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

1.   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; we also quote ^[18] for quantitative uniqueness obtained by monotonicity methods.

The monotonicity 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 ^{[1,2,9,19,27]} under homogeneous Dirichlet conditions and in ^[26] 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 ^{[9,10,11,12,13]} 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 $\Omega$ be an open subset of $\mathbb{R}^2$ with Lipschitz boundary. Let $\Gamma _n\subset \partial \Omega$ and $\Gamma _d\subset \partial \Omega$ be two nonconstant curves (open in $\partial\Omega$) such that $\overline{\Gamma _n} \cap \overline{\Gamma _d} = \{P\}$ for some $P\in\partial\Omega$. We are interested in regularity of weak solutions $u\in H^1(\Omega)$ to the mixed boundary value problem

with $f\in L^\infty(\Omega)$ and $g\in C^1(\overline{\Gamma _n})$, see Section 2 for the weak formulation. Our aim is to prove unique continuation properties from the Dirichlet-Neumann junction $\{P\} = \overline{\Gamma _n}\cap \overline{\Gamma _d}$ and sharp asymptotics of nontrivial solutions near $P$ provided $\partial \Omega$ is of class $C^{2, \gamma }$ 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 ^[16,25], 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. ^[6,20]. 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 led 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 $P\in \overline{\Gamma _n}\cap \overline{\Gamma _d}$, we will need to assume that $\partial \Omega$ is of class $C^{2, \delta }$ near $P$. The asymptotic profile of the solution will be given by the function

for some $k\in \mathbb{N}\setminus\{0\}$. We note that $F_{k}\in H^1_{\rm loc}(\mathbb{R}^2)$ and solves the equation

where here and in the following $\mathbb{R}^2_+: = \{(x_1, x_2)\in \mathbb{R}^2\, :\, x_2>0\}$.

The main result of the present paper provides an evaluation of the local behavior of weak solutions $u\in H^1(\Omega)$ 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 $\Omega \subset\mathbb{R}^2$ are satisfied. Here and in the remaining of this paper, $\Gamma _n, \Gamma _d\subset\partial \Omega$ are nonconstant curves (open as subsets of $\partial\Omega$) such that $\overline{\Gamma _n}\cap \overline{\Gamma _d} = \{0\}$ with $0\in\partial\Omega$.

(ⅰ) The domain $\Omega$ is of class $C^{2, \delta }$ in a neighborhood of $0$, for some $\delta >0$.

(ⅱ) The unit vector $e_1: = (1, 0)$ is tangent to $\partial \Omega$ at $0$ and pointed towards $\Gamma _n$. Moreover, the exterior unit normal vector to $\partial\Omega$ at $0$ is $(0, -1)$.

We are now in position to state the main result of the present paper.

Theorem 1.1. We assume that $\Omega$ satisfies the assumptions (ⅰ)-(ⅱ) above. Let $u\in H^1(\Omega)$ be a nontrivial weak solution to (1.1), with $f\in L^\infty(\Omega)$ and $g\in C^1(\overline{\Gamma _n})$. Then, there exist $k_0\in \mathbb{N}\setminus\{0\}$, $\beta \in\mathbb{R}\setminus\{0\}$ and $r>0$ such that, for every $\varrho\in (0, 1/2)$, there exists $C>0$ such that

Here, the function $\varphi: \overline {\Omega \cap B_{r_0}}\to \overline {\mathbb{R}^2_+}$ is a conformal map of class $C^2$, for some $r_0>0$ only depending on $\Omega$.

Remark 1.2. Here and in the sequel, we identify $\mathbb{R}^2$ with the complex plane $\mathbb{C}$; hence, by a conformal map on an open set $U\subset\mathbb{R}^2$ we mean a holomorphic function with complex derivative everywhere non-zero on $U$. We notice that, if $\Omega$ satisfies (ⅰ)-(ⅱ) and $\varphi: \overline {\Omega \cap B_{r_0}}\to \overline {\mathbb{R}^2_+}$ is conformal, then $D\varphi(0) = \alpha \mathop{\rm Id}$ and $\varphi'(0) = \alpha$ for some real $\alpha>0$, where $D\varphi$ denotes the jacobian matrix of $\varphi$ and $\varphi'$ denotes the complex derivative of $\varphi$.

As a direct consequence of Theorem 1.1, we derive the following Hopf-type lemma.

Corollary 1.3. Under the same assumptions as in Theorem 1.1, let $u\in H^1(\Omega)$ be a non-trivial weak solution to (1.1), with $u\geq 0$. Then

(ⅰ) for every $t\in[0, \pi)$,

where $\alpha = \varphi'(0)>0$ and $\varphi$ is as in Theorem 1.1;

(ⅱ) for every cone $\mathcal{C}\subset \mathbb{R} ^2$ satisfying $(1, 0)\in \mathcal{C}$ and $(-1, 0)\in \mathbb{R}^2\setminus \overline {\mathcal{C}}$, we have

A further relevant byproduct of our asymptotic analysis is the following unique continuation principle, whose proof follows directly from Theorem 1.1.

Corollary 1.4. Under the same assumptions as in Theorem 1.1, let $u\in H^1(\Omega)$ be a weak solution to (1.1) such that $u(x) = O(|x|^n)$ as $x\in\Omega$, $|x|\to 0$, for any $n\in \mathbb{N}$. Then $u\equiv 0$.

We observe that Theorem 1.1 provides a sharp asymptotic expansion (and consequently a unique continuation principle) at the boundary for $\frac12$-fractional elliptic equations in dimension 1. Indeed, if $v\in H^{1/2}(\mathbb{R})$ weakly solves

for some $g\in C^1([0, R])$, then its harmonic extension $V\in H^1_{\rm loc}(\overline{\mathbb{R}^2_+})$ weakly solves

see ^[4]. Theorem 1.1 and Corollary 1.4 apply to (1.5). 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 $C^{2, \delta }$ regularity on $\Omega$ 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 $C^{2, \delta }$, 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:

for every $\varrho\in (0, 1/2)$, for some $C>0$, $k_0\geq 1$ and where $a_k(r) = \frac{\left\langle { u, F_k\circ\varphi } \right\rangle _{L^2(B_r)}}{\|F_j\circ\varphi \|_{L^2(B_r) }^2}$. Now, if $u$ is nontrivial, a blow-up analysis combined with Almgren type monotonicity formula allows to depict a $k_0\geq 1$ for which $a_{k_0}(r)\to \beta \not = 0$ and $a_k(r)\to 0$ for every $k < k_0$ as $r\to0$. The proof of (1.6) uses also a blow-up analysis argument inspired by Serra ^[24], see also ^[22,23].

