factor in fact belongs to a Hardy-Smirnov class, hence is uniquely defined by its boundary values on a boundary subset of positive measure.

-A Rolle-type theorem for W 1,2 Sobolev functions on the real line.

Our uniqueness result for the Robin inverse problem generalizes that of [START_REF] Chaabane | Differentiability properties of the L 1 -tracking functional and application to the Robin inverse problem[END_REF] established in smoother cases and under the restriction that the imposed flux is non negative. The proof therein is based on positivity and monotonicity arguments established in [START_REF] Chaabane | Differentiability properties of the L 1 -tracking functional and application to the Robin inverse problem[END_REF], and does not use complex analysis. We also turn the counterexample of [START_REF] Bourgain | A remark on gradients of harmonic functions in dimension ≥ 3[END_REF] into a counterexample to uniqueness in the Robin problem in dimension 3, and raise an intriguing issue on harmonic gradients vanishing on a boundary subset of positive measure which governs uniqueness in higher dimension under mild smoothness assumptions on the sets where the Cauchy data and the Robin coefficient are defined. The paper is organized as follows. In section 2, we set some notation and we recall several results from the theory of Sobolev spaces. In Section 3, we introduce the isotropic conductivity PDE and associated Robin problems.

In Section 4, we state our uniqueness results for such equations on Lipschitz domains in dimension 2. We also give a counterexample in higher dimen-sion. Section 5 is a review of holomorphic Hardy spaces on the disk and their generalization into Smirnov spaces on Lipschitz domains, in connection with the Dirichlet problem for harmonic functions. Proofs of the results in Section 4 are provided in Section 6, along with the necessary factorization and regularity properties of solutions to the 2D Neumann problem which are of interest in their own right. Surprisingly perhaps, these seem not to have appeared before in the literature. In section 7, we indicate how our uniqueness result for the isotropic Robin problem implies a corresponding result in the anisotropic case. For this, we rely on the method of isothermal coordinates initiated in [START_REF] Sylvester | An anisotropic inverse boundary value problem[END_REF] and pursued in [START_REF] Astala | Calderòn's inverse problem for anisotropic conductivity in the plane[END_REF][START_REF] Sun | Anisotropic inverse problems in two dimensions[END_REF], allowing us to transform an anisotropic equation in the plane into an isotropic one. Section 8 contains concluding remarks.

Notation and preliminaries on Sobolev spaces

Let R and C denote the real and complex numbers. With superscript "t" to mean "transpose", we write

x = (x 1 , • • • , x n ) t to indicate the coordinates of x ∈ R n , and we identify C with R 2 on putting z = x 1 + ix 2 .
For 1 ≤ p ≤ ∞, k > 0 an integer and E ⊂ R n a Lebesgue measurable set, we let L p (E) be the space of R k -valued measurable functions on E such that

f p L p (E) = E |f | p dm n < ∞ if p < ∞, (1)
f L ∞ (E) = ess sup E |f | < +∞,
where m n stands for Lebesgue measure. In (1) above, |f | designates the Euclidean norm of f and the notation is irrespective of k, which should cause no confusion. In Section 2.1 we recall some properties of Sobolev spaces. We turn in Section 2.2 to classical definitions of non tangential convergence and maximal functions, while Section 2.3 is specifically devoted to the planar case.

Sobolev spaces

For Ω ⊂ R n an open set, we let W 1,p (Ω) be the familiar Sobolev space of complex-valued functions in L p (Ω) whose first order derivatives again lie in L p (Ω). A complete norm on W 1,p (Ω) is given by

f p W 1,p (Ω) = f p L p (Ω) + ∇f p L p (Ω) if p < ∞, (2)
f W 1,∞ (Ω) = max f L ∞ (Ω) , ∇f L ∞ (Ω) ,
where ∇f is the gradient of f defined as ∇f

= (∂ x 1 f, • • • , ∂ xn f ) t , with ∂ x j
to indicate the derivative with respect to x j .

When n = 1, we simply write f ′ instead of ∂ x 1 f . Throughout, differentiation is given in the distributional sense: Ω ∂ x j f ϕdm n = -Ω f ∂ x j ϕdm n whenever ϕ ∈ D(Ω), the space of complex-valued C ∞ smooth functions with compact support in Ω. When n = 2, which is the main (but not the sole) concern of this paper, it is often convenient to use the complex differential operators:

∂ = 1 2 (∂ x 1 -i∂ x 2 ), ∂ = 1 2 (∂ x 1 + i∂ x 2 ), (3)
so that df = ∂f dz + ∂f dz. When f is holomorphic: ∂f = 0, we also write f ′ instead of ∂f = df /dz. We put W 1,p loc (Ω) for the space of functions whose restriction to any relatively compact open subset Ω 0 of Ω lies in W 1,p (Ω 0 ). The space W 2,p (Ω) is comprised of L p -functions whose distributional derivatives of the first order lie in

W 1,p (Ω), with norm f p W 2,p (Ω) = f p L p (Ω) + j ∂ x j f p W 1,p (Ω)
. The definition of W 2,p loc (Ω) parallels that of W 1,p loc (Ω). For emphasis, we use at places a subscript "R", as in W 1,p R (Ω), to single out the real subspace of real-valued functions. The same symbol (e.g. "C") is used many times to mean different constants. We write A ∼ B to abbreviate CA ≤ B ≤ C ′ A, where C, C ′ are constants. If n = 1, then W 1,p (Ω) is just the space of locally absolutely continuous functions with derivative in L p (Ω). The corresponding characterization when n > 1 is more subtle [START_REF] Ziemer | Weakly differentiable functions[END_REF]Thm. 2.1.4], but in any case W 1,∞ (Ω) identifies with Lipschitz-continuous functions on Ω [48, Sec. V. When Ω is bounded and Lipschitz, each member of W 1,p (Ω) is the restriction to Ω of a function in W 1,p (R n ) (the extension theorem [48, Ch. VI, Thm 5]), and the space of restrictions (D Thm 3.22]. Here and below, the subscript "|E" indicates restriction to a set E. If moreover p > n then W 1,p (Ω) embeds continuously in the space of Hölder-continuous functions on Ω with exponent 1n/p; when p = n such an embedding holds in every L ℓ (Ω), 1 ≤ ℓ < ∞, and if p < n then W 1,p (Ω) embeds continuously in L p * with p * = np/(np) (the Sobolev embedding theorem [START_REF] Adams | Sobolev spaces[END_REF]Thms 4.12,4.39]). In addition, for p ≤ n and ℓ < p * (p * = ∞ if p = n), the previous embeddings are compact (the Rellich-Kondrachov theorem [START_REF] Adams | Sobolev spaces[END_REF]Thm 6.3]). Also, a distribution g on Ω whose first derivatives lie in L p (Ω) does belong to W1,p (Ω) [START_REF] Demengel | Espaces fonctionnels. Utilisation dans la résolution des équations aux dérivées partielles[END_REF]Thm 6.74] 1 , and if Ω is connected while

(R n )) |Ω is dense in W 1,p (Ω) for 1 ≤ p < ∞ [2,
E ⊂ Ω is such that m n (E) > 0, then g -g E L p (Ω) ≤ C ∇g L p (Ω) ,
where

g E := 1 m n (E) E g dm n , (4)
for some C = C(p, Ω, E) (the Poincaré inequality, apply [56, Thm 4.2.1] with L(u) = u E ). The Sobolev embedding theorem entails that W 1,p (Ω) is an algebra for p > n [2, Thm 4.39], in particular if f ∈ W 1,p (Ω) and F is entire then F (f ) ∈ W 1,p (Ω) with norm bounded in terms of Ω, p, F , and f W 1,p (Ω) . For 1 < p < ∞, the space W θ,p (Ω) of fractional order θ ∈ (0, 1) consists of those f ∈ L p (Ω) for which

f p W θ,p (Ω) = f p L p (Ω) + Ω Ω |f (x) -f (y)| p |x -y| n+θp dm n (x)dm n (y) < ∞. (5)
The space W 1+θ,p (Ω) is comprised of f ∈ L p (Ω) whose derivatives of the first order lie in W θ,p (Ω), with norm

f p W 1+θ,p (Ω) = f p L p (Ω) + j ∂ x j f p W θ,p (Ω) .
When Ω is bounded and Lipschitz, W θ,p (Ω) may also be defined via real interpolation between L p (Ω) and W 1,p (Ω) where it corresponds to the Besov space B θ,p,p (Ω); that is, using standard notation for the interpolation functor, it holds that W θ,p (Ω) = [L p (Ω), W 1,p (Ω)] θ,p , see [START_REF] Adams | Sobolev spaces[END_REF]Sec. 7.32 & Thm 7.47]. A slightly different, but equivalent interpolation method is that of trace spaces of J.-L. Lions [1,Ch. 7]. If d(x, ∂Ω) denotes Euclidean distance from x ∈ R n to the boundary of Ω, there is

C = C(Ω, θ, p) such that for all f ∈ L p (Ω) with |∇f | ∈ L p loc (Ω), f W θ,p (Ω) ≤ C d(., ∂Ω) 1-θ ∇f L p (Ω) + f L p (Ω) . (6)
In fact, [START_REF] Jerison | The inhomogeneous Dirichlet problem in Lipschitz domains[END_REF]Thm 4.1] asserts that the left and right hand sides of ( 6) are equivalent when f is harmonic (with constants depending only on Ω), and one can check that the portion of proof yielding (6) (which rests on trace space interpolation) does not depend on harmonicity.

Recall the basic property of interpolation: if A is linear and continuous both X → X ′ and Y → Y ′ where (X, X ′ ) and (Y, Y ′ ) are interpolation pairs of Banach spaces, then If Ω is connected, a variant of the Poincaré inequality involving the trace is: for p > 1 and Γ ⊂ ∂Ω a subset of strictly positive measure Σ(Γ) > 0, there is C > 0 depending only on p, Ω and Γ such that for all g ∈ W 1,p (Ω)

A is continuous [X, Y ] θ,p → [X ′ , Y ′ ] θ,
g - Γ tr ∂Ω g dΣ L p (Ω) ≤ C ∇g L p (Ω) . (7)
This follows from the continuity of the trace operator, the Rellich-Kondrachov theorem and [START_REF] Ziemer | Weakly differentiable functions[END_REF]Lem. 4.1.3]. We need mention Sobolev spaces of negative order in connection with duality of trace spaces: if 1 < p < ∞ and 1/p + 1/q = 1 then, since W 1/q,p (∂Ω) embeds in L p (∂Ω), each g ∈ L q (∂Ω) gives rise via h → ∂Ω g hdΣ to a member of (W 1/q,p (∂Ω)) ′ , the dual space of W 1/q,p (∂Ω). As W 1/q,p (∂Ω) is reflexive (for it is uniformly convex), we see as in [2, Sec. 3.13, 3.14] that the completion W -1/q,q (∂Ω) of L q (∂Ω) with respect to the norm g W -1/q,q (∂Ω) := sup h W 1/q,p (∂Ω) =1 ∂Ω g h dΣ can be identified with (W 1/q,p (∂Ω)) ′ . This we use when p = q = 2 only.

Non tangential maximal function

For ξ ∈ ∂Ω, each α > 1 defines a nontangential region of approach to ξ from Ω given by

R Ω α (ξ) = {x ∈ Ω : |x -ξ| < α d(x, ∂Ω)}. ( 8
)
When Ω is Lipschitz and bounded, R Ω α (ξ) contains a nonempty open truncated cone with vertex ξ, whose aperture and height are independent of ξ [31, Thm 1.2.2.2]. Subsequently, whenever h is R k -valued on Ω, we define its nontangential maximal function (associated with α) to be

M α h(ξ) = sup x∈R Ω α (ξ) |h(x)|, ξ ∈ ∂Ω, (9)
which is well-defined with values in [0, +∞]. Also, we say that h defined on Ω converges nontangentially to a at ξ ∈ ∂Ω if, for every α > 1,

lim x→ξ, x∈R Ω α (ξ) h(x) = a. (10) 2.3
. For f ∈ L 2 (∂Ω), define ∂ τ f ∈ (W 1,2 (∂Ω)) ′ to be the linear form h → -∂Ω f ∂ τ hdΛ, h ∈ W 1,2 (∂Ω)
. This generalizes the previous definition of ∂ τ when f ∈ W 1,2 (∂Ω), for in this case integration by parts shows that the linear form just mentioned extends to a member of (L 2 (∂Ω)) ′ ∼ L 2 (∂Ω) which is just ∂ τ f in the former sense. Thus, by interpolation, ∂ τ is continuous from W 1/2,2 (∂Ω) into the space (W 1/2,2 (∂Ω)) ′ ∼ W -1/2,2 (∂Ω). Indeed, from [42, Ch.I, Thm 6.2]:

[(L 2 (∂Ω)) ′ , (W 1,2 (∂Ω)) ′ ] 1/2,2 = ([W 1,2 (∂Ω), L 2 (∂Ω)] 1/2,2 ) ′ = (W 1/2,2 (∂Ω)) ′ .

