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Research article

The combined effect of calcium, pectin methylesterase and mild heat on frozen mango quality

  • Received: 02 August 2018 Accepted: 22 November 2018 Published: 26 November 2018
  • Both single and mixed solutions of calcium and pectin methylesterase were studied to examine how they improve the quality of frozen mango. The result showed that calcium and pectin methylesterase had a synergistic effect on the mango, helping to improve firmness and decrease the drip loss of the frozen fruit. To increase the efficiency of the mixed solution of pectin methylesterase and calcium, this study examined three immersion temperatures (25, 40 and 50 °C). Pectin methylesterase activity, calcium content, -carotene content, firmness, drip loss and color of the samples were evaluated. The treatment at 40 °C caused high pectin methylesterase activity and calcium content. The firmness of the frozen-thawed mango increased two-fold from 1.25 to 2.50 N, and the drip loss decreased from 15.19 to 9.87 g/100 g when compared with the control samples. Moreover, the frozen-thawed mango that was immersed in pectin methylesterase and calcium at 40 °C had the highest lightness and hue value and the lowest browning index.

    Citation: Arpassorn Sirijariyawat, Panita Ngamchuachit, Rungkan Boonnattakorn, Soraya Saenmuang. The combined effect of calcium, pectin methylesterase and mild heat on frozen mango quality[J]. AIMS Agriculture and Food, 2018, 3(4): 455-466. doi: 10.3934/agrfood.2018.4.455

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  • Both single and mixed solutions of calcium and pectin methylesterase were studied to examine how they improve the quality of frozen mango. The result showed that calcium and pectin methylesterase had a synergistic effect on the mango, helping to improve firmness and decrease the drip loss of the frozen fruit. To increase the efficiency of the mixed solution of pectin methylesterase and calcium, this study examined three immersion temperatures (25, 40 and 50 °C). Pectin methylesterase activity, calcium content, -carotene content, firmness, drip loss and color of the samples were evaluated. The treatment at 40 °C caused high pectin methylesterase activity and calcium content. The firmness of the frozen-thawed mango increased two-fold from 1.25 to 2.50 N, and the drip loss decreased from 15.19 to 9.87 g/100 g when compared with the control samples. Moreover, the frozen-thawed mango that was immersed in pectin methylesterase and calcium at 40 °C had the highest lightness and hue value and the lowest browning index.


    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 Ω be an open subset of R2 with Lipschitz boundary. Let ΓnΩ and ΓdΩ be two nonconstant curves (open in Ω) such that ¯Γn¯Γd={P} for some PΩ. We are interested in regularity of weak solutions uH1(Ω) to the mixed boundary value problem

    {Δu=f(x)u,in Ω,νu=g(x)u,on Γn,u=0,on Γd, (1.1)

    with fL(Ω) and gC1(¯Γn), see Section 2 for the weak formulation. Our aim is to prove unique continuation properties from the Dirichlet-Neumann junction {P}=¯Γn¯Γd and sharp asymptotics of nontrivial solutions near P provided Ω is of class C2,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 [16,25], 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¯Γn¯Γd, we will need to assume that Ω is of class C2,δ near P. The asymptotic profile of the solution will be given by the function

    Fk(rcosθ,rsinθ)=r2k12cos(2k12θ),r>0, θ(0,π), (1.2)

    for some kN{0}. We note that FkH1loc(R2) and solves the equation

    {ΔFk=0,in R2+,Fk(x1,0)=0, for x1<0,x2Fk(x1,0)=0, for x1>0, (1.3)

    where here and in the following R2+:={(x1,x2)R2:x2>0}.

    The main result of the present paper provides an evaluation of the local behavior of weak solutions uH1(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 ΩR2 are satisfied. Here and in the remaining of this paper, Γn,ΓdΩ are nonconstant curves (open as subsets of Ω) such that ¯Γn¯Γd={0} with 0Ω.

    (ⅰ) The domain Ω is of class C2,δ in a neighborhood of 0, for some δ>0.

    (ⅱ) The unit vector e1:=(1,0) is tangent to Ω at 0 and pointed towards Γn. Moreover, the exterior unit normal vector to Ω 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 Ω satisfies the assumptions (ⅰ)-(ⅱ) above. Let uH1(Ω) be a nontrivial weak solution to (1.1), with fL(Ω) and gC1(¯Γn). Then, there exist k0N{0}, βR{0} and r>0 such that, for every ϱ(0,1/2), there exists C>0 such that

    |u(x)βFk0(φ(x))|C|x|2k012+ϱ,foreveryxΩ¯B+r. (1.4)

    Here, the function φ:¯ΩBr0¯R2+ is a conformal map of class C2, for some r0>0 only depending on Ω.

    Remark 1.2. Here and in the sequel, we identify R2 with the complex plane C; hence, by a conformal map on an open set UR2 we mean a holomorphic function with complex derivative everywhere non-zero on U. We notice that, if Ω satisfies (ⅰ)-(ⅱ) and φ:¯ΩBr0¯R2+ is conformal, then Dφ(0)=αId 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 1.3. Under the same assumptions as in Theorem 1.1, let uH1(Ω) be a non-trivial weak solution to (1.1), with u0. Then

    (ⅰ) for every t[0,π),

    limr0u(rcost,rsint)r1/2=βα1/2cos(t2)>0,

    where α=φ(0)>0 and φ is as in Theorem 1.1;

    (ⅱ) for every cone CR2 satisfying (1,0)C and (1,0)R2¯C, we have

    lim infx0x¯ΩCu(x)|x|1/2>0.

    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 uH1(Ω) be a weak solution to (1.1) such that u(x)=O(|x|n) as xΩ, |x|0, for any nN. Then u0.

    We observe that Theorem 1.1 provides a sharp asymptotic expansion (and consequently a unique continuation principle) at the boundary for 12-fractional elliptic equations in dimension 1. Indeed, if vH1/2(R) weakly solves

    {(Δ)1/2v=g(x)v,in (0,R),v=0,in R(0,R),

    for some gC1([0,R]), then its harmonic extension VH1loc(¯R2+) weakly solves

    {ΔV=0,in R2+,νV=g(x)V,on (0,R)×{0},V=0,on (R(0,R))×{0}, (1.5)

    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 C2,δ regularity on Ω and C1 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 C1-boundary which is not of class C2,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:

    uk0k=1ak(r)FkφL(Br)Cr2k012+ϱ, (1.6)

    for every ϱ(0,1/2), for some C>0, k01 and where ak(r)=u,FkφL2(Br)Fjφ2L2(Br). Now, if u is nontrivial, a blow-up analysis combined with Almgren type monotonicity formula allows to depict a k01 for which ak0(r)β0 and ak(r)0 for every k<k0 as r0. 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 B1Ω near 0 and state Theorem 2.1 giving the sharp asymptotic behaviour of its solutions. Section 3 contains some Hardy-Poincaré type inequalities for H1-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 C2,δ-regularity assumption.

