Asymptotic expansions and unique continuation at Dirichlet-Neumann boundary junctions for planar elliptic equations

We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.


Introduction
The present paper deals with elliptic equations in planar domains with mixed boundary conditions and aims at proving asymptotic expansions and unique continuation properties for solutions near boundary points where a transition from Dirichlet to Neumann boundary conditions occurs.
A great attention has been devoted to the problem of unique continuation for solutions to partial differential equations starting from the paper by Carleman [5], whose approach was based on some weighted a priori inequalities. An alternative approach to unique continuation was developed by Garofalo and Lin [14] for elliptic equations in divergence form with variable coefficients, via local doubling properties and Almgren monotonicity formula. The latter approach has the advantage of giving not only unique continuation but also precise asymptotics of solutions near a fixed point, via a suitable combination of monotonicity methods with blow-up analysis, as done in [9][10][11][12][13]. The method based on doubling properties and Almgren monotonicity formula has also been successfully applied to treat the problem of unique continuation from the boundary in [1,2,9,25] under homogeneous Dirichlet conditions and in [24] under homogeneous Neumann conditions. Furthermore, in [9] a sharp asymptotic description of the behaviour of solutions at conical boundary points was given through a fine blow-up analysis. In the present paper, we extend the procedure developed in [9,[11][12][13] to the case of mixed Dirichlet/Neumann boundary conditions, providing sharp asymptotic estimates for solutions near the Dirichlet-Neumann junction and, as a consequence, unique continuation properties. In addition, comparing our result with the aforementioned papers, here we also provide an estimate of the remainder term in the difference between the solution and its asymptotic profile.
Let Ω be an open subset of R 2 with Lipschitz boundary. Let Γ n ⊂ ∂Ω and Γ d ⊂ ∂Ω be two nonconstant curves (open in ∂Ω) such that Γ n ∩ Γ d = {P } for some P ∈ ∂Ω. We are interested in regularity of weak solutions u ∈ H 1 (Ω) to the mixed boundary value problem with f ∈ L ∞ (Ω) and g ∈ C 1 (Γ n ), see Section 2 for the weak formulation. Our aim is to prove unique continuation properties from the Dirichlet-Neumann junction {P } = Γ n ∩ Γ d and sharp asymptotics of nontrivial solutions near P provided ∂Ω is of class C 2,γ in a neighborhood of P . We mention that some regularity results for solutions to second-order elliptic problems with mixed Dirichlet-Neumann type boundary conditions were obtained in [16,23], see also the references therein. Some interest in the derivation of asymptotic expansions for solutions to planar mixed boundary value problems at Dirichlet-Neumann junctions arises in the study of crack problems, see e.g. [6,18]. Indeed, if we consider an elliptic equation in a planar domain with a crack and prescribe Neumann conditions on the crack and Dirichlet conditions on the rest of the boundary, in the case of the crack end-point belonging to the boundary of the domain we are lead to consider a problem of the type described above in a neighborhood of the crack's tip (which corresponds to the Dirichlet-Neumann junction). We recall (see e.g. [6]) that, in crack problems, the coefficients of the asymptotic expansion of solutions near the crack's tip are related to the so called stress intensity factor.
In order to get a precise asymptotic expansion of u at point P ∈ Γ n ∩ Γ d , we will need to assume that ∂Ω is of class C 2,δ near P . The asymptotic profile of the solution will be given by the function (1.2) F k (r cos θ, r sin θ) = r 2k−1 2 cos 2k − 1 2 θ , r > 0, θ ∈ (0, π), for some k ∈ N \ {0}. We note that F k ∈ H 1 loc (R 2 ) and solves the equation      ∆F k = 0, in R 2 + , F k (x 1 , 0) = 0, for x 1 < 0, ∂ x2 F k (x 1 , 0) = 0, for x 1 > 0, (1.3) where here and in the following R 2 + := {(x 1 , x 2 ) ∈ R 2 : x 2 > 0}. The main result of the present paper provides an evaluation of the behavior of weak solutions u ∈ H 1 (Ω) to (1.1) at the boundary point where the boundary conditions change. In order to simplify the statement and without losing generality, we can fix the cartesian axes in such a way that the following assumptions on Ω ⊂ R 2 are satisfied. Here and in the remaining of this paper, Γ n , Γ d ⊂ ∂Ω are nonconstant curves (open as subsets of ∂Ω) such that Γ n ∩ Γ d = {0} with 0 ∈ ∂Ω.