Conductivity equation and Robin inverse problem

In Section 3.1 we introduce the conductivity equation under study. Sections 3.2 and 3.3 are dedicated to the associated forward Neumann and Robin problems. Section 3.4 concerns the inverse Robin problem.

The conductivity equation

The conductivity equation with unknown real-valued function u is

∇ • (σ ∇u) = 0 , (11)
where "∇ • X" means "divergence of the vector field X". Except in Section 7, we assume that the conductivity σ is a real-valued function on a bounded Lipschitz domain Ω ⊂ R n satisfying

σ ∈ W 1,r R (Ω), r > n, ( 12
) 0 < c ≤ σ ≤ 1/c < +∞ for some constant c. ( 13
)
The fact that σ is real means that the conduction is isotropic. Condition [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF] above means that (11) is strictly elliptic. Condition [START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF] is less restrictive than Lipschitz-regularity, but still it implies some Hölder-smoothness. Note that, since r > n, the space W 1,r (Ω) consists of multipliers on W 1,2 (Ω), see [START_REF] Zolesio | Multiplication dans les espaces de Besov[END_REF] or [START_REF] Grisvard | Elliptic problems in non-smooth domains[END_REF]Thm 1.4.4.2]. As [START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF] and ( 13) together imply that 1/σ ∈ W 1,r R (Ω), our assumptions are thus to the effect that multiplication by (the restriction to ∂Ω of) 1/σ is an isomorphism on W 1/2,2 R (∂Ω). By duality, it follows that multiplication by 1/σ is an isomorphism on W -1/2,2 R (∂Ω). This entails that each u ∈ W 1,2 R (Ω) solving for [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] has a well-defined normal derivative on ∂Ω, denoted by

∂ n u ∈ W -1/2,2 R
(∂Ω). The standard definition is the weak one: if J designates a right inverse to the trace operator

W 1,2 R (Ω) → W 1/2,2 R
(∂Ω) and , the duality

pairing on W -1/2,2 R (∂Ω) × W 1/2,2 R (∂Ω), then h → Ω σ∇u.∇(J(h)) d m n is a continuous linear form on W 1/2,2 R (∂Ω) which can be represented uniquely as φ, h for some φ ∈ W -1/2,2 R (∂Ω). Since division by σ is an isomorphism of W -1/2,2 R (∂Ω), we may set ∂ n u = φ/σ ∈ W -1/2,2 R
(∂Ω) and then it holds that

σ∂ n u , tr ∂Ω ψ = Ω σ∇u.∇ψ dm n , ψ ∈ W 1,2 R (Ω). ( 14
)
Indeed, ( 14) holds by construction when ψ ∈ Ran J, hence it is enough to check it when ψ ∈ W 1,2 0,R (Ω) in order to get it for all ψ ∈ W 1,2 R (Ω). By density, we are left to prove that Ω σ∇u.∇ψ dm n = 0 whenever ψ ∈ D R (Ω) which is nothing but the distributional meaning of [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF]. Comparing ( 11) and ( 14) with the classical Green formula, it is natural to call ∂ n u the (exterior) normal derivative of u on ∂Ω. Checking ( 14) against ψ ≡ 1, we observe in particular that ∂ n u , σ = 0. ( 15)

The Neumann problem

The Neumann problem in W 1,2 (Ω) for the conductivity equation ( 11) is:

given g ∈ W -1/2,2 R (∂Ω) such that g , σ = 0, to find u ∈ W 1,2 R (Ω) such that ∇ • (σ ∇u) = 0 in Ω, ∂ n u = g on ∂Ω. (16)
Note that the vanishing of g , σ is necessary by [START_REF] Brezis | Analyse fonctionnelle[END_REF]. A solution to [START_REF] Buttazzo | Reinforcement by a thin layer with oscillating thickness[END_REF] exists and is unique up to an additive constant. To check this well-known fact, simply observe that f → σg, tr ∂Ω f is a continuous linear form on W 1,2 R (Ω)/R (the quotient space of W 1,2 R (Ω) modulo constants), a Hilbert norm on which is given by ∇f L 2 (Ω) in view of (4). As σ 1/2 ∇f L 2 (Ω) is an equivalent norm by [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF], we see upon denoting by ǔ

∈ W 1,2 R (Ω)/R the equivalence class of u ∈ W 1,2
R (Ω) that there is a unique ǔ to meet [START_REF] Bourgain | A remark on gradients of harmonic functions in dimension ≥ 3[END_REF] with ∂ n u replaced by g, thanks to the Lax-Milgram theorem [15, Cor. V.8]. As pointed out earlier, this is equivalent to u solving [START_REF] Buttazzo | Reinforcement by a thin layer with oscillating thickness[END_REF]. Such a u is called an energy solution to the Neumann problem.

The forward Robin problem

The forward Robin problem is an implicit variation of the Neumann problem where the solution to [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] and its normal derivative have to satisfy an affine relation with functional coefficients on the boundary. In particular, the normal derivative is sought to be a function rather than a distribution on ∂Ω. Below we consider a rather simple form of the problem, arising naturally in the setting of non-destructive control, where the affine relation has L 2 righthand side and bounded coefficient. More general versions with right-hand side in L p (∂Ω), p ∈ (1, 2], are studied in [START_REF] Lanzani | On the Robin boundary condition for Laplace's equation in Lipschitz domains[END_REF]. Throughout we assume that ∂Ω is partitioned into measurable subsets Γ and Γ 0 of strictly positive arclength:

∂Ω = Γ ∪ Γ 0 , Γ ∩ Γ 0 = ∅, Σ(Γ) > 0, Σ(Γ 0 ) > 0. ( 17
)
We put for simplicity

L ∞ + (Γ) := {λ ∈ L ∞ R (Γ) , λ ≥ 0 a.e. on Γ , λ ≡ 0}. ( 18
) Given λ ∈ L ∞ + (Γ) and g ∈ L 2 (Γ 0 ), the forward Robin problem consists in seeking u ∈ W 1,2 (Ω) such that    ∇ • (σ ∇u) = 0 in Ω, ∂ n u = g on Γ 0 , ∂ n u + λu = 0 on Γ. (19)
As tr

∂Ω u ∈ W 1/2,2 R (∂Ω) ⊂ L 2 R (∂Ω), boundary conditions make sense in that g concatenated with (-λtr ∂Ω u) |Γ defines a member of L 2 R (∂Ω) ⊂ W -1/2,2 R
(∂Ω). Replacing g and λ by g/σ and λ/σ respectively, which is possible by [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF], solving [START_REF] Chaabane | Differentiability properties of the L 1 -tracking functional and application to the Robin inverse problem[END_REF]

is tantamount to obtain u ∈ W 1,2 R (Ω) satisfying    ∇ • (σ ∇u) = 0 in Ω σ ∂ n u = g on Γ 0 σ ∂ n u + λu = 0 on Γ. ( 20
)
In view of [START_REF] Bourgain | A remark on gradients of harmonic functions in dimension ≥ 3[END_REF], problem [START_REF] Chaabane | Identification of Robin coefficients by the means of boundary measurements[END_REF] is equivalent to the following weak formulation:

to find u in W 1,2 (Ω) such that Ω σ ∇u • ∇ψ dm n + Γ λuψ dΣ = Γ 0 gψ dΣ, ψ ∈ W 1,2 R (Ω). ( 21
)
As soon as σ ∈ L ∞ R (Ω), well-posedness of problem [START_REF] Cheney | Electrical impedance tomography[END_REF], that is, existence and uniqueness of a solution u ∈ W 1,2 (Ω), follows at once from the Lax-Milgram theorem and Lemma 3.1 below. Further, as a consequence of ( 21), it holds that

Γ λu dΣ = Γ 0 g dΣ . Lemma 3.1. Let Ω ⊂ R n be a bounded Lipschitz domain, σ ∈ L ∞ R (Ω) satisfy (13), and λ ∈ L ∞ + (Γ) for some Γ ⊂ ∂Ω such that Σ(Γ) > 0. Then, u → Ω σ |∇u| 2 dm n + Γ λ u 2 dΣ 1/2
is an equivalent norm on W 1,2 (Ω).

Proof. We must show that there exist two constants c, C > 0 such that

c ψ 2 W 1,2 (Ω) ≤ Ω σ |∇ψ| 2 dm n + Γ λ ψ 2 dΣ ≤ C ψ 2 W 1,2 (Ω) , ψ ∈ W 1,2 R (Ω).
The right inequality follows from the boundedness of σ, λ, together with the continuity of the trace operator and the embedding W 1/2,2 (∂Ω) → L 2 (∂Ω).

To prove the left inequality we can replace Γ by a subset on which λ ≥ ε > 0, and then the result drops out from ( 7), the Schwarz inequality, and the fact that σ is bounded away from 0 by [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF].

The inverse Robin problem

Associated to the forward Robin problem [START_REF] Chaabane | Identification of Robin coefficients by the means of boundary measurements[END_REF] is the inverse Robin problem, which consists in finding the unknown impedance λ in L ∞ + (Γ) from the knowledge of u and g on Γ 0 . Note that a solution u to [START_REF] Chaabane | Identification of Robin coefficients by the means of boundary measurements[END_REF] uniquely exists in W 1,2 (Ω), as was pointed out before Lemma 3.1 above. In the setting of nondestructive control, Γ 0 represents that part of the boundary ∂Ω which is accessible to pointwise measurement or imposition of u and ∂ n u. In this work, we consider the uniqueness issue as to whether λ is uniquely determined by g and u |Γ 0 when Ω is a bounded contractible Lipschitz domain. For general partitions of the boundary like [START_REF] Chaabane | Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems[END_REF], it will turn out that the answer is "yes" when n = 2 and "no" when n ≥ 3. Pointing out this structural difference between the planar and the higher dimensional cases is the main purpose of the present article.

Uniqueness results

The two uniqueness theorems in Section 4.1 are the main results of this work. Section 4.2 provides a counterexample in dimension 3.

Inverse Robin problem in dimension 2: uniqueness results

In this section we investigate the planar case: Ω ⊂ R 2 , in particular it is understood throughout that n = 2 in (12) and we write Λ instead of Σ in [START_REF] Chaabane | Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems[END_REF].

Theorem 4.1. Assume that Ω ⊂ R 2 is a bounded simply connected Lipschitz domain and that (17) holds. Let σ satisfy (12)-( 13)

and g ∈ L 2 (Γ 0 ) be such that g ≡ 0. Suppose λ 1 , λ 2 ∈ L ∞ + (Γ) are such that the corresponding solutions u 1 , u 2 ∈ W 1,2 R (Ω) to problem (20) satisfy u 1| Γ 0 = u 2| Γ 0 .
Then λ 1 = λ 2 . Theorem 4.1 will be a consequence of the following unique continuation result which is proved in Section 6.

Theorem 4.2. Assume that Ω ⊂ R 2 is a bounded simply connected Lipschitz domain and that σ satisfies (12)- [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF]. Let u ∈ W 1,2 R (Ω) be a solution to [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] in Ω such that ∂ n u ∈ L 2 (∂Ω). If both u and ∂ n u vanish on a subset γ ⊂ ∂Ω of strictly positive measure, then u ≡ 0 in Ω.