    For every R>0 let BR={(x1,x2)R2:x21+x22<R2} and B+R={(x1,x2)BR:x2>0}. Since Ω is of class C2,δ near zero, we can find r0>0 such that Γ:=ΩBr0 is a C2,δ curve. Here and in the following, we let B be a C2,δ simply connected open bounded set such that BΩ and BΩ=Γ. For some functions

    fL(B)andgC1(¯Γn), (2.1)

    let uH1(B) be a solution to

    {Δu=f(x)u,in B,νu=g(x)u,on Γn,u=0,on Γd. (2.2)

    We introduce the space H10,Γd(B) as the closure in H1(B) of the subspace

    C0,Γd(B):={uC(¯B):u=0 on ΓdB}.

    We say that uH1(B) is a weak solution to (2.2) if

    {uH10,Γd(B),Bu(x)v(x)dx=Bf(x)u(x)v(x)dx+Γnguvdsfor any vC0,BΓn(B)

    where C0,BΓn(B)={uC(¯B):u=0 on BΓn}. Since B is of class C2,δ, in view of the Riemann mapping Theorem and [17,Theorem 5.2.4], there exists a conformal map ˆφ:¯B¯B1 which is of class C2. Let N=ˆφ(0)B1 and let S be its antipodal. We then consider the map ˜φ:R2{S}R2{¯S} given by ˜φ(z):=2¯zS|zS|2+¯S, where, for every zR2C, ¯z denotes the complex conjugate of z. This map is conformal and ˜φ(N)=0. In addition ˜φ(¯B1{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 ˜φˆφ is a conformal map which is of class C2 from a neighborhood of the origin ¯BBr into ¯P for some r>0. It is now clear that there exists a rotation R and a real number R>0 such that, letting UR:=φ1(B+R), the map φ:=R˜φˆφ:¯UR¯B+R is an invertible conformal map of class C2 with inverse φ1:¯B+R¯UR of class C2. Moreover φ(0)=0.

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

    Dφ(0)=αId,with α=φ(0)>0, (2.3)

    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, φ1((R,0)×{0})Γd and φ1((0,R)×{0})Γn.

    Therefore letting w=uφ1:B+RR and Ψ:=φ1, we then have that wH1(B+R) solves

    {Δw(z)=p(z)w(z),in B+R,νw(x1,0)=q(x1)w(x1,0),x1(0,R),w=0,on (R,0)×{0}, (2.4)

    with

    p(z)=|Ψ(z)|2f(Ψ(z)),q(x1)=(g(Ψ(x1,0))|Ψ(x1,0)|.

    It is plain that pL(B+R) and qC1([0,R)). Here and in the following, for every r>0, we define

    Γrn:=(0,r)×{0} and Γrd:=(r,0)×{0}. (2.5)

    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

    N(r)=B+r|w|2dzB+rpw2dzr0q(x)w2(x,0)dxπ0w2(rcost,rsint)dt.

    In Section 4 we will prove that N is well defined in the interval (0,R0) for some R0>0.

    Theorem 2.1. Let w be a nontrivial solution to (2.4). Then there exists k0N, k01, such that

    limr0+N(r)=2k012. (2.6)

    Furthermore

    τ2k012w(τz)β|z|2k012cos(2k012Argz)as τ0+

    strongly in H1(B+r) for all r>0 and in C0,μloc(¯R2+{0}) for every μ(0,1), where β0 and

    β=2ππ0R2k012w(Rcoss,Rsins)cos(2k012s)ds+2ππ0[R0tk0+3/2R12k0tk0+1/22k01p(tcoss,tsins)w(tcoss,tsins)dt]cos(2k012s)ds+2πR0t1/2k0R12k0tk01/22k01q(t)w(t,0)dt. (2.7)

    In particular

    τ2k012w(τcost,τsint)βcos(2k012t)in C0,μ([0,π])as τ0+. (2.8)

    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.

    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

    {ψ"=λψ,in [0,π],ψ(0)=0,ψ(π)=0. (3.1)

    It is easy to verify that (3.1) admits the sequence of (all simple) eigenvalues

    λk=14(2k1)2,kN, k1,

    with corresponding eigenfunctions

    ψk(t)=cos(2k12t),kN, k1.

    It is well known that the normalized eigenfunctions

    {2πcos(2k12t)}k1 (3.2)

    form an orthonormal basis of the space L2(0,π). Furthermore, the first eigenvalue λ1=14 can be characterized as

    λ1=14=minψH1(0,π){0}ψ(π)=0π0|ψ(t)|2dtπ0|ψ(t)|2dt. (3.3)

    For every r>0, we let (recall (2.5) for the definition of Γrd)

    Hr={wH1(B+r):w=0 onΓrd}.

    As a consequence of (3.3) we obtain the following Hardy-Poincaré inequality in Hr.

    Lemma 3.1. For every r>0 and wHr, we have that

    B+r|w(z)|2dz14B+r|w(z)|2|z|2dz.

    Proof. Let wC(¯B+r) with w=0 on ¯Γrd=[r,0]×{0}. Then, in view of (3.3),

    B+r|w(z)|2dz=r0π0ρ(|ρ(w(ρcost,ρsint))|2+1ρ2|t(w(ρcost,ρsint))|2)dtdρr01ρ(π0|t(w(ρcost,ρsint))|2dt)dρ14r01ρ(π0|w(ρcost,ρsint)|2dt)dρ=14B+r|w(z)|2|z|2dz.

    We conclude by density, recalling that the space of smooth functions vanishing on [r,0]×{0} is dense in Hr, see e.g. [7].

    Lemma 3.2. For every r>0 and wHr, we have that x11w2(x1,0)L1(0,r) and

    r0w2(x1,0)x1dx1πB+r|w(z)|2dz.