We are now in position to state the main result of the present paper.
Remark 1.1. Here and in the sequel, we identify R 2 with the complex plane C; hence, by a conformal map on an open set U ⊂ R 2 we mean a holomorphic function with complex derivative everywhere non-zero on U . We notice that, if Ω satisfies (i)-(ii) and ϕ : Ω ∩ B r0 → R 2 + is conformal, then Dϕ(0) = α Id and ϕ ′ (0) = α for some real α > 0, where Dϕ denotes the jacobian matrix of ϕ and ϕ ′ denotes the complex derivative of ϕ.
As a direct consequence of Theorem 1.1, we derive the following Hopf-type lemma.
Corollary 1.2. Under the same assumptions as in Theorem 1.1, let u ∈ H 1 (Ω) be a non-trivial weak solution to (1.1), with u 0. Then (i) for every t ∈ [0, π), lim r→0 u(r cos t, r sin t) A further relevant byproduct of our asymptotic analysis is the following unique continuation principle, whose proof follows directly from Theorem 1.1. Corollary 1.3. Under the same assumptions as in Theorem 1.1, let u ∈ H 1 (Ω) be a weak solution to (1.1) such that u(x) = O(|x| n ) as x ∈ Ω, |x| → 0, for any n ∈ N. Then u ≡ 0.
We observe that Theorem 1.1 provides a sharp asymptotic expansion (and consequently a unique continuation principle) at the boundary for 1 2 -fractional elliptic equations in dimension 1. Indeed, if v ∈ H 1/2 (R) weakly solves see [4]. Theorem 1.1 and Corollary 1.3 apply to (1.6). Hence, V (and in particular its restriction v) satisfies expansion (1.4) and a strong unique continuation principle from 0 (i.e. from a boundary point of the domain of v). We mention that unique continuation principles from interior points for fractional elliptic equations were established in [8].
We do not know if the C 2,δ regularity on Ω and C 1 regularity of the boundary potential g in Theorem 1.1 can be weakened in order to obtain a unique continuation property. On the other hand, we can conclude that a regularity assumption on the boundary is crucial for excluding the presence of logarithms in the asymptotic expansion at the junction. Indeed, in Section 8 we produce an example of a harmonic function on a domain with a C 1 -boundary which is not of class C 2,δ , satisfying null Dirichlet boundary conditions on a portion of the boundary and null Neumann boundary conditions on the other portion, but exhibiting dominant logarithmic terms in its asymptotic expansion.
The proof of Theorem 1.1 combines the use of an Almgren type monotonicity formula, blow-up analysis and sharp regularity estimates. Indeed regularity estimates yield the expansion of u near zero as follows: . Now, if u is nontrivial, a blow-up analysis combined with Almgren type monotonicity formula allows to depict a k 0 1 for which a k0 (r) → β = 0 and a k (r) → 0 for every k < k 0 as r → 0. The proof of (1.7) uses also a blow-up analysis argument inspired by Serra [22], see also [20,21]. The paper is organized as follows. In Section 2 we introduce an auxiliary equivalent problem obtained by a conformal diffeomorphic deformation straightening B 1 ∩∂Ω near 0 and state Theorem 2.1 giving the sharp asymptotic behaviour of its solutions. Section 3 contains some Hardy-Poincaré type inequalities for H 1 -functions vanishing on a portion of the boundary of half-balls. In Section 4 we develop an Almgren type monotonicity formula for the auxiliary problem which yields good energy estimates for rescaled solutions thus allowing the fine blow-up analysis performed in Section 5 and hence the proof of Theorem 2.1. Section 7 contains the proof of the main Theorem 1.1, which is based on Theorem 2.1 and on some regularity and approximation results established in Section 6. Finally, Section 8 is devoted to the construction of an example of a solution with logarithmic dominant term in a domain violating the C 2,δ -regularity assumption.