Proof. (Theorem 4.1) By assumption, u 1 and u 2 have the same Cauchy data on Γ 0 ⊂ ∂Ω with Λ(Γ 0 ) > 0, so Theorem 4.2 implies that u 1 ≡ u 2 in Ω whence (λ 1λ 2 ) u 1 = 0 on Γ by the Robin boundary condition. Assume for a contradiction that λ 1 = λ 2 a.e. on Γ. Then, there exists a subset γ ⊂ Γ, Λ(γ) > 0, such that λ 1λ 2 = 0 on γ, and of necessity u 1 vanishes on γ by what precedes. In turn ∂ n u 1 = -λ 1 u 1 /σ vanishes identically on γ, therefore Theorem 4.2 implies that u 1 ≡ 0 in Ω. Consequently ∂ n u 1 = 0 a.e. on ∂Ω, thereby contradicting the assumption that g ≡ 0 in Γ 0 .

The proof of Theorem 4.2 (see Section 6 and Theorem 6.1) ultimately rests on the fact that, in dimension 2, a harmonic gradient (i.e. the conjugate of a holomorphic function if we identify C with R 2 ) which has nontangential limit zero on a subset of ∂Ω of positive measure is identically zero (see Section 5). This is no longer true in higher dimension, as illustrated in the next section.

Examples of non uniqueness in higher dimension

An initial example was constructed in [54, Thm 1] of a nonconstant harmonic function on a half space in R 3 , with Hölder-continuous derivatives up to the boundary, whose gradient vanishes on a boundary set E with m 2 (E) > 0, see also [START_REF] Aleksandrov | Hardy classes of functions that are harmonic in a half-space[END_REF]. In [START_REF] Bourgain | A remark on gradients of harmonic functions in dimension ≥ 3[END_REF], this construction was refined to the effect that there is a nonzero harmonic function on a half space, C 1 -smooth up to the boundary, that vanishes together with its normal derivative on a boundary set E with m 2 (E) > 0. In fact, such examples can be constructed on any open subset of R n , n ≥ 3, whose boundary is a C 1,ε manifold [START_REF] Wang | A remark on gradients of harmonic functions[END_REF]. This shows that Theorem 4.2 does not hold in dimension strictly bigger than 2, and casts doubt on whether an analog to Theorem 4.1 can hold in higher dimension. Indeed, the example below shows that it cannot, already for harmonic functions on smooth domains. Hereafter, we denote by B 3 ⊂ R 3 the open unit ball and by S 2 the boundary sphere (recall the definition (18

) of L ∞ + (Γ)). Example 4.1. Let u be a nonzero harmonic function in B 3 , of class C 1 on B 3 , such that u |E = (∂ n u) |E = 0 where E ⊂ S 2 , with Σ(E) > 0.
If in problem [START_REF] Chaabane | Differentiability properties of the L 1 -tracking functional and application to the Robin inverse problem[END_REF] we set:

Γ 0 = {ξ ∈ S 2 , u 2 (ξ) + ∂ n u 2 (ξ) = 0}, g = ∂ n u |Γ 0 , then Γ := S 2 \ Γ 0 contains E hence it has strictly positive Σ-measure, but clearly λ can be arbitrary in L ∞ + (Γ) since u |Γ = ∂ n u |Γ ≡ 0. Example 4.
1 shows that a solution to [START_REF] Chaabane | Differentiability properties of the L 1 -tracking functional and application to the Robin inverse problem[END_REF] may be associated to all Robin functions. This is an extreme example of non uniqueness which, however, is not fully satisfactory in that it is highly non generic and will be destroyed by small perturbations of the Neumann boundary data g on Γ 0 . The theorem below gives another example of non uniqueness which is easily seen to be stable under L p (Γ 0 )-small perturbations of g, for p > 2.

Theorem 4.3. Set σ ≡ 1 on B 3 . Then, there is a partition

S 2 = Γ ∪ Γ 0 of the form (17), along with functions g ∈ L 2 R (Γ 0 ) and λ 1 = λ 2 ∈ L ∞ + (Γ) such that the corresponding solutions u 1 , u 2 to (19) on B 3 , though distinct, satisfy (u 1 ) |Γ 0 = (u 2 ) |Γ 0 . Proof. Let Γ 0 ⊂ S 2 , Σ(Γ 0 ) > 0, have the property that there is a nonzero harmonic function u in B 3 , of class C 1 on B 3 , with u |Γ 0 = (∂ n u) |Γ 0 = 0. Such a Γ 0 exists by [14]. Let h ∈ L ∞ R (S 2
) be such that 0 < c < -h < C on Γ = S 2 \ Γ 0 for some constants c, C, and moreover S 2 hdΣ = 0. In addition, we pick c large enough that h < -|∂ n u| -1 on Γ. Let v be a solution to the Neumann problem [START_REF] Buttazzo | Reinforcement by a thin layer with oscillating thickness[END_REF] where σ ≡ 1, Ω = B 3 , and ∂ n v = h. On the sphere, the "Riesz tranform" mapping the normal derivative of a harmonic function w in B 3 to its tangential gradient vector field is continuous in L pnorm for 1 < p < ∞; this follows easily by dominated convergence from the fact that, for each α > 1,

M α ∇w L p (S 2 ) ≤ C α ∂ n w L p (S 2 ) , see [27, Thm 2.6]. Therefore v |S 2 ∈ W 1,p R (S 2
) for all p ∈ (1, ∞), hence it is bounded by the Sobolev embedding theorem. Thus, upon adding a positive constant to v, we may assume that v > |u| + 1 on Γ and that the function v∂ n uhu does not identically vanish on Γ. Now, letting λ 1 = -h/v and λ 2 = -(h+∂ n u)/(u+v) on Γ, we have that λ 1 , λ 2 ∈ L ∞ + (Γ), λ 1 ≡ λ 2 , while the functions u 1 = v and u 2 = v + u coincides together with their normal derivatives on Γ 0 , as desired.

Counterexamples similar to the one in Theorem 4.3 can be constructed in any dimension greater than 3.

Remark 4.1. Whenever σ ∈ W 1,∞
R (Ω) and γ contains an open subset of ∂Ω, it is not difficult to deduce from the unique continuation result in [START_REF] Garofalo | Monotonicity properties of variational integrals, A p weights and unique continuation[END_REF] that the analog of Theorem 4.2 holds for any n ≥ 2. However, Example 4.1 shows that assuming Γ 0 open cannot rescue a higher dimensional analog of Theorem 4.1. The situation becomes more interesting if we assume that the interiors of Γ 0 and Γ fill S 2 up to a set of Σ-measure zero. Then, proving or disproving the analog of Theorem 4.1 when n ≥ 3 is tantamount to decide if a solution to (16) that vanishes together with its normal derivative on some E ⊂ ∂Ω, with Σ(E) > 0, can be such that ∂ n u/u is (essentially) bounded and nonnegative in a neighborhood of E in ∂Ω. This question seems to be open, even for harmonic functions in a ball.

Hardy-Smirnov classes of holomorphic functions

In Section 5.1 we review Hardy spaces and conjugate functions on the disk, as well as conformal maps onto simply connected Lipschitz domains. This we use in Section 5.2 to discuss Smirnov spaces on Lipschitz domains, in particular of exponent 2. There, we bridge classical material from complex analysis with known results from elliptic regularity theory to characterize Smirnov functions in terms of Sobolev smoothness (Theorem 5.1). Roughly speaking, Smirnov spaces consist of holomorphic functions with Lebesgue integrable boundary values with respect to arclength, and as such they are basic to solve Dirichlet and Neumann problems for the Laplace equation in dimension 2. In Section 5.3, we dwell on this connection to prove wellposedness of the Dirichlet problem with W 1,2 -data which we could not find in the literature (Proposition 5.2). This well-posedness and the fact that a nonzero Smirnov function cannot vanish on a boundary subset of positive measure are fundamental to the proof of Theorem 4.2 in Section 4.

Hardy spaces of the disk

We set D(ξ, ρ) and T(ξ, ρ) to designate the disk and the circle of radius ρ, centered at ξ in the complex plane. When ξ = 0 we simply write D ρ and T ρ , and if ρ = 1 we omit subscripts. Arclength on T ρ will be denoted by m, irrespective of ρ, which should cause no confusion. Thus, dm(ρe iθ ) = ρdθ. Given a function f on D and ρ ∈ [0, 1), we write f ρ to mean the function on D given by f ρ (z) = f (ρz). For p ∈ [1, ∞), the Hardy space H p consists of functions f which are holomorphic in the unit disk and satisfy the growth condition:

f H p = sup 0<ρ<1 Tρ |f (ξ)| p dm(ξ) 1/p < +∞. ( 22
)
The space H ∞ is comprised of bounded holomorphic functions endowed with the sup norm. Note that f ρ L p (T) is non-decreasing with ρ by subharmonicity of |f | p , see [START_REF] Rudin | Real and complex analysis[END_REF]Thm 17.6], hence the sup in ( 22) is really a limit as ρ → 1 -. It is well-known (see [START_REF] Duren | Theory of H p spaces[END_REF]Ch. 2] or [28, Ch. 2]) that each f ∈ H p has a non-tangential limit f (ξ) at m-a.e. ξ ∈ T, which makes for a definition of f on the unit circle. The map f → f |T is an isometry from H p onto the closed subspace of L p (T) consisting of functions whose Fourier coefficients of strictly negative index do vanish. This allows us to regard H p both as a space of holomorphic functions on D and as a space of L p -functions on T, upon identifying f with f |T . Every f ∈ H p can be represented as the Cauchy as well as the Poisson integral of its non-tangential limit:

f (z) = 1 2iπ T f (ξ) ξ -z dξ, f (z) = 1 2π T 1 -|z| 2 |ξ -z| 2 f (ξ) dm(ξ), z ∈ D.
Hereafter, the Poisson integral of a function ψ ∈ L 1 (T) will be abbreviated as

P [ψ]. If f ∈ H p , 1 ≤ p < ∞, then (f ρ ) |T -f |T L p (T) → 0 as ρ → 1 -. As for the nontangential maximal function, it holds if p > 1 and f ∈ H p that, for any α > 1, M α f L p (T) ≤ C f L p (T) (23)
where the constant C depends on α and p [28, Ch. II, Thm 3.1].

Clearly H p ⊂ L p (D), moreover one can see from the Cauchy formula that if f ∈ H p and ε > 0 then the derivative f ′ satisfies f ′ ρ L p (T) ≤ C(1ρ) -1-ε where C depends only on ε and p [26, Thm 5.5]. Thus, using Fubini's theorem to evaluate the right hand side of (6), we deduce that H p embeds in W θ,p (D) for θ ∈ (0, 1/p) and we get by the Sobolev embedding theorem that

f L λ (D) ≤ C f H p , p ≤ λ < 2p, (24)
where C = C(p, λ). When p = 2 these estimates can be sharpened, for in this case Green's formula yields that (1 25) is a member of H p . The multiplicative decomposition f = JE |f | is called the inner-outer factorization of f . We shall need that if f ∈ H p , g ∈ H q , and |f |T g |T | ∈ L r for some p, q, r ≥ 1, then f g ∈ H r . Indeed, one has inner-outer factorizations

-|z| 2 ) 1/2 ∇f (z) L 2 (D) ∼ f -f (0) H 2 [
E h (z) = exp 1 2π T ξ + z ξ -z log h(ξ) dm(ξ) , z ∈ D, (25)
f = J 1 E |f | and g = J 2 E |g| , so that f g = J 1 J 2 E |f | E |g| = J 1 J 2 E |f g| ; now, J 1 J 2 is inner and E |f g| ∈ H r since log |f |T g |T | = log |f |T | + log |g |T | ∈ L 1 (T) and |f g| |T ∈ L r (T), whence f g ∈ H r .
Every real-valued harmonic function u on D has a harmonic conjugate, that is, a real-valued harmonic function v on D such that u + iv is holomorphic; this follows by simple connectedness of D from the fact that ∆u = 0 makes -∂ x 2 udx 1 + ∂ x 1 udx 2 an exact differential. The conjugate function is defined up to an additive constant, and we customarily normalize it so that v(0) = 0. When 1 ≤ p ≤ ∞ and real ψ ∈ L p (T), then u(z) = P [ψ](z) is harmonic on D and it is a theorem of Fatou that it has nontangential limit ψ a.e. on T. Also, it holds that u L p (Tρ) ≤ ψ L p (T) for 0 ≤ ρ < 1. Under the stronger assumption that 1 < p < ∞,

then v = P [ ψ]
where

ψ(e iθ ) := lim ε→0 1 2 π ε<|θ-t|<π ψ(e it ) tan( θ-t 2 ) dm(t) (26)
is called the conjugate function of ψ. It is a theorem of M. Riesz that the conjugation operator ψ → ψ is an isomorphism of L p (T) when 1 < p < ∞. Thus, we see that if ψ ∈ L p R (T) and 1 < p < ∞, then there exists g ∈ H p (namely g = u + iv) such that Re g = ψ on T [28, Ch. III]. Such a g is unique up to addition of a pure imaginary constant, and if we normalize it so that Im g(0) = 0, then g H p ≤ C ψ L p (T) with C = C(p). When ψ ∈ L 1 (T), the conjugate function ψ is still defined pointwise almost everywhere via (26) but it may no longer belong to L 1 (T).