Remark 1.5. The extension of our results to higher dimensions are the object of current investigation. First of all, the implementation of the monotonicity argument for Dirichlet-Neumann problems exhibits substantial additional difficulties due to the positive dimension of the junction set and some role played by the geometry of the domain. Moreover, further technical difficulties appear in higher dimension since, in such a situation, we can no more make use of conformal transformations like the ones employed in Section 2 which are based on the Riemann mapping Theorem.

Remark 1.6. For the sake of simplicity of the exposition, in the present paper we considered an elliptic problem with the Laplacian and a linear term with a bounded potential; a possible extension to more general elliptic problems with variable coefficients and first order terms could be obtained with a more sophisticated monotonicity approach like in ^[9].

The paper is organized as follows. In Section 2 we introduce an auxiliary equivalent problem obtained by a conformal diffeomorphic deformation straightening $B_1\cap\partial\Omega$ 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 $C^{2, \delta }$-regularity assumption.

2.   The auxiliary problem

For every $R>0$ let $B_R = \{(x_1, x_2)\in\mathbb{R}^2:x_1^2+x_2^2 < R^2\}$ and $B_R^+ = \{(x_1, x_2)\in B_R:x_2>0\}$. Since $\partial \Omega$ is of class $C^{2, \delta }$ near zero, we can find $r_0>0$ such that $\Gamma : = \partial \Omega \cap B_{ r_0}$ is a $C^{2, \delta }$ curve. Here and in the following, we let $\mathcal{B}$ be a $C^{2, \delta }$ simply connected open bounded set such that $\mathcal{B}\subset \Omega$ and $\partial\mathcal B\cap\partial\Omega = \Gamma$. For some functions

let $u\in H^1(\mathcal{B})$ be a solution to

We introduce the space $H^1_{0, \Gamma _d}(\mathcal B)$ as the closure in $H^1(\mathcal B)$ of the subspace

We say that $u\in H^1(\mathcal B)$ is a weak solution to (2.2) if

where $C^\infty_{0, \partial\mathcal B\setminus\Gamma _n}(\mathcal B) = \{u\in C^\infty(\overline{\mathcal B}):u = 0\text{ on }\partial\mathcal B\setminus\Gamma _n\}$. Since $\mathcal{B}$ is of class $C^{2, \delta }$, in view of the Riemann mapping Theorem and [17,Theorem 5.2.4], there exists a conformal map $\hat \varphi :\overline { \mathcal{B}} \to \overline {B}_1$ which is of class $C^2$. Let $N = \hat\varphi (0)\in \partial B_1$ and let $S$ be its antipodal. We then consider the map ${\tilde \varphi } : \mathbb{R}^2\setminus\{S\} \to \mathbb{R}^2\setminus\{\overline S\}$ given by ${\tilde \varphi } (z): = 2\frac{\overline{z-S}}{|z-S|^2}+\overline{S}$, where, for every $z\in\mathbb{R}^2\simeq\mathbb{C}$, $\overline{z}$ denotes the complex conjugate of $z$. This map is conformal and ${\tilde \varphi } (N) = 0$. In addition ${\tilde \varphi } (\overline {B_1} \setminus\{S\}) \subset \overline { \mathcal{P}}$ where $\mathcal P$ is the half plane not containing $\overline S$ whose boundary is the line passing through the origin orthogonal to $\overline S$.

Then the map ${\tilde \varphi } \circ \hat \varphi$ is a conformal map which is of class $C^2$ from a neighborhood of the origin $\overline{{\mathcal B}\cap B_r}$ into $\overline { \mathcal{P}}$ for some $r>0$. It is now clear that there exists a rotation $\mathcal{R}$ and a real number $R>0$ such that, letting $\mathcal{U}_R: = \varphi ^{-1}(B_R^+)$, the map $\varphi : = \mathcal{R} \circ {\tilde \varphi } \circ \hat \varphi : \overline {\mathcal{U}_R}\to \overline {B_R^+}$ is an invertible conformal map of class $C^2$ with inverse $\varphi ^{-1}:\overline {B_R^+}\to \overline {\mathcal{U}_R}$ of class $C^2$. Moreover $\varphi(0) = 0$.

Since $\varphi$ is a conformal diffeomorphism, in view of Remark1.2 we have that, under the assumptions of Theorem 1.1,

being $\varphi'(0)$ the complex derivative of $\varphi$ at $0$, which turns out to be real because of the assumption that $(1, 0)$ is tangent to $\partial\Omega$ at $0$ and strictly positive because of the assumption that the exterior unit normal vector to $\partial\Omega$ at $0$ is $(0, -1)$. In addition, (2.3) implies that, if $R$ is chosen sufficiently small, $\varphi ^{-1}((-R, 0) \times\{0\})\subset\Gamma _d$ and $\varphi ^{-1}((0, R)\times\{0\})\subset \Gamma _n$.

Therefore letting $w = u\circ \varphi^{-1}: B_R^+ \to \mathbb{R}$ and $\Psi: = \varphi ^{-1}$, we then have that $w\in H^1(B_R^+)$ solves

with

It is plain that $p\in L^\infty(B_R^+)$ and $q \in C^1([0, R))$. Here and in the following, for every $r>0$, we define

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 $\mathcal N$ 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 $k_0\in \mathbb{N}$, $k_0\geq1$, such that

Furthermore

strongly in $H^1(B_r^+)$ for all $r>0$ and in $C^{0, \mu}_{\rm loc}(\overline{\mathbb{R}^2_+}\setminus\{0\})$ for every $\mu\in (0, 1)$, where $\beta\neq0$ and

In particular

The proof of Theorem 2.1 is based on the study of the monotonicity properties of the Almgren function $\mathcal N$ and on a fine blow-up analysis which will be performed in Sections 4 and 5.

3.   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

It is well known that the normalized eigenfunctions

form an orthonormal basis of the space $L^2(0, \pi)$. Furthermore, the first eigenvalue $\lambda_1 = \frac14$ can be characterized as

For every $r>0$, we let (recall (2.5) for the definition of $\Gamma _d^r$)

As a consequence of (3.3) we obtain the following Hardy-Poincaré inequality in $\mathcal H_r$.

Lemma 3.1. For every $r>0$ and $w\in \mathcal H_r$, we have that

Proof. Let $w\in C^\infty(\overline{B_r^+})$ with $w = 0$ on $\overline { \Gamma _d^r} = [-r, 0]\times\{0\}$. Then, in view of (3.3),

We conclude by density, recalling that the space of smooth functions vanishing on $[-r, 0]\times\{0\}$ is dense in $\mathcal H_r$, see e.g. ^[7].