The auxiliary problem
Since ∂Ω is of class C 2,δ near zero, we can find r 0 > 0 such that Γ := ∂Ω ∩ B r0 is a C 2,δ curve. Here and in the following, we let B be a C 2,δ simply connected open bounded set such that B ⊂ Ω and ∂B ∩ ∂Ω = Γ. For some functions We introduce the space H 1 0,Γ d (B) as the closure in H 1 (B) of the subspace Since B is of class C 2,δ , in view of the Riemann mapping Theorem and [17,Theorem 5.2.4], there exists a conformal mapφ : B → B 1 which is of class C 2 . Let N =φ(0) ∈ ∂B 1 and let S be its antipodal. We then consider the map ϕ : R 2 \ {S} → R 2 \ {S} given by ϕ(z) := 2 z−S |z−S| 2 + S, where, for every z ∈ R 2 ≃ C, z denotes the complex conjugate of z. This map is conformal and ϕ(N ) = 0. In addition ϕ(B 1 \ {S}) ⊂ P where P is the half plane not containing S whose boundary is the line passing through the origin orthogonal to S.
Then the map ϕ •φ is a conformal map which is of class C 2 from a neighborhood of the origin B ∩ B r into P for some r > 0. It is now clear that there exists a rotation R and a real number Moreover ϕ(0) = 0. Since ϕ is a conformal diffeomorphism, in view of Remark 1.1 we have that, under the assumptions of Theorem 1.1, being ϕ ′ (0) the complex derivative of ϕ at 0, which turns out to be real because of the assumption that (1, 0) is tangent to ∂Ω at 0 and strictly positive because of the assumption that the exterior unit normal vector to ∂Ω at 0 is (0, −1). In addition, (2.3) implies that, if R is chosen sufficiently small, ϕ −1 ((−R, 0) × {0}) ⊂ Γ d and ϕ −1 ((0, R) × {0}) ⊂ Γ n . Therefore letting w = u • ϕ −1 : B + R → R and Ψ := ϕ −1 , we then have that w ∈ H 1 (B + R ) solves It is plain that p ∈ L ∞ (B + R ) and q ∈ C 1 ([0, R)). Here and in the following, for every r > 0, we define (2.5) Γ r n := (0, r) × {0} and Γ r d := (−r, 0) × {0}. The following theorem describes the behaviour of w at 0 in terms of the limit of the Almgren quotient associated to w, which is defined as In Section 4 we will prove that N is well defined in the interval (0, R 0 ) for some R 0 > 0.
Theorem 2.1. Let w be a nontrivial solution to (2.4). Then there exists k 0 ∈ N, k 0 1, such that Arg z as τ → 0 + strongly in H 1 (B + r ) for all r > 0 and in C 0,µ loc (R 2 + \ {0}) for every µ ∈ (0, 1), where β = 0 and p(t cos s, t sin s)w(t cos s, t sin s) dt cos 2k0−1 2 s ds In particular The proof of Theorem 2.1 is based on the study of the monotonicity properties of the Almgren function N and on a fine blow-up analysis which will be performed in Sections 4 and 5.

Hardy-Poincaré type inequalities
In the description of the asymptotic behavior at the Dirichlet-Neumann junction of solutions to equation (2.4) a crucial role is played by eigenvalues and eigenfunctions of the angular component of the principal part of the operator.
Let us consider the eigenvalue problem It is easy to verify that (3.1) admits the sequence of (all simple) eigenvalues with corresponding eigenfunctions ψ k (t) = cos 2k−1 2 t , k ∈ N, k 1. It is well known that the normalized eigenfunctions  form an orthonormal basis of the space L 2 (0, π). Furthermore, the first eigenvalue λ 1 = 1 4 can be characterized as For every r > 0, we let (recall (2.5) for the definition of Γ r d ) As a consequence of (3.3) we obtain the following Hardy-Poincaré inequality in H r . Lemma 3.1. For every r > 0 and w ∈ H r , we have that We conclude by density, recalling that the space of smooth functions vanishing on [−r, 0] × {0} is dense in H r , see e.g. [7].
Lemma 3.2. For every r > 0 and w ∈ H r , we have that It follows that We conclude by density.