For p ∈ (1, ∞), a non-negative function w ∈ L 1 (T) is said to satisfy Muck- enhoupt condition A p if {w} Ap := sup I 1 m(I) I w dm 1 m(I) I w -1/(p-1) dm p-1 < +∞, ( 27
)
where the supremum is taken over all arcs I ⊂ T. A theorem of Hunt, Muckenhoupt and Wheeden [28, Ch. VI, Thm 6.2]4 asserts that w satisfies condition A p if and only if there is C > 0 independent of φ for which

T | φ| p w dm ≤ C T |φ| p w dm, φ ∈ L 1 (T), (28)
and also that ( 28) is equivalent to

T |Mφ| p w dm ≤ C 1 T |φ| p w dm (29)
for C 1 > 0 and where Mφ is the Hardy-Littlewood maximal function of φ:

Mφ(ξ) = sup I∋ξ 1 m(I) I |φ| dm, ξ ∈ T, (30)
the supremum being taken over all subarcs of T that contain ξ. In (28), the assumption φ ∈ L 1 (T) is just a means to ensure that φ is well defined m-a.e. and the constants C, C 1 can be chosen to depend only on {w} Ap . Condition A 2 is fundamental to function theory on Lipschitz (and more generally chord-arc 5 ) domains, as was first pointed out in the seminal work [START_REF] Kenig | Weighted H p spaces on Lipschitz domains[END_REF], see also [START_REF] Jerison | Hardy spaces, A ∞ and singular integrals on chord-arc domains[END_REF][START_REF] Wolff | Counterexamples to two variants of the Helson-Szegö theorem[END_REF][START_REF] Zinsmeister | Courbes de Jordan vérifiant une condition corde-arc[END_REF]. Recall from the Riemann mapping theorem that to each simply connected domain Ω ⊂ C there is a conformal map ϕ from D onto Ω, which is unique if we impose for instance ϕ(0) ∈ Ω and arg ϕ ′ (0) ∈ [0, 2π). The precise normalization is unimportant in what follows.

Lemma 5.1. Let Ω ⊂ C be a bounded simply connected Lipschitz domain and ϕ : D → Ω a conformal map. Then ϕ extends homeomorphically from D onto Ω and preserves nontangential regions of approach in that, to every α, β > 1, there are α ′ , β ′ > 1 such that:

R Ω α (ϕ(ξ)) ⊂ ϕ R D α ′ (ξ) and ϕ R D β (ξ) ⊂ R Ω β ′ (ϕ(ξ)), ξ ∈ T. (31)
The derivative ϕ ′ as well as its reciprocal 1/ϕ ′ lie in H p for some p > 1, and it holds for any measurable E ⊂ ∂Ω that Λ(E) = ϕ -1 (E) |ϕ ′ |dm. Moreover, the weights |ϕ ′ |T | and 1/|ϕ ′ |T | satisfy condition A 2 . Proof. Since ∂Ω is a Jordan curve, ϕ extends to a homeomorphism from D onto Ω mapping T to ∂Ω by Carathéodory's theorem [START_REF] Ch | Boundary behaviour of conformal maps[END_REF]Thm 2.6]. To prove [START_REF] Grisvard | Elliptic problems in non-smooth domains[END_REF], we follow the argument (attributed to F. Gehring) outlined in [34, Prop. 1.1] for conformal maps from a half-plane onto unbounded chord-arc domains. Observe first that Ω is a fortiori chord-arc since it is Lipschitz, in particular Ω is a quasi-disk 6 [44, Prop. 7.7], and consequently ϕ extends to a quasi-conformal homeomorphism 7 of C [44, Thm 5.17]. Such a map is quasisymmetric [9, Def. 3.2.1, Thm3.5.3], meaning that there is an increasing

5 A Jordan domain Ω is chord-arc (or Lavrentiev) if Λ(J(ξ 1 , ξ 2 )) ≤ M |ξ 1 -ξ 2 | whenever ξ 1 , ξ 2 ∈ ∂Ω, where J(ξ 1 , ξ 2 )
is the smaller arc of ∂Ω between ξ 1 and ξ 2 and M is a constant. 6 Same definition as a chord-arc domain except that Λ gets replaced by "diameter". 7 An orientation-preserving homeomorphism

ϕ ∈ W 1,2 loc (C) is quasi-conformal if ∂ϕ/∂ϕ L ∞ (C) < 1, see [9, Def. 2.5.2, Thm 2.5.4]. homeomorphism η of [0, ∞) such that ϕ(z 0 ) -ϕ(z 1 ) ϕ(z 0 ) -ϕ(z 2 ) ≤ η z 0 -z 1 z 0 -z 2 , z 0 , z 1 , z 2 ∈ C. (32)
Now, fix β > 1, z 1 ∈ T and let z 0 range over R D β (z 1 ). If we choose z 2 ∈ T such that |ϕ(z 0 )ϕ(z 2 )| = d(ϕ(z 0 ), ∂Ω), then it follows from (32) that

|ϕ(z 0 ) -ϕ(z 1 )| d(ϕ(z 0 ), ∂Ω) ≤ η z 0 -z 1 z 0 -z 2 ≤ η |z 0 -z 1 | d(z 0 , T) ≤ η(β), (33)
hence ϕ(z 0 ) ∈ R Ω β ′ (ϕ(z 1 )) with β ′ = η(β). This proves the second inclusion in [START_REF] Grisvard | Elliptic problems in non-smooth domains[END_REF] and the first follows in the same manner, replacing ϕ by its inverse which is also quasi-conformal [

|} A 2 = {1/|ϕ ′ |T |} A 2 .
As Ω is chord-arc, it is in particular a Smirnov domain, meaning that ϕ ′ is outer [START_REF] Ch | Boundary behaviour of conformal maps[END_REF]Sec. 7.3,7.4]. Hence 1/ϕ ′ is also outer and since 1/|ϕ ′ |T | ∈ L 1+δ (T) for some δ > 0 because it satisfies A 2 , we find that in turn 1/ϕ ′ ∈ H p for some p > 1.

Smirnov classes of a Lipschitz plane domain

On an arbitrary simply connected domain Ω (whose boundary contains more than one point), there are at least two generalizations of the Hardy space H p of the disk. One which goes by the name of Hardy space, but is of no concern to us here, requires |f | p to have a harmonic majorant on Ω. The other, which is the one we are interested in, is the so-called Smirnov space, denoted as S p (Ω). It consists of functions f , holomorphic in Ω, for which there is a sequence of relatively compact Jordan domains ∆ n ⊂ Ω with rectifiable boundary such that each compact

K ⊂ Ω is contained in ∆ n for n ≥ n(K) and sup n∈N f L p (∂∆n) < ∞. (34)
By the maximum principle S ∞ (Ω) consists of bounded holomorphic functions on Ω. When 1 ≤ p < ∞ it is not immediately clear that S p (Ω) is a Banach space, but this is nevertheless true and there is in fact a fixed sequence ∆ n such that (34) holds for all f ∈ S p (Ω). Such a sequence can be taken to be ϕ(D ρn ) where ρ n → 1 -and ϕ is a conformal map from D onto Ω [26, Thm As soon as ∂Ω is rectifiable, so that ϕ ′ ∈ H 1 , the previous characterization together with Lemma 5.1 and the discussion in Section 5.1 imply that each f ∈ S p (Ω) has nontangential limits a.e. on ∂Ω with respect to arclength, and that the boundary function thus defined lies in L p (∂Ω). Moreover, this boundary function cannot vanish on a set of positive arclength unless f ≡ 0, and its norm in L p (∂Ω) coincides with f S p (Ω) , thereby identifying S p (Ω) with a closed subspace of L p (∂Ω). Again f is recovered from its boundary function by a Cauchy integral [START_REF] Duren | Theory of H p spaces[END_REF]Thm 10.4], but the (analog of the) Poisson representation may now fail. Our interest in Smirnov spaces is here limited to S 2 (Ω) for Ω a bounded simply connected Lipschitz domain. Theorem 5.1 below gives two alternative descriptions of this space.

We mention that the analog of point (i) for unbounded chord-arc domains is contained in [START_REF] Jerison | Hardy spaces, A ∞ and singular integrals on chord-arc domains[END_REF]Thm 2.2]. First, we need a lemma: Lemma 5.2. Let Ω ⊂ C be a bounded simply connected Lipschitz domain, and ϕ map D conformally onto Ω.

If f ∈ S 2 (Ω), then f • ϕ ∈ H 1 .
Proof. Set for simplicity F = f •ϕ. Since f ∈ S 2 (Ω), we have that F (ϕ ′ ) 1/2 ∈ H 2 , and we know from Lemma 5.1 that 1/ϕ ′ lies in H 1 . Therefore, by the Schwarz inequality and the monotonicity of ρ → g ρ L p (T) for holomorphic g, we get that

Tρ |F |dm 2 ≤ Tρ |F | 2 |ϕ ′ |dm Tρ |1/ϕ ′ |dm ≤ F (ϕ ′ ) 1/2 2 H 2 1/ϕ ′ H 1 .
Theorem 5.1. Let Ω ⊂ C be a bounded simply connected Lipschitz domain.

(i) For each α > 1, the space S 2 (Ω) coincides with holomorphic functions

f in Ω such that M α f ∈ L 2 (∂Ω) and f → M α f L 2 (∂Ω) is an equivalent norm on S 2 (Ω).
(ii) S 2 (Ω) is the closed subspace of W 1/2,2 (Ω) consisting of holomorphic functions, with equivalence of norms.

Proof. Let ϕ map D conformally onto Ω, put ψ for the inverse map, and pick α > 1. By [START_REF] Grisvard | Elliptic problems in non-smooth domains[END_REF], there is

β > 1 such that M α f ≤ (M β F ) • ψ for f ∈ S 2 (Ω) and F = f • ϕ. From Lemma
∂Ω (M α f ) 2 dΛ ≤ ∂Ω (M β F • ψ) 2 dΛ = T (M β F ) 2 |ϕ ′ |dm ≤ C 2 T (MF ) 2 |ϕ ′ |dm, (35)
where the change of variable is justified by Lemma 5.1. Now, as |ϕ ′ | satisfies condition A 2 , we get in view of (29) that

T (MF ) 2 |ϕ ′ | dm ≤ C 2 1 T |F | 2 |ϕ ′ | dm = C 2 1 f 2 S 2 (Ω) (36)
for some C 1 depending only on {|ϕ ′ |} A 2 . From ( 35) and ( 36), it follows that

M α f L 2 (∂Ω) ≤ C 2 f S 2 (Ω) (37)
with

C 2 = C 2 (Ω, α). Conversely, assume that f is holomorphic in Ω with M α f L 2 (∂Ω) < ∞.
Whenever δ ∈ (1, ∞) and z 0 ∈ Ω, it is a famous estimate for harmonic functions on Lipschitz domains (in any dimension) that

M δ (f -f (z 0 )) L 2 (∂Ω) ∼ d(., ∂Ω) 1/2 ∇f L 2 (Ω) ( 38
)
where the constants depend only on Ω, δ and z 0 [24, Thm 1, Cor. 1]. Assume first that f (ϕ(0)) = 0, in which case it follows from [START_REF] Kenig | The Neumann problem for elliptic equations with non-smooth coefficients[END_REF] that

M δ f L 2 (∂Ω) ≤ C M α f L 2 (∂Ω) < ∞, (39)
where C = C(α, δ, Ω). Pick δ = η(2), where η is as in [START_REF] Hrbacek | Introduction to set theory[END_REF]; note that indeed η(2) > 1, since η is strictly increasing and η(1) ≥ 1. Now, the argument in ( 33) can be reversed (set β = 2 there) so that, if

z 1 ∈ T, then R Ω δ (ϕ(z 1 )) ⊃ ϕ(R D 2 (z 1 )). Hence M 2 F ≤ (M δ f ) • ϕ, F = f • ϕ, and for ρ ∈ [0, 1) we get from the obvious inequality |F (ρe iθ )| ≤ M 2 F (e iθ ) that T |F ρ | 2 |ϕ ′ | dm ≤ T |M 2 F | 2 |ϕ ′ | dm ≤ T |(M δ f )•ϕ| 2 |ϕ ′ | dm = M δ f 2 L 2 (∂Ω) .
8

The proof given there on the half-plane carries over immediately to the disk.