    Proof. Let wC(¯B+r) with w=0 on [r,0]×{0}. Then for any 0<x1<r

    |w(x1,0)|=|π0ddtw(x1cost,x1sint)dt|=|π0x1w(x1cost,x1sint)(sint,cost)dt|ππ0x21|w(x1cost,x1sint)|2dt.

    It follows that

    r0w2(x1,0)x1dx1πr0π0x1|w(x1cost,x1sint)|2dtdx1=πB+r|w(z)|2dz.

    We conclude by density.

    Let wH1(B+R) be a non trivial solution to (2.4). For every r(0,R] we define

    D(r)=B+r|w|2dzB+rpw2dzr0q(x1)w2(x1,0)dx1 (4.1)

    and

    H(r)=1rS+rw2ds=π0w2(rcost,rsint)dt, (4.2)

    where S+r:={(x1,x2):x21+x22=r2 and x2>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(0,R) we have

    r2S+r|w|2ds=rS+r|wν|2ds12r0(q(x1)+x1q(x1))w2(x1,0)dx1+r2q(r)w2(r,0)+B+rpwzwdz (4.3)

    and

    B+r|w|2dz=B+rpw2dz+S+rwνwds+r0q(x1)w2(x1,0)dx1. (4.4)

    Proof. We observe that, by elliptic regularity theory, wH2(B+rB+ε) for all 0<ε<r<R. Furthermore, the fact that w has null trace on ΓRd implies that wx1 has null trace on ΓRd. Then, testing (2.4) with zw and integrating over B+rB+ε, we obtain that

    r2S+r|w|2dsε2S+ε|w|2ds=B+rB+εpwzwdz+rS+r|wν|2dsεS+ε|wν|2ds+rεq(x1)w(x1,0)x1wx1(x1,0)dx1. (4.5)

    An integration by parts, which can be easily justified by an approximation argument, yields that

    rεq(x1)w(x1,0)x1wx1(x1,0)dx1=r2q(r)w2(r,0)ε2q(ε)w2(ε,0)12rε(q+x1q)w2(x1,0)dx1. (4.6)

    We observe that there exists a sequence εn0+ such that

    limn[εnw2(εn,0)+εnS+εn|w|2ds]=0.

    Indeed, if no such sequence exists, there would exist ε0>0 such that

    w2(r,0)+S+r|w|2dsCrfor all r(0,ε0),for some C>0;

    integration of the above inequality on (0,ε0) would then contradict the fact that wH1(B+R) and, by trace embedding, wL2(Γε0n). Then, passing to the limit in (4.5) and (4.6) with ε=εn 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. HW1,1loc(0,R) and

    H(r)=2π0w(rcost,rsint)wν(rcost,rsint)dt=2rS+rwwνds, (4.7)

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

    H(r)=2rD(r),fora.e.r(0,R). (4.8)

    Proof. Let ϕCc(0,R). Since w,wL2(B+R) and wC1(B+R), using twice Fubini's Theorem we obtain that

    R0H(r)ϕ(r)dr=R0(π0w2(rcost,rsint)dt)ϕ(r)dr=π0(R0w2(rcost,rsint)ϕ(r)dr)dt=π0(R0ddr(w2(rcost,rsint))ϕ(r)dr)dt=π0(R0(2w(rcost,rsint)wν(rcost,rsint))ϕ(r)dr)dt=R0(π0(2w(rcost,rsint)wν(rcost,rsint))dt)ϕ(r)dr

    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 W1,1(0,R) and

    D(r)=2S+r|wν|2ds1rr0(q(x1)+x1q(x1))w2(x1,0)dx1+2rB+rpwzwdzS+rpw2ds (4.9)

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

    Proof. From the fact that wH1(B+R) and w|ΓRnL2(ΓRn), we deduce that D belongs to W1,1(0,R) and

    D(r)=S+r|w|2dsS+rpw2dsq(r)w2(r,0) (4.10)

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

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

    Lemma 4.4. There exists R0(0,R) such that H(r)>0 for any r(0,R0).

    Proof. Let R0(0,R) be such that

    4pL(B+R)R20+πqL(ΓRn)R0<1. (4.11)

    Assume by contradiction that there exists r0(0,R0) such that H(r0)=0, so that w=0 a.e. on S+r0. From (4.4) it follows that

    B+r0|w|2dzB+r0pw2dzr00q(x1)w2(x1,0)dx1=0.

    From Lemmas 3.1 and 3.2, we get

    0=B+r0|w|2dzB+r0pw2dzr00q(x1)w2(x1,0)dx1[14pL(B+R)r20πqL(ΓRn)r0]B+r0|w|2dz,

    which, together with (4.11) and Lemma 3.1, implies w0 in B+r0. 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

    N:(0,R0)R,N(r)=D(r)H(r), (4.12)

    is well defined. Using Lemmas 4.2 and 4.3, we now compute the derivative of N.

    Lemma 4.5. The function N defined in (4.12) belongs to W1,1loc(0,R0) and

    N(r)=ν1(r)+ν2(r) (4.13)

    in a distributional sense and for a.e. r(0,R0), where

    ν1(r)=2r[(S+r|wν|2ds)(S+rw2ds)(S+rwwνds)2](S+rw2ds)2 (4.14)

    and

    ν2(r)=r0(q(x)+xq(x))w2(x,0)dxS+rw2ds+2B+rpwzwdzS+rw2dsrS+rpw2dsS+rw2ds. (4.15)

    Proof. From Lemmas 4.2, 4.4, and 4.3, it follows that NW1,1loc(0,R0). From (4.8) we deduce that

    N(r)=D(r)H(r)D(r)H(r)(H(r))2=D(r)H(r)12r(H(r))2(H(r))2

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

    We now prove that N(r) admits a finite limit as r0+.

    Lemma 4.6. There exists γ[0,+) such that limr0+N(r)=γ.

    Proof. From Lemmas 3.1 and 3.2 it follows that

    D(r)[14pL(B+R)r2πqL(ΓRn)r]B+r|w|2dz,

    hence there exist ˉr(0,R0) and C1>0 such that

    D(r)C1B+r|w|2dz,for all r(0,ˉr).