Lemma 3.2. For every $r>0$ and $w\in \mathcal H_r$, we have that $x_1^{-1}w^2(x_1, 0)\in L^1(0, r)$ and

Proof. Let $w\in C^\infty(\overline{B_r^+})$ with $w = 0$ on $[-r, 0]\times\{0\}$. Then for any $0 < x_1 < r$

It follows that

We conclude by density.

4.   The monotonicity formula

Let $w\in H^1(B_R^+)$ be a non trivial solution to (2.4). For every $r\in (0, R]$ we define

and

where $S_r^+: = \{(x_1, x_2):x_1^2+x_2^2 = r^2\text{ and }x_2>0\}$.

In order to differentiate the functions $D$ and $H$, the following Pohozaev type identity is needed.

Theorem 4.1. Let $w$ solve (2.4). Then for a.e. $r\in (0, R)$ we have

and

Proof. We observe that, by elliptic regularity theory, $w\in H^2(B_r^+\setminus B_\varepsilon ^+)$ for all $0 < \varepsilon < r < R$. Furthermore, the fact that $w$ has null trace on $\Gamma _d^R$ implies that $\frac{\partial w}{\partial x_1}$ has null trace on $\Gamma _d^R$. Then, testing (2.4) with $z\cdot\nabla w$ and integrating over $B_r^+\setminus B_\varepsilon ^+$, 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 $\varepsilon _n\to 0^+$ such that

Indeed, if no such sequence exists, there would exist $\varepsilon _0>0$ such that

integration of the above inequality on $(0, \varepsilon _0)$ would then contradict the fact that $w\in H^1(B_{R}^+)$ and, by trace embedding, $w\in L^2(\Gamma _n^{\varepsilon _0})$. Then, passing to the limit in (4.5) and (4.6) with $\varepsilon = \varepsilon _n$ 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 $H$.

Lemma 4.2. $H\in W^{1, 1}_{\rm loc}(0, R)$ and

in a distributional sense and for a.e. $r\in (0, R)$, and

Proof. Let $\phi\in C^\infty_{\rm c}(0, R)$. Since $w, \nabla w\in L^2(B_R^+)$ and $w\in C^1(B_R^+)$, using twice Fubini's Theorem we obtain that

thus proving (4.7). Identity (4.8) follows directly from (4.7) and (4.4).

Let us now study the regularity of the function $D$.

Lemma 4.3. The function D defined in (4.1) belongs to $W^{1, 1}(0, R)$ and

in a distributional sense and for a.e. $r\in (0, R)$.

Proof. From the fact that $w\in H^1(B_R^+)$ and $w\big|_{\Gamma _n^R}\in L^2(\Gamma _n^R)$, we deduce that $D$ belongs to $W^{1, 1}(0, R)$ and

for a.e. $r\in (0, R)$ and in the distributional sense.

The conclusion follows combining (4.10) and (4.3).

Lemma 4.4. There exists $R_0\in(0, R)$ such that $H(r)>0$ for any $r\in (0, R_0)$.

Proof. Let $R_0\in (0, R)$ be such that

Assume by contradiction that there exists $r_0\in(0, R_0)$ such that $H(r_0) = 0$, so that $w = 0$ a.e. on $S_{r_0}^+$. 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 $w\equiv 0$ in $B_{r_0}^+$. From classical unique continuation principles for second order elliptic equations with locally bounded coefficients (see e.g. ^[28]) 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 ${\mathcal N}$.

Lemma 4.5. The function ${\mathcal N}$ defined in (4.12) belongs to $W^{1, 1}_{{\rm loc}}(0, R_0)$ and

in a distributional sense and for a.e. $r\in (0, R_0)$, where

and

Proof. From Lemmas 4.2, 4.4, and 4.3, it follows that ${\mathcal N}\in W^{1, 1}_{{\rm loc}}(0, R_0)$. From (4.8) we deduce that

and the proof of the lemma easily follows from (4.7) and (4.9).

We now prove that ${\mathcal N}(r)$ admits a finite limit as $r\to 0^+$.

Lemma 4.6. There exists $\gamma\in[0, +\infty)$ such that $\lim_{r\to 0^+}\mathcal N(r) = \gamma$.

Proof. From Lemmas 3.1 and 3.2 it follows that

hence there exist $\bar r\in (0, R_0)$ and $C_1>0$ such that

In particular

Moreover, using again Lemmas 3.1 and 3.2 we can estimate $\nu_2$ in $(0, \bar r)$ as follows

Since $\nu_1\geq 0$ by Schwarz's inequality, from Lemma 4.5 and the above estimate it follows that there exists $C_2>0$ such that

which implies that

It follows that the limit of $r\mapsto e^{C_2r}(1+\mathcal N(r))$ as $r\to0^+$ exists and is finite; hence the function $\mathcal N$ has a finite limit $\gamma$ as $r\to 0^+$. From (4.16) we deduce that $\gamma\geq 0$.

The function $H$ defined in (4.2) can be estimated as follows.

Lemma 4.7. Let $\gamma: = \lim_{r\rightarrow 0^+} {\mathcal N}(r)$ be as in Lemma 4.6. Then

Moreover, for any $\sigma>0$,

Proof. From Lemma 4.6 we have that

hence from (4.18) it follows that $\mathcal N'\geq -C_3$ 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).

Since $\gamma = \lim_{r\rightarrow 0^+} {\mathcal N}(r)$, for any $\sigma>0$ there exists $r_\sigma>0$ such that ${\mathcal N}(r) < \gamma+\sigma/2$ for any $r\in (0, r_\sigma)$ and hence $\frac{H'(r)}{H(r)} = \frac{2\, {\mathcal N}(r)}{r} < \frac{2\gamma+\sigma}{r}$ for all $r\in (0, r_\sigma)$. By integration we obtain (4.20).