In view of (39), the previous inequality shows that F ρ (ϕ ′ ) 1/2 H 2 is bounded independently of ρ, so there is a sequence ρ k → 1 -such that F ρ k (ϕ ′ ) 1/2 converges weakly in H 2 to some function G. Since F ρ k (z) = F (ρ k z) converges to F (z) locally uniformly in D, passing to the weak limit in the Cauchy formula yields G = F (ϕ ′ ) 1/2 . As the norm of the weak limit cannot exceed the lim inf of the norms, we deduce on using (39) that

f S 2 (Ω) = F (ϕ ′ ) 1/2 H 2 ≤ M δ f L 2 (∂Ω) ≤ C M α f L 2 (∂Ω) . (40)
Finally, if f (ϕ(0)) = 0, we apply [START_REF] Lanzani | Szegő and Bergman projections on nonsmooth planar domains[END_REF] to ψf which has the same S 2 (Ω)-norm as f and does vanish at ϕ(0)

(that ψf S 2 (Ω) = f S 2 (Ω) is clear from the relation (ψf ) • ϕ(z) = zf (ϕ(z))). Since M α ≤ M α f because |ψ| ≤ 1, we get that f S 2 (Ω) ≤ C M α f L 2 (∂Ω)
where C = C(α, Ω), thereby proving (i).

As for (ii), since holomorphic functions are harmonic, we know from [START_REF] Jerison | The inhomogeneous Dirichlet problem in Lipschitz domains[END_REF]Thm 4.1] that

f L 2 (D) + d(., ∂Ω) 1/2 ∇f L 2 (Ω) ∼ f W 1/2,2 (Ω) , f ∈ S 2 (Ω), ( 41
)
where the constants depend only on Ω. Pick z 0 ∈ Ω. As f (z 0 ) is the mean of f over some disk

D z 0 ,ρ 0 ⊂ Ω, we get that |f (z 0 )| ≤ C 3 f L 2 (Ω)
where

C 3 = C 3 (z 0 , Ω).
From this, together with ( 41) and ( 38), it follows that

M α f L 2 (∂Ω) ≤ |f (z 0 )| + M α (f -f (z 0 )) L 2 (∂Ω) (42)
≤ C 4 f L 2 (D) + d(., ∂Ω) 1/2 ∇f L 2 (Ω) ∼ f W 1/2,2 (Ω) .
Conversely, the Schwarz inequality implies that

f 2 L 2 (Ω) = D |(f • ϕ)| 2 |ϕ ′ | 2 dm 2 ≤ (f • ϕ)(ϕ ′ ) 1/2 2 L 4 (Ω) ϕ ′ 2 L 2 (Ω) ,
and since H 2 embeds in L 4 (D) (see discussion after [START_REF] Dahlberg | Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain[END_REF]) while (ϕ ′ ) 1/2 ∈ H 2 by Lemma 5.1, we get that f L 2 (Ω) ≤ C 5 f S 2 (Ω) where C 5 = C 5 (Ω, ϕ).

From this together with ( 41), [START_REF] Kenig | The Neumann problem for elliptic equations with non-smooth coefficients[END_REF], and the inequality |f

(z 0 )| ≤ C 3 f L 2 (Ω)
already mentioned, we obtain:

f W 1/2,2 (Ω) ∼ f L 2 (Ω) + d(., ∂Ω) 1/2 ∇f L 2 (Ω) (43)
≤ C 6 f S 2 (Ω) + M α f L 2 (∂Ω) + |f (z 0 )| ≤ C 7 f S 2 (Ω) + M α f L 2 (∂Ω) .
Now, point (ii) follows from ( 42), [START_REF] Necǎs | Les méthodes directes en théorie des équations elliptiques[END_REF] and point (i).

Smirnov spaces and Dirichlet problems for the Laplacian

Let t ∈ L ∞ (∂Ω) be the tangent vector field to ∂Ω written in complex form: (Ω) be a harmonic function. Then ∂H ∈ S 2 (Ω) and

t = τ x 1 + iτ x 2 Λ-a.
∂ τ H = 2Re (∂H t) . ( 44
)
In particular, tr

∂Ω H ∈ W 1,2 R (∂Ω). Proof. Since H is harmonic, ∂H is holomorphic, and ∂H ∈ W 1/2,2 (Ω) because H ∈ W 3/2,2 R
(Ω). Thus, ∂H ∈ S 2 (Ω) by Theorem 5.1 (ii). Also, by the Sobolev embedding theorem, H is continuous on Ω. Let ϕ map D conformally onto Ω. Lemma 5.1 implies that u := H • ϕ is harmonic on D and continuous on D. Moreover, the complex chain rule [4, Ch. 1, Sec. C] gives us, since ∂ϕ = 0, that

du = (∂H • ϕ) ϕ ′ dz + ( ∂H • ϕ) ϕ ′ dz = 2Re (∂H • ϕ) ϕ ′ dz , (45)
where we used that ∂H = ∂H. Now, to say that ∂H ∈ S 2 (Ω) is equivalent to say that (∂H • ϕ)(ϕ ′ ) 1/2 ∈ H 2 , and Lemma 5.1 implies that (ϕ ′ ) 1/2 ∈ H 2 , hence F := (∂H • ϕ)ϕ ′ ∈ H 1 . In particular, F ρ converges to F |T in L 1 (T) as ρ → 1 -and therefore, by integration, we get from (45) upon setting ϕ(e iθ j ) = ζ j ∈ ∂Ω, j = 1, 2, that

H(ζ 1 ) -H(ζ 2 ) = lim ρ→1 -(u(ρe iθ 1 ) -u(ρe iθ 2 )) = lim ρ→1 -2 θ 2 θ 1 Re F (ρe iθ )ie iθ ρdθ = 2 θ 2 θ 1 Re (∂H • ϕ)(e iθ )ϕ ′ (e iθ )ie iθ dθ. (46)
Since ϕ ′ (e iθ )ie iθ /|ϕ ′ (e iθ )| = t(ϕ(e iθ )) and dΛ = |ϕ ′ (e iθ )|dθ by Lemma 5.1, we may rewrite [START_REF] Schwartz | Théorie des distributions[END_REF] as

H(ζ 1 ) -H(ζ 2 ) = 2 [ζ 1 ,ζ 2 ]
Re (∂Ht) dΛ,

where [ζ 1 , ζ 2 ]
is the oriented arc from ζ 1 to ζ 2 on ∂Ω. This proves (44).

Remark 5.1. That tr ∂Ω H belongs to W 1,2 R (∂Ω) in Proposition 5.1 depends on the fact that H is harmonic, and is not a general property of W 3/2,2 (Ω)functions, see the discussion before [START_REF] Jerison | The inhomogeneous Dirichlet problem in Lipschitz domains[END_REF]Prop. 3.2] for a counterexample credited to G. David.

In view of Theorem 5.1, the next proposition stands analog in the planar case to a well-known result on the Dirichlet problem obtained in [52, Thm 5.1] for n ≥ 3. The proof we give here in the planar case is quite different, and uses conformal mapping and the M. Riesz theorem as global tools9 . Proposition 5.2. Let Ω ⊂ C be a bounded simply connected Lipschitz domain, and ψ ∈ L 2 R (∂Ω) be such that ∂Ω ψdΛ = 0. Then, there is a harmonic function

U ∈ W 3/2,2 R
(Ω) such that ∂ τ tr ∂Ω U = ψ. Such a function is unique up to an additive real constant and ∂U ∈ S 2 (Ω) with ∂U S 2 (Ω) ≤ C ψ L 2 (∂Ω) , where C depends only on Ω.

Proof. As U is continuous on Ω by the Sobolev embedding theorem, and tr ∂Ω U ∈ W 1,2 (∂Ω) by Proposition 5.1, uniqueness follows from the maximum principle for harmonic functions. Next, let ϕ map D conformally onto Ω and Υ be the inverse map. Define

h = (ψ • ϕ)|ϕ ′ | on T. By Lemma 5.1 ψ L 2 (∂Ω) = h(ϕ ′ ) -1/2 |T L 2 (T) and (ϕ ′ ) 1/2
|T ∈ L ℓ (T) for some ℓ > 2. Therefore h ∈ L p (T) for some p > 1, by Hölder's inequality. Moreover T hdm = ∂Ω ψdΛ = 0, hence by the M. Riesz theorem there is G ∈ H p such that G(0) = 0 and Re

G |T = h. Because ψ L 2 (∂Ω) = h(ϕ ′ ) -1/2 |T L 2 (T) and 1/|ϕ ′ | meets condition A 2 by Lemma 5.1, we get from (28) that G |T (ϕ ′ ) -1/2 |T L 2 (T) ≤ C ψ L 2 (∂Ω) with C = C(ϕ). Therefore, as G ∈ H p while (ϕ ′ ) -1/2 ∈ H 2 by Lemma 5.1, the product H = G(ϕ ′ ) -1/2 lies in H 2 and H H 2 ≤ C ψ L 2 (∂Ω)
. Since H(0) = 0 the function H 1 (z) = H(z)/(iz) in turn lies in H 2 with same norm as H, and consequently

F (ζ) := H 1 (Υ(ζ)) (ϕ ′ (Υ(ζ))) 1/2 = G(Υ(ζ)) iΥ(ζ)ϕ ′ (Υ(ζ)) ∈ S 2 (Ω) (47)
with F S 2 (Ω) ≤ C ψ L 2 (∂Ω) . In view of Lemma 5.1, ζ ∈ Ω converges nontangentially to ξ ∈ ∂Ω if, and only if z = Υ(ζ) ∈ D converges nontangentially to e iθ = Υ(ξ) ∈ T. Since ie iθ ϕ ′ (e iθ )/|ϕ ′ (e iθ )| = t(ϕ(e iθ )), with t the tangent vector field in complex form as defined before Proposition 5.1, we see from equation [START_REF] Sincich | Lipschitz stability for the Robin problem[END_REF] and the definition of

G that Re (F (ξ) t(ξ)) = ψ(ξ), Λ -a.e. ξ ∈ ∂Ω. (48)
Let U be harmonic and real-valued in Ω with ∂U = F/2. Clearly U exists, for F (z)dz + F (z)dz is a closed real-valued differential on the simply connected domain Ω. Moreover,

U ∈ W 3/2,2 R (Ω) because ∂U ∈ W 1/2,2
(Ω) by Theorem 5.1 (ii). Then, it follows from (48) and Proposition 5.1 that ∂ τ U = ψ.

Proof of Theorem 4.2

In Section 6.1, we state Theorem 6.1 which is instrumental for the proof of Theorem 4.2 but is also of independent interest. It is proved in Section 6.2, along with generalizations of results from Section 5.3 to more general conductivity equations, and a version of Rolle's theorem in W 1,2 (R). Finally, the proof of Theorem 4.2 is given in Section 6.3.