The monotonicity formula
Let w ∈ H 1 (B + R ) be a non trivial solution to (2.4). For every r ∈ (0, R] we define In order to differentiate the functions D and H, the following Pohozaev type identity is needed. Proof. We observe that, by elliptic regularity theory, w ∈ H 2 (B + r \ B + ε ) for all 0 < ε < r < R. Furthermore, the fact that w has null trace on Γ R d implies that ∂w ∂x1 has null trace on Γ R d . Then, testing (2.4) with z · ∇w and integrating over B + r \ B + ε , we obtain that An integration by parts, which can be easily justified by an approximation argument, yields that We observe that there exists a sequence ε n → 0 + such that Indeed, if no such sequence exists, there would exist ε 0 > 0 such that integration of the above inequality on (0, ε 0 ) would then contradict the fact that w ∈ H 1 (B + R ) and, by trace embedding, w ∈ L 2 (Γ ε0 n ). Then, passing to the limit in (4.5) and (4.6) with ε = ε n yields (4.3). Finally (4.4) follows by testing (2.4) with w and integrating by parts in B + r .
In the following lemma we compute the derivative of the function H.
Let us now study the regularity of the function D.
Assume by contradiction that there exists r 0 ∈ (0, R 0 ) such that H(r 0 ) = 0, so that w = 0 a.e. on S + r0 . From (4.4) it follows that From Lemmas 3.1 and 3.2, we get which, together with (4.11) and Lemma 3.1, implies w ≡ 0 in B + r0 . From classical unique continuation principles for second order elliptic equations with locally bounded coefficients (see e.g. [26]) we can conclude that w = 0 a.e. in B + R , a contradiction. Thanks to Lemma 4.4, the frequency function is well defined. Using Lemmas 4.2 and 4.3, we now compute the derivative of N .
It follows that the limit of r → e C2r (1 + N (r)) as r → 0 + exists and is finite; hence the function N has a finite limit γ as r → 0 + . From (4.16) we deduce that γ 0.
The function H defined in (4.2) can be estimated as follows.
Lemma 4.7. Let γ := lim r→0 + N (r) be as in Lemma 4.6. Then Moreover, for any σ > 0, Proof. From Lemma 4.6 we have that (4.21) N is bounded in a neighborhood of 0, hence from (4.18) it follows that N ′ −C 3 for some positive constant C 3 in a neighborhood of 0. Then in a neighborhood of 0. From (4.8), (4.12), and (4.22) we deduce that, in a neighborhood of 0, which, after integration, yields (4.19).
Taking r fixed, we deduce that ψ is necessarily an eigenfunction of the eigenvalue problem (3.1).
Then there exists k 0 ∈ N \ {0} such that ψ(t) = ± 2 π cos( 2k0−1 2 t) and ϕ(r) solves the equation Hence ϕ(r) is of the form Hence, in view of (4.8), The proof of the lemma is thereby complete.
Proof. In view of (4.19) it is sufficient to prove that the limit exists. By (4.2), (4.8), and Lemma 4.6 we have that d dr and then, by integration over (r, R 0 ), where ν 1 and ν 2 are as in (4.14) and (4.15). Since, by Schwarz's inequality, ν 1 0, we have that On the other hand, from Lemma 4.6 N is bounded and hence from (4.17) we deduce that ν 2 is bounded close to 0 + . Hence, in view of (4.19), the function ρ → ρ −2γ−1 H(ρ) ρ 0 ν 2 (t)dt is bounded and hence integrable near 0. We conclude that both terms at the right hand side of (5.13) admit a limit as r → 0 + thus completing the proof.
The following lemma provides some pointwise estimate for solutions to (2.4). Then there exist C 4 , C 5 > 0 and r ∈ (0, R 0 ) such that (i) sup S + r |w| 2 C4 r S + r |w(z)| 2 ds for every 0 < r <r, (ii) |w(z)| C 5 |z| γ for all z ∈ B + r , with γ as in Lemma 4.6. Proof. We first notice that (ii) follows directly from (i) and (4.19). In order to prove (i), we argue by contradiction and assume that there exists a sequence τ n → 0 + such that sup t∈[0,π] w τ n 2 cos t, τ n 2 sin t 2 > nH τ n 2 with H as in (4.2), i.e., defining w τ as in (5.2) (5.14) sup From Lemma 5.1, there exists a subsequence τ n k such that w τn k → w in C 0 (S + 1/2 ) with w being as in (5.1), hence passing to the limit in (5.14) a contradiction arises.
To obtain a sharp asymptotics of H(r) as r → 0 + , it remains to prove that lim r→0 + r −2γ H(r) is strictly positive. w(r cos t, r sin t) cos 2k−1 2 t dt.