5.   Blow-up analysis for the auxiliary problem

Lemma 5.1. Let $w\in H^1(B_R^+)$ be a non trivial solution to (2.4). Let $\gamma: = \lim_{r\rightarrow 0^+} {\mathcal N}(r)$ be as in Lemma 4.6. Then there exists $k_0\in \mathbb{N}$, $k_0\geq1$, such that

Furthermore, for every sequence $\tau_n\to0^+$, there exist a subsequence $\{\tau_{n_k}\}_{k\in\mathbb{N}}$ such that

strongly in $H^1(B_r^+)$ and in $C^{0, \mu}_{\rm loc}(\overline{B_r^+}\setminus\{0\})$ for every $\mu\in (0, 1)$ and all $r\in(0, 1)$, where

Proof. Let us set

We notice that, for all $\tau\in (0, R)$, $w^\tau\in \mathcal H_1$ and $\int_{S_1^+}|w^{\tau}|^2ds = \int_0^\pi |w^\tau(\cos t, \sin t)|^2\, dt = 1$. Moreover, by scaling and (4.21),

as $\tau\to 0^+$, whereas from Lemmas 3.1 and 3.2 it follows that

for every $\tau\in(0, R_0)$, being $R_0$ as in (4.11). From (5.4), (5.5), and Lemma 3.1 we deduce that

Therefore, for any given sequence $\tau_n\to 0^+$, there exists a subsequence $\tau_{n_k}\to0^+$ such that $w^{\tau _{n_k}}\rightharpoonup \widetilde w$ weakly in $H^1(B_1^+)$ for some $\widetilde w\in H^1(B_1^+)$. Due to compactness of trace embeddings, we have that $\widetilde w = 0$ on $\Gamma _d^1$ and

In particular $\widetilde w\not\equiv 0$. For every small $\tau\in (0, R_0)$, $w^\tau$ satisfies

in a weak sense, i.e.

for all $\varphi \in H^1(B_1^+)$ s.t. $\varphi = 0$ on $S_1^+\cup\Gamma _d^1$. From weak convergence $w^{\tau _{n_k}}\rightharpoonup \widetilde w$ in $H^1(B_1^+)$, we can pass to the limit in (5.8) along the sequence $\tau _{n_k}$ and obtain that $\widetilde w$ weakly solves

From (5.6) it follows that $\{\tau q(\tau x) w^\tau(x, 0)\}_{\tau\in(0, R_0)}$ is bounded in $H^{1/2}(\Gamma ^1_n)$. Then, by elliptic regularity theory, for every $0 < r_1 < r_2 < 1$ we have that $\{w^\tau\}_{\tau\in(0, R_0)}$ is bounded in $H^2(B^+_{r_2}\setminus \overline{B^+_{r_2}})$. From compactness of trace embeddings we have that, up to passing to a further subsequence, $\frac{\partial w^{\tau_{n_k}}}{\partial\nu}\to \frac{\partial \widetilde w}{\partial\nu}$ in $L^2(S_r^+)$ for every $r\in (0, 1)$. Testing equation (5.8) for $\tau = \tau_{n_k}$ with $w^{\tau}$ on $B_r^+$ we obtain that

thus proving that $\|w^{\tau _{n_k}}\|_{H^1(B_r^+)}\to \|\widetilde w\|_{H^1(B_r^+)}$ for all $r\in(0, 1)$, and hence

for every $r\in (0, 1)$. Furthermore, by compact Sobolev embeddings, we also have that, up to extracting a further subsequence,

for every $r\in (0, 1)$ and $\mu\in(0, 1)$.

For any $r\in (0, 1)$ and $k\in \mathbb{N}$, let us define the functions

and ${\mathcal N}_k(r): = \frac{D_k(r)}{H_k(r)}$. Direct calculations yield that ${\mathcal N}_k(r) = {\mathcal N}(\tau_{n_k}r)$ for all $r\in (0, 1)$. From (5.10) it follows that, for any fixed $r\in (0, 1)$,

From classical unique continuation principles for harmonic functions it follows that $\widetilde D(r)>0$ and $\widetilde H(r)>0$ for all $r\in(0, 1)$ (indeed $\widetilde D(r) = 0$ or $\widetilde H(r) = 0$ for some $r\in(0, 1)$ would imply that $\widetilde w \equiv 0$ in $B_r^+$ and, by unique continuation, $\widetilde w \equiv 0$ in $B_1^+$, a contradiction). Hence, by Lemma 4.6,

for all $r\in (0, 1)$. Therefore $\widetilde {\mathcal N}$ is constant in $(0, 1)$ and hence $\widetilde {\mathcal N}'(r) = 0$ for any $r\in (0, 1)$. By (5.9) and Lemma 4.5 with $p\equiv 0$ and $q\equiv0$, we obtain

which implies that $\widetilde w$ and $\frac{\partial \widetilde w}{\partial \nu}$ are parallel as vectors in $L^2(S_r^+)$. Hence there exists $\eta = \eta(r)$ such that $\frac{\partial \widetilde w}{\partial \nu}(r\cos t, r\sin t) = \eta(r) \widetilde w(r\cos t, r\sin t)$ for all $r\in(0, 1)$ and $t\in[0, \pi]$. It follows that

where $\varphi(r) = e^{\int_1^r \eta(s)ds}$ and $\psi(t) = \widetilde w(\cos t, \sin t)$. From (5.7) we have that $\int_0^\pi \psi^2 = 1$. From (5.9) and (5.12) we can conclude that

Taking $r$ fixed, we deduce that $\psi$ is necessarily an eigenfunction of the eigenvalue problem (3.1). Then there exists $k_0\in\mathbb{N}\setminus\{0\}$ such that $\psi(t) = \pm\sqrt{\frac2\pi}\cos(\frac{2k_0-1}{2}t)$ and $\varphi(r)$ solves the equation

Hence $\varphi(r)$ is of the form

for some $c_1, c_2\in\mathbb{R}$. Since the function $r^{-\frac{2k_0-1}2}\psi(t)\notin H^1(B_1^+)$, we deduce that necessarily $c_2 = 0$ and $\varphi(r) = c_1 r^{\frac{2k_0-1}2}$. Moreover, from $\varphi(1) = 1$, we obtain that $c_1 = 1$ and then

From (5.13) it follows that

Hence, in view of (4.8),

The proof of the lemma is thereby complete.

We observe that at this stage of our analysis we cannot exclude that the limit function $\widetilde w$ found in Lemma 5.1 depends on the subsequence. In order to prove that the convergence in (5.1) actually holds as $\tau\to 0^+$ we need to univocally identify the limit profile $\widetilde w$.

Lemma 5.2. Let $w\not\equiv 0$ satisfy (2.4), $H$ be defined in (4.2), and $\gamma: = \lim_{r\rightarrow 0^+} {\mathcal N}(r)$ be as in Lemma 4.6. Then the limit $\lim_{r\to0^+}r^{-2\gamma}H(r)$ exists and it is finite.