    In particular

    N(r)0,for all r(0,ˉr). (4.16)

    Moreover, using again Lemmas 3.1 and 3.2 we can estimate ν2 in (0,ˉr) as follows

    |ν2(r)|q+xqL(ΓRn)πrB+r|w|2dzS+rw2ds+pL(B+R)r(1+4r2)B+r|w(z)|2dzS+rw2ds+rpL(B+R)1C1(q+xqL(ΓRn)π+pL(B+R)(1+4ˉr2))N(r)+ˉrpL(B+R). (4.17)

    Since ν10 by Schwarz's inequality, from Lemma 4.5 and the above estimate it follows that there exists C2>0 such that

    N(r)C2(N(r)+1)for all r(0,ˉr), (4.18)

    which implies that

    ddr(eC2r(1+N(r)))0.

    It follows that the limit of reC2r(1+N(r)) as r0+ exists and is finite; hence the function N has a finite limit γ as r0+. From (4.16) we deduce that γ0.

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

    Lemma 4.7. Let γ:=limr0+N(r) be as in Lemma 4.6. Then

    H(r)=O(r2γ)asr0+. (4.19)

    Moreover, for any σ>0,

    r2γ+σ=O(H(r))asr0+. (4.20)

    Proof. From Lemma 4.6 we have that

    Nis bounded in a neighborhood of0, (4.21)

    hence from (4.18) it follows that NC3 for some positive constant C3 in a neighborhood of 0. Then

    N(r)γ=r0N(ρ)dρC3r (4.22)

    in a neighborhood of 0. From (4.8), (4.12), and (4.22) we deduce that, in a neighborhood of 0,

    H(r)H(r)=2N(r)r2γr2C3,

    which, after integration, yields (4.19).

    Since γ=limr0+N(r), for any σ>0 there exists rσ>0 such that N(r)<γ+σ/2 for any r(0,rσ) and hence H(r)H(r)=2N(r)r<2γ+σr for all r(0,rσ). By integration we obtain (4.20).

    Lemma 5.1. Let wH1(B+R) be a non trivial solution to (2.4). Let γ:=limr0+N(r) be as in Lemma 4.6. Then there exists k0N, k01, such that

    γ=2k012.

    Furthermore, for every sequence τn0+, there exist a subsequence {τnk}kN such that

    w(τnkz)H(τnk)˜w(z) (5.1)

    strongly in H1(B+r) and in C0,μloc(¯B+r{0}) for every μ(0,1) and all r(0,1), where

    ˜w(rcost,rsint)=±2πr2k012cos(2k012t),forallr(0,1)andt[0,π]. (5.2)

    Proof. Let us set

    wτ(z)=w(τz)H(τ). (5.3)

    We notice that, for all τ(0,R), wτH1 and S+1|wτ|2ds=π0|wτ(cost,sint)|2dt=1. Moreover, by scaling and (4.21),

    B+1(|wτ(z)|2τ2p(τz)|wτ(z)|2)dzτ10q(τx)|wτ(x,0)|2dx=N(τ)=O(1) (5.4)

    as τ0+, whereas from Lemmas 3.1 and 3.2 it follows that

    N(τ)1H(τ)[14pL(B+R)τ2πqL(ΓRn)τ]B+τ|w|2dz=[14pL(B+R)τ2πqL(ΓRn)τ]B+1|wτ|2dz (5.5)

    for every τ(0,R0), being R0 as in (4.11). From (5.4), (5.5), and Lemma 3.1 we deduce that

    {wτ}τ(0,R0)is bounded in H1(B+1). (5.6)

    Therefore, for any given sequence τn0+, there exists a subsequence τnk0+ such that wτnk˜w weakly in H1(B+1) for some ˜wH1(B+1). Due to compactness of trace embeddings, we have that ˜w=0 on Γ1d and

    S+1|˜w|2ds=1. (5.7)

    In particular ˜w0. For every small τ(0,R0), wτ satisfies

    {Δwτ=τ2p(τz)wτ,in B+1,νwτ=τq(τx1,0)wτ,on Γ1n,wτ=0,on Γ1d, (5.8)

    in a weak sense, i.e.

    B+1wτ(z)φ(z)dz=τ2B+1p(τz)wτ(z)φ(z)dz+τ10q(τx)wτ(x,0)φ(x,0)dx

    for all φH1(B+1) s.t. φ=0 on S+1Γ1d. From weak convergence wτnk˜w in H1(B+1), we can pass to the limit in (5.8) along the sequence τnk and obtain that ˜w weakly solves

    {Δ˜w=0,in B+1,ν˜w=0,on Γ1n,˜w=0,on Γ1d. (5.9)

    From (5.6) it follows that {τq(τx)wτ(x,0)}τ(0,R0) is bounded in H1/2(Γ1n). Then, by elliptic regularity theory, for every 0<r1<r2<1 we have that {wτ}τ(0,R0) is bounded in H2(B+r2¯B+r2). From compactness of trace embeddings we have that, up to passing to a further subsequence, wτnkν˜wν in L2(S+r) for every r(0,1). Testing equation (5.8) for τ=τnk with wτ on B+r we obtain that

    B+r|wτnk(z)|2dz=S+rwτnkνwτnkds+τ2nkB+rp(τnkz)|wτnk(z)|2dz+τnkr0q(τnkx)|wτnk(x,0)|2dxk+S+r˜wν˜wds=B+r|˜w(z)|2dz,

    thus proving that wτnkH1(B+r)˜wH1(B+r) for all r(0,1), and hence

    wτnk˜win H1(B+r) (5.10)

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

    wτnk˜win C0,μloc(¯B+r{0}),

    for every r(0,1) and μ(0,1).

    For any r(0,1) and kN, let us define the functions

    Dk(r)=B+r|wτnk|2dzτ2nkB+rp(τnkz)|wτnk(z)|2dzτnkr0q(τnkx)|wτnk(x,0)|2dx,Hk(r)=1rS+r|wτnk|2ds,

    and Nk(r):=Dk(r)Hk(r). Direct calculations yield that Nk(r)=N(τnkr) for all r(0,1). From (5.10) it follows that, for any fixed r(0,1),

    Dk(r)˜D(r):=B+r|˜w|2dzandHk(r)˜D(r):=1rS+r|˜w|2ds.