Factorization and regularity

Theorem 6.1. Assume that Ω ⊂ R 2 is a bounded Lipschitz domain and that σ satisfies (12)- [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF]. Let u ∈ W 1,2 R (Ω) be a solution to [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] which is such that ∂ n u ∈ L 2 R (∂Ω). Then:

(i) u ∈ W 3/2,2 R
(Ω) and ∂u = e Ψ Φ where Ψ ∈ W 1,r (Ω) and Φ ∈ S 2 (Ω). Moreover ∇u converges nontangentially to ∂ τ u τ + ∂ n u n on ∂Ω, and if u gets normalized so that u(z 0 ) = 0 for some z 0 ∈ Ω, there is a constant C depending only on Ω, z 0 , r, σ W 1,r (Ω) and c in (13) such that

u W 3/2,2 (Ω) ≤ C ∂ n u L 2 (∂Ω) . (49)
(ii) For each α > 1, it holds that

∂ τ u L 2 (∂Ω) ∼ ∂ n u L 2 (∂Ω) ∼ M α ∇u L 2 (∂Ω) , (50)
where constants depend only on Ω, r, σ W 1,r (Ω) , c in [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF], and also on α as to the second equivalence.

(iii) We have that u ∈ W 2,r R,loc (Ω) and that

j=1,2 d(., ∂Ω) 1/2 ∂ x j ∇u 2 L 2 (Ω) + u 2 W 1,2 (Ω) ∼ u 2 W 3/2,2 (Ω) , (51)
where constants depend only on Ω, r, σ W 1,r (Ω) , and c in [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF].

The proof of Theorem 4.2 dwells on the factorization ∂u = e Ψ Φ introduced in Theorem 6.1 (i) and on a generalized form of Rolle's theorem given in Proposition 6.1. Roughly speaking, the latter shows that if both u and ∂ n u vanish on a subset of positive measure of ∂Ω, then the full gradient ∇u = ∂ τ u τ + ∂ n u n also has to vanish on such a set. Consequently, Theorem 6.1 shows that the gradient vanishes everywhere in Ω, because ∂u factors through a holomorphic function of Smirnov class which cannot vanish on a subset of positive measure of ∂Ω if it is not identically zero.

The regularity results needed to put this approach to work are set forth in Theorem 6.1 points (i)-(ii). Point (ii) is known, even in higher dimension and with less regular σ, provided Ω is starlike [START_REF] Kenig | The Neumann problem for elliptic equations with non-smooth coefficients[END_REF]. Point (iii) is not used but mentioned for its own sake, as it generalizes to more general conductivities, in the case where n = 2, the equivalence between the two hand sides of ( 6) established for harmonic functions in [START_REF] Jerison | The inhomogeneous Dirichlet problem in Lipschitz domains[END_REF]Thm 4.1].

6.2. Proof of Theorem 6.1 6.2.1. The σ-harmonic conjugate function When Ω ⊂ R 2 is simply connected, we observe that ( 11) is a compatibility condition for the generalized Cauchy-Riemann system:

∂ x 1 v = -σ∂ x 2 u, ∂ x 2 v = σ∂ x 1 u, (52)
with unknown real-valued functions u, v. In fact, [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] is equivalent to the Schwarz rule 52), thus there is a distribution v to meet the latter whenever u ∈ W 1,2 R (Ω) satisfies [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] [46, Ch. II, Sec. 6, Thm VI]. From ( 13) and [START_REF] Verchota | Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains[END_REF]

∂ x 2 ∂ x 1 v = ∂ x 1 ∂ x 2 v in (
we get that |∇v| ∈ L 2 R (Ω), hence v ∈ W 1,2 R (Ω) and ∂ τ tr ∂Ω v exists in W -1/2,2 R
(∂Ω). The function v is a so-called σ-harmonic conjugate to u, and it is unique up to an additive constant by (4). Because (52) entails that ∇v is the rotation of σ∇u by π/2 on Ω, it may be surmised that

∂ n u = ∂ τ tr ∂Ω v/σ ∈ W -1/2,2 R
(∂Ω). This is indeed the case, as follows from the Green formula on Lipschitz domains [START_REF] Necǎs | Les méthodes directes en théorie des équations elliptiques[END_REF]Ch. 3,Thm 1.1]:

Ω (h ∂ x i g + g ∂ x i h)dm = ∂Ω g h n x i dΛ , g, h ∈ W 1,2 (Ω) , i = 1, 2 , ( 53
)
where we have put n = (n x 1 , n x 2 ) t . In fact, since τ = (-n x 2 , n x 1 ), it holds for ϕ ∈ D(R 2 ) that

∂Ω (∂ τ v)ϕ dΛ = - ∂Ω v∇ϕ.τ dΛ = ∂Ω v(n x 2 ∂ x 1 ϕ -n x 1 ∂ x 2 ϕ) dΛ
so that, by [START_REF] Wang | A remark on gradients of harmonic functions[END_REF],

∂Ω ∂ τ v ϕ dΛ = Ω (∂ x 2 v ∂ x 1 ϕ -∂ x 1 v ∂ x 2 ϕ) dm = Ω σ ∇u . ∇ ϕ dm. (54)
By density, we conclude on comparing ( 14) and ( 54) that ∂ n u = ∂ τ v/σ, as announced. It is easy to check that v satisfies [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] with σ replaced by 1/σ, so the previous discussion also yields that ∂ n v = -σ ∂ τ u on ∂Ω. In fact, the peculiarity of the planar case is that solving the Neumann problem in W 1,2 R (Ω) for the conductivity equation [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF], with normal derivative

g ∈ W -1/2,2 R (∂Ω), is tantamount to solve the Dirichlet problem in W 1,2
R (Ω) for a conductivity equation having conductivity 1/σ with tangential derivative σg ∈ W -1/2,2 R (∂Ω), and then compute the σ-harmonic conjugate. In particular uniqueness-up-to-a-constant of energy solutions implies that a solution u ∈ W 1,2 (Ω) to [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] meeting ∂ τ u = 0 is a constant. The functions v and f = u + iv, which lie respectively in W 1,2 R (Ω) and W 1,2 (Ω), will be instrumental to our analysis. For definiteness, we normalize v (initially defined up to an additive real constant) so that ∂Ω v dΛ = 0. A short computation (see [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF]Sec. 3.1]) shows that f satisfies on Ω the conjugate Beltrami equation:

∂f = ν∂f , ν = (1 -σ)/(1 + σ). (55)
Note that ν L ∞ (Ω) < 1 and that ν ∈ W 1,r R (Ω) because of ( 4), ( 12) and ( 13). Interior regularity estimates for [START_REF] Wolff | Counterexamples to two variants of the Helson-Szegö theorem[END_REF] imply that f ∈ W 2,r loc (Ω) [START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF]Cor. 3.3] (see also Section 6.2.2), hence also u, v ∈ W 2,r R,loc (Ω). In particular, by the Sobolev embedding theorem, ∇u, ∇v are locally Hölder continuous on Ω. Let {Ω k } be a sequence of open subsets of Ω with smooth boundary such that Ω k ⊂ Ω k+1 and ∪ n Ω k = Ω. Whenever u ∈ W 1,2 (Ω) satisfies [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] and g ∈ W -1/2,2 (∂Ω), it follows from ( 14) (and its analog on Ω k ) by means of the Schwarz inequality, and since ∇u L 2 (Ω\Ωn) → 0 as k → ∞, that

∂ n u = g ⇐⇒ lim k→∞ ∂Ω k σ∇u.n ψ dΛ = σg , tr ∂Ω ψ , ψ ∈ W 1,2 R (Ω), ( 56
)
where n denotes the unit normal on ∂Ω k , irrespective of k. Elaborating on this, we let n indicate the complex number n x + in y where (n x , n y ) t = n, and we observe upon making use of ( 52) that

σ ∂u n = σ ∂ x 1 u -i∂ x 2 u 2 n = (σ ∇u + i ∇v) . n 2 ,
where "." indicates the Euclidean scalar product. In view of ( 56) and its analog for v (remember v satisfies [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] with σ replaced by 1/σ), we obtain:

u ∈ W 1,2 (Ω) satisfies (11) with ∂ n u = g and ∂ τ u = -h ⇐⇒ 2 lim k→∞ ∂Ω k σ ∂u n ψ dΛ = σ (g + ih) , tr ∂Ω ψ , ψ ∈ W 1,2 (Ω) , (57)
where we complexified the space of test functions (i.e. from ψ ∈ W 1,2 R (Ω) to ψ ∈ W 1,2 (Ω)) upon extending the pairing , in a complex-linear manner.

Factorization of the complex derivative

The lemma below substantially reduces the study of solutions to [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF], when n = 2 and ( 12)-( 13) hold, to that of harmonic functions. Lemma 6.1. Let u ∈ W 1,2 R (Ω) satisfy (11) on a bounded simply connected Lipschitz domain Ω ⊂ R 2 , with σ subject to [START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF] and [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF]. Then, there exists a holomorphic function F ∈ L 2 (Ω), a number r 1 ∈ (2, r], a function Υ ∈ W 1,r 1 (Ω) with real-valued tr ∂Ω Υ whose norm is bounded solely in terms of Ω, σ W 1,r (Ω) , r, and ellipticity constants in [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF], such that ∂u = e Υ F . Moreover F = ∂H where H ∈ W 1,2 R (Ω) is harmonic in Ω and satisfies on ∂Ω:

∂

n H = e -Υ ∂ n u, ∂ τ H = e -Υ ∂ τ u. (58)
Proof. Let f = u + iv where v is the σ-harmonic conjugate to u. Since f ∈ W 2,r loc is a fortiori locally bounded and satisfies [START_REF] Wolff | Counterexamples to two variants of the Helson-Szegö theorem[END_REF] with ν ∈ W 1,r R (Ω) and ν L ∞ (Ω) < 1, a short computation as in the proof of [START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF]Cor. 3.3] or [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF]Lem. 5] shows that w := (1ν 2 ) 1/2 ∂f satisfies ∂w = (∂ν/(1ν 2 )) w.

As ∂ν/(1ν 2 ) ∈ L r (Ω) and r > 2, the Bers similarity principle for pseudoholomorphic functions (see e.g. [START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF]Prop. 3.2]) entails that there exist s ∈ W 1,r (Ω), whose norm10 is bounded in terms of r and ∂ν/(1ν 2 ) L r (Ω) only, and also a holomorphic function F 1 on Ω such that w = e s F 1 . Hence ∂f = e s 1 F 1 , where s 1 = slog(1ν 2 ) 1/2 belongs to W 1,r (Ω) by ( 12), ( 13) and ( 4). Now, it is straightforward to check using (52) that ∂f = (1 + σ)∂u. Therefore, if we set Υ 1 = s 1log(1 + σ) and appeal again to [START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF], ( 13) and ( 4), we get that

∂u = e Υ 1 F 1 , Υ 1 ∈ W 1,r (Ω), F 1 holomorphic in Ω , (59)
where we notice that Υ 1 W 1,r (Ω) is bounded in terms of Ω, r, the constants in ( 13) and σ W 1,r (Ω) . Factorization (59) is not yet what we need, for tr ∂Ω Υ 1 may not be real-valued. To remedy this, we will trade r for a possibly smaller exponent r 1 > 2. Specifically, it follows from [START_REF] Jerison | The inhomogeneous Dirichlet problem in Lipschitz domains[END_REF]Thm 5.1]11 that the Dirichlet problem for harmonic functions with boundary values in W θ,p (∂Ω) is solvable in W θ+1/p,p (Ω), as soon as 0 < θ < 1 and p ∈ [2, 2 + ε) where ε > 0 depends on the Lipschitz constant of ∂Ω. Since tr ∂Ω Υ 1 ∈ W 1-1/r,r (∂Ω) and the latter space increases as r decreases, there is r 1 ∈ (2, r] (depending on Ω and r) and a harmonic function h ∈ W 1,r 1 (Ω) such that tr ∂Ω h = tr ∂Ω ImΥ 1 . Let g be a harmonic conjugate to h, normalized so that g E = 0 for some E ⊂ Ω with m 2 (E) > 0. Since |∇h| = |∇g| pointwise by the Cauchy-Riemann equations, it follows from (4) that g lies in W 1,r 1 (Ω), and so do the holomorphic functions G := -g + ih and e G since exp is entire. Setting

Υ = Υ 1 -G and F = F 1 e G , (60)
we have that Υ ∈ W 1,r 1 (Ω) is real-valued on ∂Ω and that F is holomorphic, while ∂u = e Υ F , as desired.