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

and then, by integration over $(r, R_0)$,

where $\nu_1$ and $\nu_2$ are as in (4.14) and (4.15). Since, by Schwarz's inequality, $\nu_1\geq 0$, we have that $\lim_{r\to 0^+} \int_r^{R_0} \rho^{-2\gamma-1} H(\rho) \left(\int_0^\rho \nu_1(t) dt \right) d\rho$ exists. On the other hand, from Lemma 4.6 $\mathcal N$ is bounded and hence from (4.17) we deduce that $\nu_2$ is bounded close to $0^+$. Hence, in view of (4.19), the function $\rho\mapsto \rho^{-2\gamma-1} H(\rho) \left(\int_0^\rho \nu_2(t) dt \right)$ is bounded and hence integrable near $0$. We conclude that both terms at the right hand side of (5.14) admit a limit as $r\to 0^+$ thus completing the proof.

The following lemma provides some pointwise estimate for solutions to (2.4).

Lemma 5.3. Let $w\in H^1(B_R^+)$ be a nontrivial solution to (2.4). Then there exist $C_4, C_5>0$ and $\bar r\in(0, R_0)$ such that

(ⅰ) $\sup_{S_{r}^+}|w|^2\leq \frac{C_4}{r}\int_{S_r^+}|w(z)|^2\, ds$ for every $0 < r < \bar r$,

(ⅱ) $|w(z)|\leq C_5|z|^\gamma$ for all $z\in B_{\bar r}^+$, with $\gamma$ 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 $\tau_n\to 0^+$ such that

with $H$ as in (4.2), i.e., defining $w^\tau$ as in (5.3)

From Lemma 5.1, there exists a subsequence $\tau_{n_k}$ such that $w^{\tau_{n_k}}\to\widetilde w$ in $C^{0}(S_{1/2}^+)$ with $\widetilde w$ being as in (5.2), hence passing to the limit in (5.15) a contradiction arises.

To obtain a sharp asymptotics of $H(r)$ as $r\to 0^+$, it remains to prove that $\lim_{r\to 0^+} r^{-2\gamma} H(r)$ is strictly positive.

Lemma 5.4. Under the same assumptions as in Lemmas 5.2 and 5.3, we have that

Proof. From Lemma 5.1 there exists $k_0\in\mathbb{N}$, $k_0\geq 1$ such that $\gamma = \frac{2k_0-1}{2}$. Let us expand $w$ as

where

The Parseval identity yields

From (4.19) and (5.18) it follows that, for all $k\geq1$,

Let $\eta\in C^\infty_c(0, R)$. Testing (2.4) with the function $\eta(r)\cos\left(\frac{2k-1}2 t\right)$, by (5.16) we obtain

Integrating by parts in the first in integral on the left hand side of (5.20) and exploiting the fact that $\eta\in C^\infty_c(0, R)$ is an arbitrary test function, we infer

where

Then, by a direct calculation, there exist $c_1^k, c_2^k\in\mathbb{R}$ such that

From Lemma 5.3 it follows that

and hence the functions

belong to $L^1(0, R)$. Hence

and then, by (5.19), there must be

From (5.23), we then deduce that

as $r\to0^+$. From (5.22) and (5.24), we obtain that

Let us assume by contradiction that $\lim_{r \to 0^+} r^{-2\gamma}H(r) = 0$. Then (5.18) would imply that

and hence, in view of (5.25), we would have that

which, together with (5.23), implies

as $r\to0^+$. From (5.25) and (5.26), we conclude that $\varphi_{k_0}(r) = O(r^{\frac12+k_0})$ as $r\to0^+$, namely,

where $w^\tau$ is defined in (5.3). From (4.20), there exists $C>0$ such that $\sqrt{H(\tau)}\geq C\tau^{\gamma+\frac12}$ for $\tau$ small, and therefore

From Lemma 5.1, for every sequence $\tau_n\to0^+$, there exist a subsequence $\{\tau_{n_k}\}_{k\in\mathbb{N}}$ such that

From (5.27) and (5.28), we infer that

thus reaching a contradiction.

Proof of Theorem 2.1. Identity (2.6) follows from Lemma 5.1, thus there exists $k_0\in \mathbb{N}$, $k_0\geq 1$, such that $\gamma = \lim_{r\to 0^+}{\mathcal N}(r) = \frac{2k_0-1}{2}$.

Let $\{\tau_n\}_{n\in\mathbb{N}}\subset (0, +\infty)$ be such that $\lim_{n\to+\infty}\tau_n = 0$. Then, from Lemmas 5.1 and 5.4, scaling and a diagonal argument, there exists a subsequence $\{\tau_{n_k}\}_{k\in\mathbb{N}}$ and $\beta\neq0$ such that

strongly in $H^1(B_r^+)$ for all $r>0$ and in $C^{0, \mu}_{\rm loc}(\overline{\mathbb{R}^2_+}\setminus\{0\})$ for every $\mu\in (0, 1)$. In particular

in $C^{0, \mu}([0, \pi])$. To prove that the above converge occurs as $\tau\to0^+$ and not only along subsequences, we are going to show that $\beta$ depends neither on the sequence $\{\tau_n\}_{n\in\mathbb{N}}$ nor on its subsequence $\{\tau_{n_k}\}_{k\in\mathbb{N}}$.

Defining $\varphi_{k_0}$ and $\zeta_{k_0}$ as in (5.17) and (5.21), from (5.30) it follows that

as $k\to+\infty$. On the other hand, from (5.22), (5.24), and (5.25) we know that that

Choosing $\tau = R$ in the first line of (5.32), we obtain

Hence, from the second line of (5.32), we obtain that

as $\tau\to0^+$. Then, from (5.31) we deduce that

In particular $\beta$ depends neither on the sequence $\{\tau_n\}_{n\in\mathbb{N}}$ nor on its subsequence $\{\tau_{n_k}\}_{k\in\mathbb{N}}$, thus implying that the convergence in (5.29) actually holds as $\tau\to 0^+$ and proving the theorem. We observe that (2.7) follows by replacing (5.17) and (5.21) into (5.33).

6.   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 $\beta F_{k_0}$.