    From classical unique continuation principles for harmonic functions it follows that ˜D(r)>0 and ˜H(r)>0 for all r(0,1) (indeed ˜D(r)=0 or ˜H(r)=0 for some r(0,1) would imply that ˜w0 in B+r and, by unique continuation, ˜w0 in B+1, a contradiction). Hence, by Lemma 4.6,

    ˜N(r)=˜D(r)˜H(r)=limkNk(r)=limkN(τnkr)=γ (5.11)

    for all r(0,1). Therefore ˜N is constant in (0,1) and hence ˜N(r)=0 for any r(0,1). By (5.9) and Lemma 4.5 with p0 and q0, we obtain

    (S+r|˜wν|2ds)(S+r˜w2ds)(S+r˜w˜wνds)2=0for all r(0,1),

    which implies that ˜w and ˜wν are parallel as vectors in L2(S+r). Hence there exists η=η(r) such that ˜wν(rcost,rsint)=η(r)˜w(rcost,rsint) for all r(0,1) and t[0,π]. It follows that

    ˜w(rcost,rsint)=φ(r)ψ(t),r(0,1), t[0,π], (5.12)

    where φ(r)=er1η(s)ds and ψ(t)=˜w(cost,sint). From (5.7) we have that π0ψ2=1. From (5.9) and (5.12) we can conclude that

    {φ"(r)ψ(t)+1rφ(r)ψ(t)+1r2φ(r)ψ"(t)=0,r(0,1),t[0,1],ψ(π)=0,ψ(0)=0.

    Taking r fixed, we deduce that ψ is necessarily an eigenfunction of the eigenvalue problem (3.1). Then there exists k0N{0} such that ψ(t)=±2πcos(2k012t) and φ(r) solves the equation

    φ"(r)+1rφ(2k01)24r2φ(r)=0.

    Hence φ(r) is of the form

    φ(r)=c1r2k012+c2r2k012

    for some c1,c2R. Since the function r2k012ψ(t)H1(B+1), we deduce that necessarily c2=0 and φ(r)=c1r2k012. Moreover, from φ(1)=1, we obtain that c1=1 and then

    ˜w(rcost,rsint)=±2πr2k012cos(2k012t),for all r(0,1) and t[0,π]. (5.13)

    From (5.13) it follows that

    ˜H(r)=π0˜w2(rcost,rsint)dt=r2k01.

    Hence, in view of (4.8),

    γ=˜N(r)=r2˜H(r)˜H(r)=r2(2k01)r2k02r2k01=2k012.

    The proof of the lemma is thereby complete.

    We observe that at this stage of our analysis we cannot exclude that the limit function ˜w found in Lemma 5.1 depends on the subsequence. In order to prove that the convergence in (5.1) actually holds as τ0+ we need to univocally identify the limit profile ˜w.

    Lemma 5.2. Let w0 satisfy (2.4), H be defined in (4.2), and γ:=limr0+N(r) be as in Lemma 4.6. Then the limit limr0+r2γH(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

    ddrH(r)r2γ=2r2γ1(D(r)γH(r))=2r2γ1H(r)r0N(ρ)dρ,

    and then, by integration over (r,R0),

    H(R0)R2γ0H(r)r2γ=2R0rH(ρ)ρ2γ+1(ρ0ν1(t)dt)dρ+2R0rH(ρ)ρ2γ+1(ρ0ν2(t)dt)dρ (5.14)

    where ν1 and ν2 are as in (4.14) and (4.15). Since, by Schwarz's inequality, ν10, we have that limr0+R0rρ2γ1H(ρ)(ρ0ν1(t)dt)dρ exists. On the other hand, from Lemma 4.6 N is bounded and hence from (4.17) we deduce that ν2 is bounded close to 0+. Hence, in view of (4.19), the function ρρ2γ1H(ρ)(ρ0ν2(t)dt) is bounded and hence integrable near 0. We conclude that both terms at the right hand side of (5.14) admit a limit as r0+ thus completing the proof.

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

    Lemma 5.3. Let wH1(B+R) be a nontrivial solution to (2.4). Then there exist C4,C5>0 and ˉr(0,R0) such that

    (ⅰ) supS+r|w|2C4rS+r|w(z)|2ds for every 0<r<ˉr,

    (ⅱ) |w(z)|C5|z|γ for all zB+ˉr, 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 τn0+ such that

    supt[0,π]|w(τn2cost,τn2sint)|2>nH(τn2)

    with H as in (4.2), i.e., defining wτ as in (5.3)

    supxS+1/2|wτn(z)|2>2nS+1/2|wτn(z)|2ds. (5.15)

    From Lemma 5.1, there exists a subsequence τnk such that wτnk˜w in C0(S+1/2) with ˜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 r0+, it remains to prove that limr0+r2γH(r) is strictly positive.

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

    limr0+r2γH(r)>0.

    Proof. From Lemma 5.1 there exists k0N, k01 such that γ=2k012. Let us expand w as

    w(rcost,rsint)=k=1φk(r)cos(2k12t) (5.16)

    where

    φk(r)=2ππ0w(rcost,rsint)cos(2k12t)dt. (5.17)

    The Parseval identity yields

    H(r)=π2k=1φ2k(r),for all 0<rR. (5.18)

    From (4.19) and (5.18) it follows that, for all k1,

    φk(r)=O(rγ)as r0+. (5.19)

    Let ηCc(0,R). Testing (2.4) with the function η(r)cos(2k12t), by (5.16) we obtain

    π2R0rφk(r)η(r)dr+π2R0(2k1)241rφk(r)η(r)dr=R0q(r)w(r,0)η(r)dr+R0rη(r)(π0p(rcost,rsint)w(rcost,rsint)cos(2k12t)dt)dr. (5.20)

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

    φk"(r)1rφk(r)+14(2k1)2φk(r)r2=ζk(r),in (0,R),

    where

    ζk(r)=2πrq(r)w(r,0)+2ππ0p(rcost,rsint)w(rcost,rsint)cos(2k12t)dt. (5.21)

    Then, by a direct calculation, there exist ck1,ck2R such that

    φk(r)=r2k12(ck1+Rrt12k2+12k1ζk(t)dt)+r12k2(ck2+Rrt2k12+112kζk(t)dt). (5.22)

    From Lemma 5.3 it follows that

    ζk0(r)=O(r2k0121)as r0+, (5.23)

    and hence the functions

    tt12k02+1ζk0(t)andtt2k012+1ζk0(t)

    belong to L1(0,R). Hence

    r2k012(ck01+Rrt12k02+12k01ζk0(t)dt)=o(r12k02)as r0+,

    and then, by (5.19), there must be

    ck02=R0t2k012+12k01ζk0(t)dt.