The second of these has already been checked. By [START_REF] Zolesio | Multiplication dans les espaces de Besov[END_REF], the first reduces to 2Re(∂Ht) = ∂ τ H which holds good by [START_REF] Ch | Boundary behaviour of conformal maps[END_REF]. That u ∈ W 3/2,2 R

(Ω) follows from the relation ∂u = e Υ F , the membership F ∈ W 1/2,2 (Ω), and the fact that e Υ is a multiplier of W 1/2,2 (Ω) for it lies in W 1,r 1 (Ω) with r 1 > 2. Because Υ W 1,r 1 (Ω) depends only on Ω, σ W 1,r (Ω) , r, and ellipticity constant c in [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF], as asserted by Lemma 6.1, so does the norm of this multiplier. Combining this with Theorem 5.1 (ii), we obtain:

∇u W 1/2,2 (Ω) ≤ C ∂H W 1/2,2 (Ω) = C ∂G W 1/2,2 (Ω) ≤ C ′ ∂G S 2 (Ω) , (63)
where C ′ depends only on the above-mentioned parameters. Besides, we get from Proposition 5.2 (applied with U = G) and ( 58) that

∂G S 2 (Ω) ≤ C ′′ ∂ τ G L 2 (∂Ω) = C ′′ ∂ n H L 2 (∂Ω) ≤ C ′′ e Υ L ∞ (∂Ω) ∂ n u L 2 (∂Ω)
where C ′′ depends only on Ω. The latter estimate and (63) together yield

∇u W 1/2,2 (Ω) ≤ C 0 ∂ n u L 2 (∂Ω) , (64)
where C 0 depends on the same parameters as C ′ . Now, if we normalize u so that u(z 0 ) = 0 for some z 0 ∈ Ω and let ρ 0 > 0 be such that D(z 0 , ρ 0 ) ⊂ Ω, we deduce from ( 11), ( 13), the Green formula (which is valid since u ∈ W 2,r R,loc (Ω)) and Hölder's inequality that, for ρ ∈ (0, ρ 0 ),

d dρ 1 ρ T(z 0 ,ρ) u dm = 1 ρ T(z 0 ,ρ) ∂ n u dm ≤ 1 cρ T(z 0 ,ρ) σ∂ n u dm = 1 cρ D(z 0 ,ρ) ∇u.∇σ dm 2 ≤ π 1/4 cρ 1/2 ∇u L 4 (Ω) ∇σ L 2 (Ω) . (65)
Since lim ρ→0 T(z 0 ,ρ0) udm/ρ = u(z 0 ) = 0, we infer from (64), (65) and the Sobolev embedding theorem that, for ρ ∈ (0, ρ 0 ),

1 ρ T(z 0 ,ρ) u dm ≤ C 1 ρ 1/2 ∂ n u L 2 (∂Ω) (66)
where C 1 depends on Ω, σ W 1,r (Ω) , r, and c in [START_REF] Baratchart | Hardy spaces of the conjugate Beltrami equation[END_REF]. Integrating (66) yields

1 πρ 2 0 D(z 0 ,ρ 0 ) u dm 2 ≤ C 2 ρ 1/2 0 ∂ n u L 2 (∂Ω) (67)
where C 2 depends on the same parameters as C 1 . Then, (49) follows from ( 4), ( 64) and (67). Finally, remember from (60) the factorization ∂u = e Υ 1 F 1 where Υ 1 ∈ W 1,r (Ω) and

F 1 = F e -G with G ∈ W 1,r 1 (Ω), G holomorphic. As G ∈ L ∞ (Ω)
by the Sobolev embedding theorem, we see that F 1 ∈ S 2 (Ω) because F does, so we may set Ψ = Υ 1 and Φ = F 1 , thereby completing the proof of (i).

In view of ( 58), the factorization ∂u = e Υ F , and the boundedness of e Υ , the proof of ( 50) reduces to the case where u is harmonic (i.e. σ ≡ 1), and then it follows from Theorem 5.1 and Proposition 5.2 applied to u and its conjugate function. This shows (ii).

As to (iii), we already mentioned that u ∈ W 2,r R,loc (Ω) (this is now obvious anyway since ∂u = e Ψ Φ) and we need to prove [START_REF] Vekua | Generalized analytic functions[END_REF] which is equivalent to

j=1,2 d(., ∂Ω) 1/2 ∂ x j (e Υ F ) 2 L 2 (Ω) + e Υ F 2 L 2 (Ω) ∼ e Υ F 2 W 1/2,2 (Ω) . (68)
We already know from ( 6) that the right hand side of ( 68) is less than a constant (depending only on Ω) times the left hand side. To prove the reverse inequality, let ϕ conformally map D onto Ω, so that f :

= (F • ϕ)(ϕ ′ ) 1/2 ∈ H 2 , and recall that |f (z)| ≤ c f H 2 /(1 -|z|)
d(., ∂Ω) 1/2 (∂ x j Υ)e Υ F 2 L 2 (Ω) + j=1,2 d(., ∂Ω) 1/2 e Υ (∂ x j F ) 2 L 2 (Ω)
which is less than

e Υ 2 L ∞ (Ω) 2 c 2 ∇Υ 2 L 2 (Ω) F 2 S 2 (Ω) + j=1,2 d(., ∂Ω) 1/2 (∂ x j F ) 2 L 2 (Ω)
.

By [START_REF] Necǎs | Les méthodes directes en théorie des équations elliptiques[END_REF] and Theorem 5.1 (ii), this quantity is majorized by c

′ F 2 W 1/2,2 (Ω)
where c ′ = c ′ (Ω, r 1 , Υ W 1,r 1 (Ω) ), and since e Υ F W 1/2,2 (Ω) ∼ F W 1/2,2 (Ω) we are done with the proof.

w(t) = v(t), w ′ (t) = v ′ (t), ∀t ∈ R \ U.
We use Lemma 6.2 to prove the following generalization of Rolle's theorem.

Proposition 6.1. Let Ω ⊂ R 2 be a bounded Lipschitz domain and v ∈ W 1,2 (∂Ω). Assume that v = 0 at Λ-a.e. point of a set B ⊂ ∂Ω with Λ(B) > 0.

Then, there is

B ′ ⊂ B, with Λ(B ′ ) > 0, such that ∂ τ v = 0 at Λ-a.e. point of B ′ .
Proof. As Ω is bounded and Lipschitz, ∂Ω can be covered with open parallelepiped

Q 1 , • • • , Q N of the form Q j = R j (Q a,b )
, where R j is an affine isometry of R 2 and Q a,b = (-a, a) × (-b, b), with a, b > 0, in such a way that

R -1 j (Ω) ∩ Q a,b = {x ∈ Q a,b : x 2 > ψ j (x 1 )},
where ψ j is a Lipschitz function. Denote by P 1 the projection onto the first component in R 2 , and for

E ⊂ ∂Ω ∩ Q j set E j = P 1 (R -1 j (E)) ⊂ (-a, a) so that Λ(E) = E j |(1, ψ ′ j ) t |dm 1 .
Since Lipschitz changes of variables preserve Sobolev classes [56, Thm 2.2.2], it holds that v ∈ W 1,2 (∂Ω) if and only if v(R j (x 1 , ψ j (x 1 ))) belongs to W 1,2 ((-a, a)) for all j. Thus, as Λ(B ∩ Q j ) > 0 for at least one j, it is enough to prove the analog of the proposition on the real interval (-a, a) instead of ∂Ω. By the extension theorem we may assume that v is defined over the whole real line. Then, taking ε small enough in Lemma 6.2, we conclude that it is enough to prove Proposition 6.1 when v has continuous derivative. Assume it is the case and let A be the set of accumulation points of B. Note that m 1 (A) = m 1 (B) > 0 because B \ A is countable [START_REF] Hrbacek | Introduction to set theory[END_REF]. Moreover, to each t ∈ A, there exists a non-stationary sequence (t n ) ⊂ B, n ∈ N, such that t n → t. Without loss of generality, we may assume that (t n ) is monotone, say t n < t n+1 (the case t n > t n+1 is similar) for all n ∈ N. Since v(t n ) = v(t n+1 ) = 0, there is s n ∈ [t n , t n+1 ] such that v ′ (s n ) = 0 by Rolle's theorem. Since s n → t and v ′ in continuous, we get that v ′ (t) = 0, as desired.

6.3. Proof of Theorem 4.2 Proof. From Theorem 6.1 (ii), we get that u |∂Ω lies in W 1,2 (∂Ω), hence Proposition 6.1 implies that both ∂ n u and ∂ τ u vanish on some E ⊂ γ with Λ(E) > 0. By Theorem 6.1 (i), we now see that (e Ψ Φ) |∂Ω vanishes a.e. on E, and since e -Ψ is bounded, by the Sobolev embedding theorem, we must have that F |∂Ω = 0 a.e. on E. As F belongs to the Smirnov class S 2 (Ω), we deduce that F ≡ 0, hence ∂u ≡ 0 and thus ∇u ≡ 0 since u is real. Therefore u is a constant, and in fact u ≡ 0.

The anisotropic case

We consider in this section a conductivity equation of the form [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] where σ is valued in the set of real symmetric matrices and the ellipticity condition ( 13) is replaced by c I n ≤ σ ≤ c -1 I n for some constant c ∈ (0, +∞), (

with I n the identity matrix of order n. Isotropic equations correspond to the case where the image of σ consists of scalar matrices; otherwise the conduction is said to be anisotropic. Existence and uniqueness of solutions to the Neumann problem and the forward Robin problem proceed as before provided that the normal derivative gets replaced by n.σ∇u. Questions about uniqueness of the Robin coefficient for the inverse problem may be raised as in Section 3, namely: given 0 ≡ g ∈ L 2 (Γ 0 ) and u the solution to the forward Robin problem with Robin coefficient λ ∈ L ∞ + (Γ), subject to the boundary condition n.σ∇u = g, does the knowledge of u |Γ 0 determine λ uniquely? Of course when n ≥ 3, uniqueness cannot prevail in general as we saw it may not even hold for the ordinary Laplacian. But if n = 2 uniqueness does hold: this follows from Theorem 4.2 and the fact that an anisotropic equation on a bounded Lipschitz domain with W 1,r coefficients, r > 2, is the diffeomorphic image of an isotropic one (on another Lipschitz domain). More precisely, if Θ is a diffeomorphism of R 2 of class C 1 and if we set Ω 1 = Θ(Ω), a computation shows (see e.g. [START_REF] Astala | Calderòn's inverse problem for anisotropic conductivity in the plane[END_REF]) that u solves for [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] (Ω) and satisfies (69), we can extend it into a symmetric matrix-valued function with entries in W 1,r loc (R 2 ) meeting (69) and equal to I 2 outside of a compact set; this only requires the extension theorem, continuity of W 1,r -functions when r > 2, and a smooth partition of unity. Denoting this extension by σ again, define the complex function µ 1 = (-σ 11 + σ 2,2 -2iσ 12 )/(σ 11 + σ 22 + 2 |σ|). As µ 1 is compactly supported and |µ 1 | < C < 1, the solution Θ to the Beltrami equation ∂Θ = µ 1 ∂Θ which is z + O(1/z) at infinity is a homeomorphism of C of class W 2,r loc (a fortiori it is C 1 -smooth) and σ given by (70) satisfies σ = |σ • Θ -1 | 1/2 [START_REF] Sun | Anisotropic inverse problems in two dimensions[END_REF], see also [START_REF] Sylvester | An anisotropic inverse boundary value problem[END_REF] where this technique was initiated for smoother coefficients and the nice exposition in [START_REF] Astala | Calderòn's inverse problem for anisotropic conductivity in the plane[END_REF] which deals with bounded coefficients (but less smooth Θ).

Because Ω 1 = Θ(Ω) is Lipschitz and the scalar-valued function σ satisfies [START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF] and (13) (with n = 2), we can apply Theorem 4.2 to v on Ω 1 . Thus, we deduce from the relations between u and v that if u ≡ 0 then u and n.σ∇u cannot vanish together on a subset of positive measure of ∂Ω. The proof of Theorem 4.1 can now be repeated to give us: Corollary 7.1. Theorem 4.1 still holds in the anisotropic case when σ is real symmetric 2 × 2-valued with entries in W 1,r (Ω) and meets (69), r > 2, provided the normal derivative ∂ n u in (20) gets replaced by n.σ∇u.