Proposition 6.1. Let $f\in L^\infty (B_4^+)$, $g\in L^\infty (\Gamma _n^4)$ and let $v\in H^1(B_4)\cap L^\infty(B_4^+)$ solve

Then, for every $\varepsilon >0$, there exists a constant $C>0$ (independent of $v$, $f$, and $g$) such that

Proof. 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 $\Omega '$ such that $B_3^+\subset \Omega '\subset B_4^+$ and $\Gamma ^3_n\cup \Gamma ^3_d\subset \partial \Omega '.$ We define the functions (obtained uniquely by minimization arguments) $v_1 \in H^1 (\Omega ')$ satisfying

and $\tilde{ v}_2\in H^{1/2}(\mathbb{R})$ satisfying

Therefore by (fractional) elliptic regularity theory (see e.g. [21,Proposition 1.1]), we deduce that

Consider the Poisson kernel $P(x_1, x_2) = \frac 1\pi x_2 |x|^{-2}$ with respect to the half-space $\mathbb{R}^2_+$, see [4,Section 2.4]. We define

where with the symbol $\star$ we denoted the convolution product with respect to the first variable. One can check that $v_2\in H^1_{\rm loc}(\overline{\mathbb{R}^2_+})$ (see for example [3,Subsection 2.1]) and

It is easy to see that

Moreover by (6.3), for $x, y\in \overline {\mathbb{R}^2_+}$ we get

It follows that

By [25,Theorem 1] and continuous embeddings of Besov spaces into Hölder spaces, we get

Multiplying (6.2) 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), (6.2) and (6.4), the function $V: = v-(v_1+v_2)\in H^1(\Omega ')$ solves the equation

By elliptic regularity theory, we have that

where $r$ is a fixed radius satisfying $\frac 52 < r < 3$ and $C>0$ is independent of $V$. Let $\eta$ a radial cutoff function compactly supported in $B_3$ satisfying $\eta\equiv 1$ in $B_r$; testing (6.7) with $\eta V$, we infer that $\|V\|_{H^1(B_r^+)}\le C\|V\|_{L^2(\Omega ')}$ for some constant $C>0$ independent of $V$. Hence by (6.8) we obtain

Let ${\tilde \eta }\in C^\infty_c(B_{5/2})$ be a radial function, with ${\tilde \eta }\equiv 1$ on $B_2$. Then the function ${\tilde V }: = {\tilde \eta } V\in H^1(\mathbb{R}^2_+)$ solves

Then by [25,Theorem 1], the arguments above, (6.9), (6.5) and (6.6), we deduce that

This, combined again with (6.5) and (6.6) completes the proof.

Recalling (1.2), for every $k\in \mathbb{N}$ with $k\geq1$, we consider the finite dimensional linear subspace of $L^2(B_r^+)$, given by

For every $r>0$, $k\geq1$, and $u\in L^2(B_r^+)$, we let

be the $L^2(B_r^+)$-projection of $u$ on $\mathcal{S}_k$, so that

and

Next, we estimate the $L^\infty$ norm of the difference between a solution of a mixed boundary value problem on $B_1^+$ and its projection on $\mathcal{S}_k$.

Proposition 6.2. Let $u\in H^1(B_1^+)\cap L^\infty(\mathbb{R}^2_+)$ solve

where, for some $k\in \mathbb{N}\setminus\{0\}$ and $\overline {C} >0$,

and $\gamma_k = \frac{2k-1}{2}$. Then, for every $\alpha \in (0, 1/2)$, we have that

Proof. In the sequel, $C>0$ stands for a positive constant, only depending on $\alpha, \overline {C}$ and $k$, which may vary from line to line. Assume by contradiction that, there exists $\alpha \in(0, 1/2)$ such that

We consider the nonincreasing function

It is clear from our assumption that

Then there exists a sequence $r_n \to 0$ such that

We define

so that

Moreover, by a change of variable in (6.10), we get

Claim: For $R = 2^m$ and $r>0$, we have

Indeed, by definition, for every $\overline {r} > r>0,$ we have

and thus, using the monotonicity of $\Theta$, for every $x\in B_r^+$ we get

Letting $F_{k, r}^u = \sum_{j = 1}^k a_j(r)F_j$ and $\gamma_j = \frac{2j-1}{2}$, by taking the $L^2(B_r^+)$-norm in (6.15), we get

Then

This proves the claim.

From the definition of $\Theta$ and (6.14), for $R = 2^m\geq 1$, we have

Consequently, letting $R\geq 1$ and $m_0\in \mathbb{N}$ be the smallest integer such that $2^{m_0}\geq R$, we obtain that

with $C$ being a positive constant independent of $R$. Thanks to (1.3) and (6.11), it is plain that

By assumption, we have that $\frac{r_n^{2-\gamma_k-\alpha }}{\Theta(r_n)} f (r_n x)$ and $\frac{r_n^{1-\gamma_k-\alpha }}{\Theta(r_n)} g (r_n x_1)$ are bounded in $L^\infty(B_M^+)$ and $L^\infty(\Gamma ^M_n)$ respectively, for every $M>0$. Hence, by Proposition 6.1 and (6.16), we have that $v_n$ is bounded in $C^{\delta }(\overline{B_M^+})$ for every $M>0$ and $\delta \in(0, 1/2)$. Furthermore, it is easy to verify that $v_n$ is bounded in $H^1(B_M^+)$ for every $M>0$. Then, for every $M>0$ and $\delta \in(0, 1/2)$, $v_n$ converges in $C^{\delta }(\overline{B_M^+})$ (and weakly in $H^1(B_M^+)$) to some $v\in C^{\delta }_{\rm loc}(\overline {\mathbb{R}^2_+})\cap H^1_{\rm loc}(\mathbb{R}^2_+)$ satisfying

and by (6.16), for every $R>1$,

By Lemma 6.3 (below), we deduce that necessarily

This clearly yields a contradiction when passing to the limit in (6.12) and (6.13).

The following Liouville type result was used in the proof of Proposition 6.2.

Lemma 6.3 (Liouville theorem). Let $v\in C(\overline {\mathbb{R}^2_+})\cap H^1_{\rm loc}(\mathbb{R}^2_+)$ satisfy

and, for some $\alpha \in (0, 1/2)$ and $C>0$,

where $\gamma_k = \frac{2k-1}{2}$, $k\in\mathbb{N}\setminus\{0\}$. Then

Proof. Arguing as in the proof of Lemma 5.4, we expand $v$ in Fourier series with respect to the orthonormal basis of $L^2(0, \pi)$ given in (3.2) as

where $\varphi_j(r) = \frac2\pi\int_{0}^\pi v(r\cos t, r\sin t) \cos\left(\tfrac{2j-1}{2}t\right)\, dt$. From assumption (6.17) and the Parseval identity we have that

It follows that

for some ${\rm const\, }>0$ independent of $j$ and $r$.