    From (5.23), we then deduce that

    r12k02(ck02+Rrt2k012+112k0ζk0(t)dt)=r12k02r0t2k012+12k01ζk0(t)dt=O(rk0+12) (5.24)

    as r0+. From (5.22) and (5.24), we obtain that

    φk0(r)=r2k012(ck01+Rrt12k02+12k01ζk0(t)dt+O(r))as r0+. (5.25)

    Let us assume by contradiction that limr0+r2γH(r)=0. Then (5.18) would imply that

    limr0+r2k012φk0(r)=0,

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

    ck01+R0t12k02+12k01ζk0(t)dt=0,

    which, together with (5.23), implies

    r2k012(ck01+Rrt12k02+12k01ζk0(t)dt)=r2k012r0t12k02+112k0ζk0(t)dt=O(r12+k0) (5.26)

    as r0+. From (5.25) and (5.26), we conclude that φk0(r)=O(r12+k0) as r0+, namely,

    H(τ)π0wτ(cost,sint)cos(2k012t)dt=O(τ12+k0)as τ0+,

    where wτ is defined in (5.3). From (4.20), there exists C>0 such that H(τ)Cτγ+12 for τ small, and therefore

    π0wτ(cost,sint)cos(2k012t)dt=O(τ12)as τ0+. (5.27)

    From Lemma 5.1, for every sequence τn0+, there exist a subsequence {τnk}kN such that

    wτnk(cost,sint)±2πcos(2k012t)in L2(0,π). (5.28)

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

    0=limk+π0wτnk(cost,sint)cos(2k012t)dt=±2ππ0cos2(2k012t)dt=±π2,

    thus reaching a contradiction.

    Proof of Theorem 2.1. Identity (2.6) follows from Lemma 5.1, thus there exists k0N, k01, such that γ=limr0+N(r)=2k012.

    Let {τn}nN(0,+) be such that limn+τn=0. Then, from Lemmas 5.1 and 5.4, scaling and a diagonal argument, there exists a subsequence {τnk}kN and β0 such that

    w(τnkz)τγnkβ|z|2k012cos(2k012Argz) (5.29)

    strongly in H1(B+r) for all r>0 and in C0,μloc(¯R2+{0}) for every μ(0,1). In particular

    τγnkw(τnk(cost,sint))βcos(2k012t) (5.30)

    in C0,μ([0,π]). To prove that the above converge occurs as τ0+ and not only along subsequences, we are going to show that β depends neither on the sequence {τn}nN nor on its subsequence {τnk}kN.

    Defining φk0 and ζk0 as in (5.17) and (5.21), from (5.30) it follows that

    φk0(τnk)τγnk=2ππ0w(τnkcost,τnksint)τγnkcos(2k012t)dt2πβπ0cos2(2k012t)dt=β (5.31)

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

    φk0(τ)=τ2k012(ck01+Rτt12k02+12k01ζk0(t)dt)+τ12k02τ0t2k012+12k01ζk0(t)dt=τ2k012(ck01+Rτt12k02+12k01ζk0(t)dt+O(τ))as τ0+. (5.32)

    Choosing τ=R in the first line of (5.32), we obtain

    ck01=R2k012φk0(R)R12k0R0t2k012+12k01ζk0(t)dt.

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

    τγφk0(τ)R2k012φk0(R)R12k0R0t2k012+12k01ζk0(t)dt+R0t12k02+12k01ζk0(t)dt,

    as τ0+. Then, from (5.31) we deduce that

    β=R2k012φk0(R)R12k0R0t2k012+12k01ζk0(t)dt+R0t12k02+12k01ζk0(t)dt. (5.33)

    In particular β depends neither on the sequence {τn}nN nor on its subsequence {τnk}kN, thus implying that the convergence in (5.29) actually holds as τ0+ and proving the theorem. We observe that (2.7) follows by replacing (5.17) and (5.21) into (5.33).

    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 βFk0.

    Proposition 6.1. Let fL(B+4), gL(Γ4n) and let vH1(B4)L(B+4) solve

    {Δv=f,inB+4,νv=g,onΓ4n,v=0,onΓ4d. (6.1)

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

    vC1/2ε(¯B+2)C(fL(B+4)+gL(Γ4n)+vL(B+4)).

    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 C2 domain Ω such that B+3ΩB+4 and Γ3nΓ3dΩ. We define the functions (obtained uniquely by minimization arguments) v1H1(Ω) satisfying

    {Δv1=f,in Ω,νv1=0, on Γ3n ,v1=0, on ΩΓ3n , (6.2)

    and ˜v2H1/2(R) satisfying

    {(Δ)12˜v2=g,in (0,4),˜v2=0,onR(0,4).

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

    ˜v2C1/2(R)CgL(Γ4n). (6.3)

    Consider the Poisson kernel P(x1,x2)=1πx2|x|2 with respect to the half-space R2+, see [4,Section 2.4]. We define

    v2(x1,x2)=(P(,x2)˜v2)(x1)=1πx2R˜v2(t)x22+(x1t)2dt=1πR˜v2(x1rx2)1+r2dr

    where with the symbol we denoted the convolution product with respect to the first variable. One can check that v2H1loc(¯R2+) (see for example [3,Subsection 2.1]) and

    {Δv2=0,in R2+,νv2=g,onΓ4n,v2=0,onR(0,4). (6.4)

    It is easy to see that

    v2L(R2+)C˜v2L(R).

    Moreover by (6.3), for x,y¯R2+ we get

    |v2(x)v2(y)|CgL(Γ4n)|xy|1/2Rmax(1,|r|1/2)1+r2drCgL(Γ4n)|xy|1/2.

    It follows that

    v2C1/2(¯R2+)CgL(Γ4n). (6.5)

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

    v12C1/2ε(¯Ω)Cv1H1(Ω)(fL(B+4)+v1H1(Ω)).