Concluding remarks

In the notation of ( 19), stable determination with respect to u |Γ 0 , of a smooth Robin coefficient λ has been studied in [START_REF] Alessandrini | Stable determination of corrosion by a single electrostatic boundary measurement[END_REF][START_REF] Chaabane | Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems[END_REF][START_REF] Chaabane | Identification of Robin coefficients by the means of boundary measurements[END_REF][START_REF] Cheng | Stable determination of a boundary coefficient in an elliptic equation[END_REF][START_REF] Sincich | Lipschitz stability for the Robin problem[END_REF] for the Laplace equation. When n = 2, the factorization of ∂u given in Theorem 6.1 may help dealing with this issue for more general conductivities and less smooth λ.

We further mention that stability of the Cauchy problem in dimension 2, for general anisotropic conductivity equations with bounded conductivity, has been extensively studied in [START_REF] Alessandrini | The stability for the Cauchy problem for elliptic equations[END_REF] using tools from complex analysis. In this connection, we point out that the factorization technique of Lemma 6.1 enjoys some generalization to the anisotropic case which rests on the method of isothermal coordinates that we recalled in Section 7. This suggests a research path worth exploring when dealing with stability for Sobolevsmooth conductivities. It is also natural to ask whether results from the present paper remain valid when σ is merely bounded. At present, the derivation of a factorization for ∂u requires some smoothness and it is not even clear if it holds for σ ∈ L ∞ ∩W 1,r when r < 2. The case where σ ∈ W 1,2 (Ω) deserves special mentioning: although the equation may no longer be strictly elliptic and solutions need not even be locally bounded, it is possible to make sense out of the Dirichlet problem for L p data and the factorization ∂u = e Ψ Φ still holds in slightly modified form where Ψ is pure imaginary on ∂Ω [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF]. Therefore we expect some of our theorems to remain valid, at least if Ω is smooth. Yet another generalization concerns with complex-valued σ, which arise in impedance tomography [START_REF] Cheney | Electrical impedance tomography[END_REF]. In this case (52) becomes a system of complex equations, and the factorization of ∂u has apparently not been investigated. Since the negative result of [START_REF] Bourgain | A remark on gradients of harmonic functions in dimension ≥ 3[END_REF], weaker unique continuation issues have been raised in dimension 3 and higher. One of them is: does a harmonic function in Ω, the trace of which vanishes on a non-empty open subset O of ∂Ω and whose normal derivative vanishes on a subset of positive measure in O, have to vanish identically? This question is still open in general, and we refer the reader to [START_REF] Adolfsson | C 1,α domains and unique continuation at the boundary[END_REF][START_REF] Kenig | A note on boundary unique continuation for harmonic functions in non-smooth domains[END_REF] for advances on the subject. In the setting of Robin inverse problems, the issue raised in Remark 4.1 as to whether ∂ n u/u can remain non-negative and bounded in a neighborhood of a set of positive measure where u, ∂ n u both vanish, seems to be more relevant and deserves further study. Finally, we did not touch upon multiply connected domains Ω, where similar uniqueness properties can be proved.

  in W 1,2 (Ω) if and only if v = u • Θ -1 solves for ∇ • (σ ∇v) = 0 in W 1,2 (Ω 1 ), where σ(Θ(z)) = 1 |DΘ(z)| DΘ(z)σ(z)DΘ t (z)(70)and |DΘ| indicates the determinant of the Jacobian matrix DΘ. Moreover, using a subscript 1 for the unit tangent and normal vectors to Ω 1 , it holds by construction that∂ τ 1 v • Θ = ∂ τ u/|DΘτ |,and from the weak formulation of the Neumann problem we get that (n 1 .σ∇v) • Θ = n.σ∇u/|DΘτ |. Now, since σ = (σ ij ) has entries in W 1,r

  6.2]. An open set Ω ⊂ R n is called Lipschitz if, in a neighborhood of each boundary point, it is isometric to the epigraph of a Lipschitz function [31, Def. 1.2.1.1].

  28, Ch. VI, Lem. 3.2], hence it follows from (6) that H 2 ⊂ W 1/2,2 (D) and subsequently, by the Sobolev embedding theorem, thatf L 4 (D) ≤ C f H 2 .In fact, since both sides of (6) are equivalent quantities when f is harmonic (see discussion after (6)), H 2 is precisely the space of holomorphic functions in W 1/2,2 (D) with equivalence of norms. A nonzero f ∈ H p is such that log |f |T | ∈ L 1 (T)[START_REF] Duren | Theory of H p spaces[END_REF] Thm 2.2], in particular, a nonzero H

p -function cannot vanish on a subset of T of strictly positive measure

3

. Conversely, if h ∈ L p (T) is non-negative and log h ∈ L 1 (T), then

  9, Thm 3.7.7]. Next, since ∂Ω is rectifiable, ϕ ′ lies in H 1 and Λ(E) = ϕ -1 (E) |ϕ ′ |dm for every measurable E ⊂ ∂Ω [44, Thm 6.8]. The fact that |ϕ ′ |T | meets A 2 is a consequence of [55, Prop. 15] (which deals more generally with local chord arc graphs), see also [40, Sec. 2] and the references therein or [29, Ch. VII, Thm 4.2] for a proof when Ω is star-shaped. The fact that |ϕ ′ |T | satisfies condition A 2 implies that it belongs to L 1+δ (T) for some δ > 0 [28, Ch. VI, Cor. 6.10], hence it holds in fact that ϕ ′ ∈ H p for some p > 1. Clearly {|ϕ ′ |T

  10.1]. Consequently f belongs to S p (Ω) if and only if (f • ϕ)(ϕ ′ ) 1/p belongs to H p , and (f • ϕ)(ϕ ′ ) 1/p p will serve as a norm on S p (Ω) [26, Ch. 10, Sec. 1, Cor. to Thm 10.1].

  5.2 we get that F ∈ H 1 , hence it is the Poisson integral of F |T . It is known, however, that M β F ≤ CMF |T pointwise on T for some constant C depending only on β [28, Ch. I, Thm 4.2] 8 . Consequently,

  e. in ∂Ω.

	Proposition 5.1. Let Ω ⊂ C be a bounded simply connected Lipschitz do-3/2,2 main and H ∈ W R

  S 2 (Ω) . Now, by the Leibniz rule and the triangle inequality, the first summand in the left hand side of (68) is bounded above by

	j=1,2

1/2 for z ∈ D and some absolute constant c, by a classical inequality of Hardy and Littlewood [26, Thm 5.9]. Since d(ϕ(z), ∂Ω) ≤ (1 -|z| 2 )|ϕ ′ (z)| by standard properties of conformal maps [44, Ch.1, Cor. 1.4], we get that d(., ∂Ω) 1/2 F L ∞ (Ω) ≤ √ 2c F

  6.2.4. A generalized Rolle's theoremWe recall below the 1-dimensional version of a Lusin-type theorem for Sobolev functions, to be found in[START_REF] Ziemer | Weakly differentiable functions[END_REF] Thm 3.10.5]. More precisely, we state the case n = 1, ℓ = k = 1 and p = 2 of the result just quoted. The latter is in terms of Bessel capacities that we did not introduce, but we use here that the Bessel capacity B 0,2 is just Lebesgue measure, see [56, Def. 2.6.2]. Lemma 6.2. [56, Thm 3.10.5] Let v ∈ W 1,2 (R) and ε > 0. There exists an open set U ⊂ R and a function w ∈ C 1 (R) such that m 1 (U) < ε and

The proof given there for bounded C 1 -smooth Ω carries over to the Lipschitz case.

Appealing to[START_REF] Lions | Sur les espaces d'interpolation[END_REF] Ch. II, Thm 1.1] instead of [42, Ch. I, Thm 6.2], the same reasoning shows that ∂ τ is continuous W 1/q,p (∂Ω) → W -1/p,p (∂Ω) for 1 < p < ∞, 1/p + 1/q = 1.

More generally, it is a theorem of Privalov that no nonzero meromorphic function on D has nontangential limit zero on a set of strictly positive measure on T, see[START_REF] Ch | Boundary behaviour of conformal maps[END_REF] Sec. 6.1].

The proof given there on the half-plane easily carries over to the disk.

The restriction to n ≥ 3 in[START_REF] Verchota | Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains[END_REF] may be due to the fact that it dwells on the method of layer potentials, where the discrepancy between Riesz and logarithmic potentials makes it cumbersome to treat both in a single stroke.

Reference[START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF] deals with Dini-smooth Ω but this assumption is not used in the proof of equations (18), (19) loc. cit. The similarity principle is called representation of pseudoanalytic functions of the first kind in [51, Ch. III, Sec. 4], and later appeared in many works.

[START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] The result is stated there for n ≥ 3 only, which may be confusing, but the proof is valid for n = 2 as well. In fact, all we need is [35, Thm 5.15, (a),(b)] for Besov spaces, along with interpolation arguments on top of[35, p. 200]. Since that part of the proof of[START_REF] Jerison | The inhomogeneous Dirichlet problem in Lipschitz domains[END_REF] Thm 5.15] depends only on[START_REF] Dahlberg | Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain[END_REF], complex interpolation and multiplier theory for singular integral operators, the restriction n ≥ 3 is easily seen to be superfluous.

Acknowledgments

The authors thank the reviewer for his constructive remarks.

[1] R. Adams. Sobolev spaces. Academic Press, 1975.

Because u ∈ W 1,2 R (Ω) by assumption and Υ ∈ L ∞ (Ω) by the Sobolev embedding theorem, we get that F ∈ L 2 (Ω). Being holomorphic, F can be written as ∂H for some real-valued harmonic H, and necessarily H ∈ W 1,2 R (Ω) for its complex derivatives ∂H and ∂H = ∂H are in L 2 (Ω). Being harmonic, H satisfies [START_REF] Baratchart | Pseudo-holomorphic functions at the critical exponent[END_REF] with σ replaced by 1, so we get the following analog to [START_REF] Zinsmeister | Courbes de Jordan vérifiant une condition corde-arc[END_REF]:

Substituting ∂u = e Υ ∂H in [START_REF] Zinsmeister | Courbes de Jordan vérifiant une condition corde-arc[END_REF] and reckoning that ψ

Taking into account in ( 62) that e Υ is real-valued on ∂Ω yields (58).

6.2.3. Proof of Theorem 6.1 Proof. Write ∂u = e Υ F as in Lemma 6.1, and let H ∈ W 1,2 R (Ω) be a harmonic function such that F = ∂H. Since e -Υ is bounded, being continuous on Ω by the Sobolev embedding theorem, we deduce from (58) that ∂ n H lies in L 2 (∂Ω). Set G to be a harmonic conjugate to H. By the discussion after (54) ( with σ ≡ 1 throughout), we get that G is, up to an additive constant, the unique harmonic function in W 1,2 R (Ω) such that ∂ τ G = ∂ n H. From Proposition 5.2, we now see that ∂G ∈ S 2 (Ω) with Re(∂G t) = e -Υ ∂ n u/2 on ∂Ω, where t is the tangent vector field to ∂Ω written in complex form. By the Cauchy-Riemann equations ∂H = i∂G, so that in turn F ∈ S 2 (Ω) and Im(

(Ω) for ∂H = F lies in W 1/2,2 (Ω) by Theorem 5.1 (ii). Let n = t/i be the normal vector field on ∂Ω, written in complex form. By definition of complex derivatives (cf. (3)), the nontangential convergence of ∇u to ∂ n u n + ∂ τ u τ is equivalent to the nontangential convergence of ∂u = e Υ F to ∂ n u n/2 + ∂ τ u t/2. Considering the existence of nontangential limits a.e. for Smirnov functions and the continuity of Υ, this is in turn equivalent to 2F = e -Υ (∂ n u n + ∂ τ u t), that is 2F t = e -Υ (i∂ n u + ∂ τ u) on ∂Ω. Taking real and imaginary parts, we are thus left to verify two real equations: 2Re(F t) = e -Υ ∂ τ u and 2Im(F t) = e -Υ ∂ n u.