Integrating by parts in the first in integral on the left hand side of (5.19) and exploiting the fact that η ∈ C ∞ c (0, R) is an arbitrary test function, we infer ζ k (r) = 2 πr q(r)w(r, 0) + 2 π π 0 p(r cos t, r sin t)w(r cos t, r sin t) cos 2k−1 2 t dt.
Then, by a direct calculation, there exist c k 1 , c k 2 ∈ R such that From (5.22), we then deduce that as r → 0 + . From (5.21) and (5.23), we obtain that Let us assume by contradiction that lim r→0 + r −2γ H(r) = 0. Then (5.17) would imply that and hence, in view of (5.24), we would have that From Lemma 5.1, for every sequence τ n → 0 + , there exist a subsequence {τ n k } k∈N such that From (5.26) and (5.27), we infer that thus reaching a contradiction.

Some regularity estimates
In this section, we prove some regularity and approximation results, which will be used to estimate the Hölder norm of the difference between a solution u to (1.1) and its asymptotic profile βF k0 .
Then, for every ε > 0, there exists a constant C > 0 (independent of v, f , and g) such that In the sequel we denote as C > 0 a positive constant independent of v, f , and g which may vary from line to line. We consider a C 2 domain Ω ′ such that B + 3 ⊂ Ω ′ ⊂ B + 4 and Γ 3 n ∪ Γ 3 d ⊂ ∂Ω ′ . We define the functions (obtained uniquely by minimization arguments) v 1 ∈ H 1 (Ω ′ ) satisfying and v 2 ∈ H 1/2 (R) satisfying Therefore by (fractional) elliptic regularity theory (see e.g. [19, Proposition 1.1]), we deduce that . Consider the Poisson kernel P (x 1 , x 2 ) = 1 π x 2 |x| −2 with respect to the half-space R 2 + , see [4,Section 2.4]. We define where with the symbol ⋆ we denoted the convolution product with respect to the first variable. One can check that v 2 ∈ H 1 loc (R 2 + ) (see for example [3, Subsection 2.1]) and It is easy to see that Moreover by (6.4), for x, y ∈ R 2 + we get . By [23, Theorem 1] and continuous embeddings of Besov spaces into Hölder spaces, we get . Multiplying (6.3) by v 1 , integrating by parts and using Young's inequality, we get where in the first estimate we have used the Poincaré inequality for functions vanishing on a portion of the boundary. We then conclude that . Now, thanks to (6.1), (6.3) and (6.5), the function V : on Γ 3 d . By elliptic regularity theory, we have that (6.9) where r is a fixed radius satisfying 5 2 < r < 3 and C > 0 is independent of V . Let η a radial cutoff function compactly supported in B 3 satisfying η ≡ 1 in B r ; testing (6.8) with ηV , we infer that V H 1 (B + r ) C V L 2 (Ω ′ ) for some constant C > 0 independent of V . Hence by (6.9) we obtain (6.10) Let η ∈ C ∞ c (B 5/2 ) be a radial function, with η ≡ 1 on B 2 . Then the function V : Then by [23, Theorem 1], the arguments above, (6.10), (6.6) and (6.7), we deduce that . This, combined again with (6.6) and (6.7) completes the proof.
Recalling (1.2), for every k ∈ N with k 1, we consider the finite dimensional linear subspace of L 2 (B + r ), given by For every r > 0, k 1, and u ∈ L 2 (B + r ), we let Next, we estimate the L ∞ norm of the difference between a solution of a mixed boundary value problem on B + 1 and its projection on S k .
Then there exists a sequence r n → 0 such that so that Moreover, by a change of variable in (6.11), we get (6.17) Claim: For R = 2 m and r > 0, we have Indeed, by definition, for every r > r > 0, we have and thus, using the monotonicity of Θ, for every x ∈ B + r we get 2 1+γ k +α r γ k +α Θ(r) Cr γ k +α Θ(r).
This proves the claim.
From the definition of Θ and (6.18), for R = 2 m 1, we have Consequently, letting R 1 and m 0 ∈ N be the smallest integer such that 2 m0 R, we obtain that with C being a positive constant independent of R. Thanks to (1.3) and (6.12), it is plain that By assumption, we have that r 2−γ k −α and by (6.21), for every R > 1, By Lemma 6.3 (below), we deduce that necessarily v ∈ S k .
This clearly yields a contradiction when passing to the limit in (6.16) and (6.17).