From the equation satisfied by $v$ it follows that the functions $\varphi_j$ satisfy

and then, for all $j\geq1$, there exist $c_1^j, c_2^j\in\mathbb{R}$ such that

The fact $v$ is continous and $v(0) = 0$ implies that $\varphi_j(r) = o(1)$ as $r\to 0^+$. As a consequence we have that $c_2^j = 0$ for all $j\geq1$. On the other hand (6.18) implies that $c_1^j = 0$ for all $j>k$. Therefore we conclude that

i.e. $v\in \mathcal{S}_k$.

7.   Asymptotics for $u$

We are now in position to prove Theorem 1.1.

Proof of Theorem 1.1. Let $w = u\circ\varphi^{-1}$, with $\varphi:\overline{\mathcal U_R}\to \overline{B_R^+}$ being the conformal map constructed in Section 2. Let $\gamma = \frac{2k_0-1}{2}$, with $k_0$ being as in Theorem 2.1. We define (recalling (5.3))

From Theorem 2.1 we have that there exists $\beta\neq0$ such that $\widetilde w^\tau\to \beta F_{k_0}$ in $H^1(B_r^+)$ for all $r>0$ and in $C^{0, \mu}_{\rm loc}(\overline{\mathbb{R}^2_+}\setminus\{0\})$ for every $\mu\in(0, 1)$.

Claim 1: We have

If this does not hold true then there exists a sequence of points $y_m\in (B_R^+\cup \Gamma _n^R)\setminus \{0\}$ and $C>0$ such that $y_m\to 0$ and

where $\tau _m = |y_m|$ and $z_m = \frac{y_m}{|y_m|}$. If $m$ is large enough, we get a contradiction with (2.8). This proves (7.1) as claimed.

Let $\varrho\in(0, 1/2)$ and let $p$ and $q$ be the functions introduced in (2.8). This proves (7.1), by the fact that $p\in L^\infty(B_R^+)$ and $q\in C^1([0, R))$, and by Proposition 6.2 applied to $w$, we have that, for every $r\in (0, R)$,

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

Claim 2: We have

Once this claim is proved, then according to (7.2), we can easily deduce that for any $r\in (0, R)$

In particular,

which finishes the proof of Theorem 1.1.

Let us now prove Claim 2. Writing $F_{k_0, r}^w(x) = \sum_{j = 1}^{k_0}a_j(r) F_j(x)$, by (7.3) we have that

Moreover by taking the $L^2(B^+_r)$-norms in (7.3), we find that

with $\gamma_j = \frac{2j-1}{2}$. This yields, for $j = 1, \ldots , k_0-1$,

From (7.2), we get, for every $x\in B_r^+$ and $R>r>0$,

Hence, for every $x\in B_{r/2}^+$, we have that

Taking the $L^2(B^+_{r/2})$-norms in the previous inequality, we find that, for every $r\in (0, R)$

This implies that

From this, (7.5) and (7.6), we obtain

This implies that, for every $x\in B_r^+$,

That is (7.4) as claimed.

Remark 7.1. (ⅰ) Since $\varphi$ is conformal, we have that ${\tilde F }: = F_{k_0}\circ\varphi$ satisfies ${\tilde F }\in H^1 (\mathcal{U}_R)$ and solves the homogeneous equation

(ⅱ) Let $\Upsilon: U^+: = \mathcal{B}\cap U\to B_\rho^+$ define a $C^2$ parametrization (e.g. given by a system of Fermi coordinates), for some open neighborhood $U$ of 0, with $\Upsilon(0) = 0$, $D \Upsilon(0) = Id$, $\Upsilon(\Gamma _n\cap U)\subset \Gamma _n^\rho$ and $\Upsilon(\Gamma _d\cap U)\subset \Gamma _d^\rho$. By Theorem 1.1, for every $\varrho\in (0, 1/2)$, there exist $C, \rho_0>0$ such that

with $\alpha>0$ as in (2.3). Indeed, to see this, we first observe that (7.8) is equivalent to

for some constant $c>0$. We then further note that

and thus

in a neighborhood of $0$, where $c>0$ is a positive constant independent of $x$ possibly varying from line to line. This, together with (1.4) and the triangular inequality, gives (7.9).

Proof of Corollary 1.3. From Theorem 1.1 and (7.8) it follows that, if $u\in H^1(\Omega)$ is a non-trivial solution to (1.1), then there exist $k_0\in \mathbb{N}\setminus\{0\}$ and $\beta\in\mathbb{R}\setminus\{0\}$ such that, for every $t\in[0, \pi)$,

Therefore, if $u\geq0$, we have that necessarily $k_0 = 1$ so that statement (ⅰ) follows. Moreover, (7.8) implies that

which easily provides statement (ⅱ).

Proof of Corollary 1.4. Let us assume by contradiction that $u\not\equiv 0$. Then, Theorem 1.1 and (7.8) imply that (7.10) holds for every $t\in[0, \pi)$ and for some $k_0\in \mathbb{N}\setminus\{0\}$ and $\beta\in\mathbb{R}\setminus\{0\}$. Taking $n>\frac{2k_0-1}2$, (7.10) contradicts the assumption that $u(x) = O(|x|^n)$ as $|x|\to 0$.

8.   An example

In this section we show that the presence of a logarithmic term in the asymptotic expansion cannot be excluded without assuming enough regularity of the boundary.

Let us define in the Gauss plane the set

and the holomorphic function $\eta:A\to\mathbb{C}$ defined as follows:

Let us consider the holomorphic function

and the set

If $z = re^{i\theta }$ with $r>0$, $\theta \in \left(-\frac \pi2, \frac{3\pi}2\right)\setminus \{-\frac \pi 4, 0, \frac \pi 4, \frac{\pi}2, \frac{3\pi}4, \pi, \frac{5\pi}4\}$, then $z\in \mathcal Z$ if and only

For some fixed $\sigma\in \left(0, \frac\pi 2\right)$, we define the curve $\Gamma _+\subset \mathcal Z$ parametrized by

If we choose $\sigma>0$ sufficiently small then $\Gamma _+$ is the graph of a function $h_+$ defined in a open right neighborhood $U_+$ of $0$. Moreover $h_+$ is a Lipschitz function in $U_+$, $h_+\in C^2(U_+)$ and

Then we define the harmonic function

In polar coordinates the function $u$ reads

From (8.1–8.2) and (8.6) we deduce that $u$ vanishes on $\Gamma _+$.