    Multiplying (6.2) by v1, integrating by parts and using Young's inequality, we get

    Cv12L2(Ω)v12L2(Ω)v1L2(Ω)fL2(B+4)εv12L2(Ω)+Cεf2L(B+4),

    where in the first estimate we have used the Poincaré inequality for functions vanishing on a portion of the boundary. We then conclude that

    v1C1/2ε(¯Ω)CfL(B+4). (6.6)

    Now, thanks to (6.1), (6.2) and (6.4), the function V:=v(v1+v2)H1(Ω) solves the equation

    {ΔV=0,in Ω,νV=0,on Γ3n,V=0,on Γ3d. (6.7)

    By elliptic regularity theory, we have that

    VC2(¯B+5/2B+1)CVH1(B+r) (6.8)

    where r is a fixed radius satisfying 52<r<3 and C>0 is independent of V. Let η a radial cutoff function compactly supported in B3 satisfying η1 in Br; testing (6.7) with ηV, we infer that VH1(B+r)CVL2(Ω) for some constant C>0 independent of V. Hence by (6.8) we obtain

    VC2(¯B+5/2B+1)CVL(Ω). (6.9)

    Let ˜ηCc(B5/2) be a radial function, with ˜η1 on B2. Then the function ˜V:=˜ηVH1(R2+) solves

    {Δ˜V=VΔ˜η2V˜η,in R2+,ν˜V(x1,0)=0,x1(0,+),˜V(x1,0)=0,x1(,0).

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

    v(v1+v2)C1/2ε(¯B+2)˜VC1/2ε(¯R2+)CVL(Ω)C(fL(B+4)+gL(Γ4n)+vL(B+4)).

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

    Recalling (1.2), for every kN with k1, we consider the finite dimensional linear subspace of L2(B+r), given by

    Sk:={kj=1ajFj:(a1,,ak)Rk}.

    For every r>0, k1, and uL2(B+r), we let

    Fuk,r:=ArgminFSkB+r(u(x)F(x))2dx

    be the L2(B+r)-projection of u on Sk, so that

    minFSkB+r(u(x)F(x))2dx=B+r(u(x)Fuk,r(x))2dx

    and

    B+r(u(x)Fuk,r(x))F(x)dx=0,for allFSk. (6.10)

    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 Sk.

    Proposition 6.2. Let uH1(B+1)L(R2+) solve

    {Δu=f,inB+1,νu=g,onΓ1n,u=0,onΓ1d, (6.11)

    where, for some kN{0} and ¯C>0,

    |f(x)|¯C|x|max(γk32,0),foreveryxB+1,|g(x1)|¯C|x1|max(γk12,0),foreveryx1(0,1),

    and γk=2k12. Then, for every α(0,1/2), we have that

    supr>0rγkαuFuk,rL(B+r)<.

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

    supr>0rγkαuFuk,rL(B+r)=.

    We consider the nonincreasing function

    Θ(r):=sup¯r>r¯rγkαuFuk,¯rL(B+¯r).

    It is clear from our assumption that

    Θ(r)+ asr0.

    Then there exists a sequence rn0 such that

    rγkαnuFuk,rnL(B+rn)Θ(rn)2.

    We define

    vn(x):=rγkαnu(rnx)Fuk,rn(rnx)Θ(rn),

    so that

    vnL(B+1)12. (6.12)

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

    B+1vn(x)F(x)dx=0 for everyFSk. (6.13)

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

    1rγk+αΘ(r)Fuk,rRFuk,rL(B+rR)CRγk+α. (6.14)

    Indeed, by definition, for every ¯r>r>0, we have

    uFuk,¯rL(B+¯r)¯rγk+αΘ(r)

    and thus, using the monotonicity of Θ, for every xB+r we get

    |Fuk,2r(x)Fuk,r(x)|uFuk,2rL(B+2r)+uFuk,rL(B+r)21+γk+αrγk+αΘ(r)Crγk+αΘ(r). (6.15)

    Letting Fuk,r=kj=1aj(r)Fj and γj=2j12, by taking the L2(B+r)-norm in (6.15), we get

    |aj(2r)aj(r)|rγjCrγk+αΘ(r) for everyr>0.

    Then

    1rγk+αΘ(r)Fuk,r2mFuk,rL(B+r2m)1rγk+αΘ(r)kj=1|aj(r2m)aj(r)|(r2m)γj1rγk+αΘ(r)kj=1mi=1|aj(2ir)aj(2i1r)|(r2m)γjCrγk+αΘ(r)kj=1mi=12γjm2(γkγj+α)(i1)rγk+αΘ(2i1r)Ckj=1mi=12γjm2(γkγj+α)(i1)Ckj=12γjm2(γkγj+α)mC2m(γk+α).

    This proves the claim.

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

    \begin{align*} \sup\limits_{x\in B_R^+}|v_n(x)|& = \frac{1}{r^{\gamma_k+\alpha }_n\Theta(r_n)}\| u-F^u_{k, r_n}\|_{L^\infty(B_{r_n R}^+)}\\ &\leq \frac{1}{r^{\gamma_k+\alpha }_n\Theta(r_n)}\| u-F^u_{k, r_n R}\|_{L^\infty(B_{r_n R}^+)}+ \frac{1}{r^{\gamma_k+\alpha }_n\Theta(r_n)}\| F^u_{k, r_n R}-F^u_{k, r_n}\|_{L^\infty(B_{ r_n R}^+)}\\ &\leq \frac{ 1}{ r^{\gamma_k+\alpha }_n \Theta(r_n) } (r_nR)^{\gamma_k+\alpha } \Theta( r_n) + CR^{\gamma_k+\alpha }\\ &\leq CR^{\gamma_k+\alpha }. \end{align*}

    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

    \sup\limits_{x\in B_R^+}|v_n(x)|\leq \sup\limits_{x\in B_{2^{m_0}}^+}|v_n(x)|\leq C2^{m_0(\gamma_k+\alpha)}\leq C(2R)^{\gamma_k+\alpha}\leq CR^{\gamma_k+\alpha }, (6.16)

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

    \begin{cases} -\Delta v_n = \frac{r_n^{2-\gamma_k-\alpha }}{\Theta(r_n)} f (r_n \cdot ), &\text{in }B_{{1}/{r_n}}^+, \\[5pt] \partial_\nu v_n = \frac{r_n^{1-\gamma_k-\alpha }}{\Theta(r_n)} g (r_n \cdot ), &\text{on }\Gamma ^{{1}/{r_n}}_n, \\[5pt] v_n = 0, &\text{on }\Gamma ^{{1}/{r_n}}_d. \end{cases}

    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

    \begin{cases} -\Delta v = 0, &\text{in } \mathbb{R}^2_+, \\ \partial_\nu v = 0, &\text{on }\Gamma ^{\infty}_n, \\ v = 0, &\text{on }\Gamma ^{\infty}_d, \end{cases}

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

    \|v\|_{L^\infty(B_R^+)}\leq C R^{\gamma_k+\alpha }.