The following Liouville type result was used in the proof of Proposition 6.2.
It follows that (6.24) |ϕ j (r)| const r γ k +α for all j 1 and r > 1, for some const > 0 independent of j and r. From the equation satisfied by v it follows that the functions ϕ j satisfy and then, for all j 1, there exist c j 1 , c j 2 ∈ R such that The fact v is continous and v(0) = 0 implies that ϕ j (r) = o(1) as r → 0 + . As a consequence we have that c j 2 = 0 for all j 1. On the other hand (6.24) implies that c j 1 = 0 for all j > k. Therefore we conclude that v(r cos t, r sin t) = i.e. v ∈ S k .

Asymptotics for u
We are now in position to prove Theorem 1.1.

Claim 1:
We have (7.1) w(y) = βF k0 (y) + o(|y| γ ) as |y| → 0 and y ∈ B + R . If this does not hold true then there exists a sequence of points y m ∈ (B + R ∪ Γ R n ) \ {0} and C > 0 such that y m → 0 and where τ m = |y m | and z m = ym |ym| . If m is large enough, we get a contradiction with (2.8). This proves (7.1) as claimed.
Let ̺ ∈ (0, 1/2) and let p and q be the functions introduced in (2.4). By (7.1), by the fact that p ∈ L ∞ (B + R ) and q ∈ C 1 ([0, R)), and by Proposition 6.2 applied to w, we have that, for every r ∈ (0, R), (7.2) |w(x) − F w k0,r (x)| Cr γ+̺ , for every x ∈ B + r , for some positive constant C > 0 independent of r, which could vary from line to line in the sequel. From (7.1) and (7.2) we deduce that
That is (7.4) as claimed.
For some fixed σ ∈ 0, π 2 , we define the curve Γ + ⊂ Z parametrized by If we choose σ > 0 sufficiently small then Γ + is the graph of a function h + defined in a open right neighborhood U + of 0. Moreover h + is a Lipschitz function in U + , h + ∈ C 2 (U + ) and Then we define the harmonic function In polar coordinates the function u reads (8.6) u(r, θ) = r 2 (log r) sin(2θ) + θ − π 2 cos(2θ) .
We can now define the functions V 1 , One may verify that V 1 , V 2 ∈ C 1 (B 1/2 ). Moreover we have Then we consider the dynamical system .
After linearization at (0, 0), by [15, Theorem IX.6.2] we deduce that the stable and unstable manifolds corresponding to the stationary point (0, 0) of (8.7), are respectively tangent to the eigenvectors (1, −1) and (1, 1) of the matrix DV (0, 0) where V is the vector field (V 1 , V 2 ). We define the curve Γ − as the stable manifold of (8.7) at (0, 0) intersected with B ε ∩ Π − where ε ∈ (0, 1 2 ) can be chosen sufficiently small in such a way that Γ − becomes the graph of a function h − defined in a open left neighborhood U − of 0. Combining the definitions of h + and h − we can introduce a function h : U + ∪ U − ∪ {0} → R such that h ≡ h + on U + , h ≡ h − on U − and h(0) = 0.
This shows that h + ∈ C 2 (U + ∪ {0}) (and a fortiori cannot be extended to be of class C 2,δ ). We observe that the reason of the appearance of a logarithmic term is not due to the presence of a corner at 0; indeed we are going to construct a domain with C 1 -boundary for which the same phenomenon occurs. In order to do this, it is sufficient to take the domain Ω and the function u defined above and to apply a suitable deformation in order to remove the angle. We recall that Ω exhibits a corner at 0 whose amplitude is 3π 4 . For this reason, we define F : C \ {ix 2 : x 2 0} → C by for any z = re iθ , r > 0 , θ ∈ − π 2 , 3π 2 .
It is immediate to verify that u = 0 on Γ + . We also prove that ∂ u ∂ν = 0 on Γ − . To avoid confusion with the notion of normal unit vectors to Γ − and Γ − we denote them respectively with ν Γ− and ν Γ− . Since u is still harmonic, ∂u ∂νΓ − = 0 on Γ − and F is a conformal mapping, for any ϕ ∈ C ∞ c ( Ω ∪ Γ − ), we have Finally we prove for h + an estimate similar to (8.11).
The above arguments show that ∂ Ω is of class C 1 but not of class C 1,δ (and a fortiori not of class C 2,δ ).