The next step is to find a curve $\Gamma _-$ on which $\frac{\partial u}{\partial\nu} = 0$ where $\nu = (\nu_1, \nu_2)$ is the unit normal to $\Gamma _-$ satisfying $\nu_2\le 0$. We observe that

From direct computation we obtain

We now define

on the set $B_1\cap \Pi_-$ where $\Pi_-: = \{(x_1, x_2)\in\mathbb{R}^2:x_1 < 0\}$. One can easily check that $H_1, H_2$ admit continuous extensions defined on $B_1\cap \overline \Pi_-$ which we still denote by $H_1$ and $H_2$ respectively. We also observe that $H_1, H_2\in C^1(B_1\cap \overline \Pi_-)$. Therefore $H_1, H_2$ may be extended also on the right of the $x_2$-axis up to restrict them to a disk of smaller radius. For example one may define

for any $(x_1, x_2)\in B_{1/2}\cap \Pi_+$ where we put $\Pi_+: = \{(x_1, x_2)\in\mathbb{R}^2:x_1>0\}$. One may check that the new functions $H_1, H_2$ belong to $C^1(B_{1/2})$.

We can now define the functions $V_1, V_2:B_{1/2}\to \mathbb{R}$ by

One may verify that $V_1, V_2\in C^1(B_{1/2})$. 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 $\Gamma _-$ as the stable manifold of (8.7) at $(0, 0)$ intersected with $B_\varepsilon \cap \Pi_-$ where $\varepsilon\in (0, \frac 12)$ can be chosen sufficiently small in such a way that $\Gamma _-$ 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 $h:U_+\cup U_-\cup \{0\}\to \mathbb{R}$ such that $h\equiv h_+$ on $U_+$, $h\equiv h_-$ on $U_-$ and $h(0) = 0$.

Then we introduce a positive number $R$ sufficiently small and a domain $\Omega \subseteq B_R$ such that

One can easily check that the function $u$ defined in (8.5) belongs to $H^1(\Omega)$. From the above construction, we deduce that $u = 0$ on $\Gamma _+\cap \partial\Omega$ and $\frac{\partial u}{\partial \nu} = 0$ on $\Gamma _-\cap \partial\Omega$. We observe that $\partial\Omega$ admits a corner at $0$ of amplitude $\frac{3\pi}4$.

The presence of a logarithmic term in $u$ can be explained since the $C^{2, \delta }$-regularity assumption is not satisfied from the right, i.e. $h_{|U_+\cup \{0\}}\not\in C^{2, \delta }(U_+\cup \{0\})$ for any $\delta \in (0, 1)$. To see this, it is sufficient to study the behavior of $h(x_1)-x_1 h'(x_1)$ in a right neighborhood of zero.

By (8.3) we know that $\theta \in \big(-\frac{\pi}2, 0\big)$ and hence, if $x_1$ belongs to a sufficiently small right neighborhood of $0$, by (8.2) we have

By (8.4) and (8.8) we have that, as $x_1\to 0^+$,

Differentiating both sides of (8.8) and multiplying by $x_1^2+(h_+(x_1))^2$ we obtain the identity

and hence (8.4) and (8.9) yield

This shows that $h_+ \not \in C^2(U_+\cup\{0\})$ (and a fortiori cannot be extended to be of class $C^{2, \delta }$).

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 $\Omega$ and the function $u$ defined above and to apply a suitable deformation in order to remove the angle. We recall that $\Omega$ exhibits a corner at $0$ whose amplitude is $\frac{3\pi}4$.

For this reason, we define $F:\mathbb{C}\setminus\{ix_2:x_2\le 0\}\to \mathbb{C}$ by

We observe that, up to shrink $R$ if necessary, the map $F:\Omega \to F(\Omega)$ is invertible so that we may define $\widetilde\Omega : = F(\Omega)$ and $\widetilde u:\widetilde \Omega \to \mathbb{R}$, $\widetilde u(y_1, y_2): = u(F^{-1}(y_1, y_2))$ for any $(y_1, y_2)\in \widetilde\Omega$.

We also define the curves $\widetilde\Gamma _+: = F(\Gamma _+)$ and $\widetilde\Gamma _-: = F(\Gamma _-)$. Up to shrink $R$ if necessary, we may assume that $\widetilde\Gamma _+$ and $\widetilde\Gamma _-$ are respectively the graphs of two functions $\widetilde h_+$ and $\widetilde h_-$.

It is immediate to verify that $\widetilde u = 0$ on $\widetilde\Gamma _+$. We also prove that $\frac{\partial\widetilde u}{\partial \nu} = 0$ on $\widetilde\Gamma _-$. To avoid confusion with the notion of normal unit vectors to $\Gamma _-$ and $\widetilde\Gamma _-$ we denote them respectively with $\nu_{\Gamma _-}$ and $\nu_{\widetilde\Gamma _-}$. Since $\widetilde u$ is still harmonic, $\frac{\partial u}{\partial \nu_{\Gamma _-}} = 0$ on $\Gamma _-$ and $F$ is a conformal mapping, for any $\widetilde \varphi\in C^\infty_c(\widetilde\Omega \cup \widetilde \Gamma _-)$, we have

where we put $\varphi(x) = \widetilde \varphi(F(x))$. This proves that $\frac{\partial\widetilde u}{\partial \nu_{\widetilde \Gamma _-}} = 0$ on $\widetilde \Gamma _-$.

Finally we prove for $\widetilde h_+$ an estimate similar to (8.11).

From the definition of $F$ it follows that $\widetilde\Gamma _+$ admits a representation in polar coordinates of the type

Proceeding exactly as for (8.8)-(8.9) one can prove that

As we did for $h_+$, also for the function $\widetilde h_+$ one can prove that

By (8.14) we have

Differentiating both sides of (8.13) and multiplying by $x_1^2+(\widetilde h_+(x_1))^2$ we obtain the identity

By (8.14), (8.15) and (8.16), we obtain

The above arguments show that $\partial\widetilde\Omega$ is of class $C^1$ but not of class $C^{1, \delta }$ (and a fortiori not of class $C^{2, \delta }$).

Acknowledgments

M.M. Fall is supported by the Alexander von Humboldt foundation. V. Felli is partially supported by the PRIN2015 grant "Variational methods, with applications to problems in mathematical physics and geometry". A. Ferrero is partially supported by the PRIN2012 grant "Equazioni alle derivate parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni" and by the Progetto di Ateneo 2016 of the University of Piemonte Orientale "Metodi analitici, numerici e di simulazione per lo studio di equazioni differenziali a derivate parziali e applicazioni". A. Ferrero and V. Felli are partially supported by the INDAM-GNAMPA 2017 grant "Stabilitàe analisi spettrale per problemi alle derivate parziali".

Conflict of interest

All authors declare no conflicts of interest in this paper.