    By Lemma 6.3 (below), we deduce that necessarily

    v\in \mathcal{S}_k.

    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

    \begin{cases} -\Delta v = 0, &\text{in } \mathbb{R}^2_+, \\ \partial_\nu v = 0, &\text{on }\Gamma ^{\infty}_n, \\ v = 0, &\text{on }\Gamma ^{\infty}_d, \end{cases}

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

    \|v\|_{L^\infty(B_R^+)}\leq C\, R^{\gamma_k+\alpha }\quad for \;every\; R \gt 1, (6.17)

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

    v\in \mathcal{S}_k.

    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

    v(r\cos t, r\sin t) = \sum\limits_{j = 1}^\infty\varphi_j(r) \cos\left(\tfrac{2j-1}{2}t\right)

    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

    \frac\pi2 \sum\limits_{j = 1}^{\infty}\varphi_j^2(r) = \int_0^\pi v^2(r\cos t, r\sin t)\, dt\leq\pi C^2 r^{2(\gamma_k+\alpha)}, \quad\text{for all }r \gt 1.

    It follows that

    |\varphi_j(r)|\leq {\rm const\, }r^{\gamma_k+\alpha}\quad\text{for all }j\geq1\text{ and }r \gt 1, (6.18)

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

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

    -\varphi_j"(r)-\frac{1}{r}\varphi_j'(r)+ \frac14(2j-1)^2 \frac{\varphi_j(r)}{r^2} = 0, \quad\text{in }(0, +\infty),

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

    \varphi_j(r) = c_1^j r^{\frac{2j-1}{2}}+ c_2^j r^{\frac{1-2j}{2}}\quad\text{for all }r \gt 0.

    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

    v(r\cos t, r\sin t) = \sum\limits_{j = 1}^k c_1^j r^{\frac{2j-1}{2}} \cos\left(\tfrac{2j-1}{2}t\right) = \sum\limits_{j = 1}^k c_1^j F_j(r\cos t, r\sin t),

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

    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))

    \widetilde w^\tau(z): = \tau^{-\gamma}w(\tau z) = \tau^{-\gamma}\sqrt{H( \tau )} w^\tau (z) .

    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

    w(y) = \beta F_{k_0}(y)+o(|y|^\gamma ) \qquad\textrm{ as}\; |y|\to 0 \;\; \textrm{and }\; y\in {B_R^+}. (7.1)

    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

    |y_m|^{-\gamma} | w(y_m)-\beta F_{k_0}(y_m)| = | {\tilde w}^{\tau _m}( z_m)-\beta F_{k_0}(z_m)|\geq C \gt 0,

    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),

    |w(x)-F_{k_0, r}^w(x)| \leq C r^{\gamma +\varrho}, \quad\textrm{for every}\; x\in B_r^+, (7.2)

    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

    \sup\limits_{x\in B_r^+} r^{-\gamma}|\beta F_{k_0}(x)-F_{k_0, r}^w(x)|\to 0, \quad\textrm{as}\; r \to 0^+. (7.3)

    Claim 2: We have

    |\beta F_{k_0}(x)-F_{k_0, r}^w(x)| \leq C r^{\gamma +\varrho}, \quad\textrm{for every} \;x\in B_r^+. (7.4)

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

    |w(x)- \beta F_{k_0}(x) |\leq |w(x)- F_{k_0, r}^w(x) |+|F_{k_0, r}^w- \beta F_{k_0}(x) | \leq C r^{\gamma +\varrho}, \quad\textrm{for every} \;x\in B_r^+.

    In particular,

    |w(x)- \beta F_{k_0}(x) | \leq C |x|^{\gamma +\varrho}, \quad\textrm{for every}\; x\in B_{R}^+

    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

    | \beta -a_{k_0}(r) |\to 0, \quad\text{as }r\to0^+ .

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

    (a_{k_0}(r)-\beta )^2 r^{2\gamma +2}+\sum\limits_{j = 1}^{k_0-1} a_j^2(r)r^{2\gamma_j+2} \leq C r^{2\gamma + 2}, \quad\textrm{for every}\; R \gt r \gt 0,

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

    |a_j(r)|\leq C r^{\gamma -\gamma_j}\to 0 \qquad\textrm{ as} \;r\to 0. (7.6)

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

    \bigg| w(x)- \sum\limits_{j = 1}^{k_0}a_j(r) F_j(x)\bigg|\leq C r^{\varrho+\gamma }.

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

    \bigg| \sum\limits_{j = 1}^{k_0}(a_j(r)-a_j(2^{-1}r)) F_j(x) \bigg|\leq | F_{k_0, r}^w(x) - w (x) |+ | F_{k_0, 2^{-1}r}^w(x) - w(x) |\leq C r^{\varrho+\gamma }.

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

    \sum\limits_{j = 1}^{k_0}| a_{j}(r) - a_{j}(2^{-1}r)| r^{\gamma_j} \leq C r^{\gamma +\varrho}.

    This implies that

    | a_{j}(r) - a_{j}(2^{-1}r)| \leq C r^{\varrho+\gamma -\gamma_j}\qquad\textrm{for all}\; 1\leq j\leq k_0 \textrm{ and }\;r\in(0, R).

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

    |\beta - a_{k_0}(r)| r^{-\varrho}+ \sum\limits_{j = 1}^{k_0-1} |a_j(r)| r^{-\varrho-\gamma +\gamma_j}\leq \sum\limits_{j = 1}^{k_0} \sum\limits_{i = 0}^\infty |a_{j}(r 2^{-i-1} )- a_{j}(r 2^{-i})|r^{-\varrho-\gamma +\gamma_j} \leq C \sum\limits_{i = 0}^\infty 2^{-i\varrho}.

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

    |\beta F_{k_0}(x)-F_{k_0, r}^w(x)|\leq |\beta - a_{k_0}(r)| r^{\gamma }+ \sum\limits_{j = 1}^{k_0-1} |a_j(r)| r^{ \gamma_j} \leq C r^{\gamma +\varrho}.

    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

    \begin{cases} \Delta {\tilde F } = 0, & \textrm{in}\; \mathcal{U}_R, \\ {\tilde F } = 0, & \textrm{on }\; \Gamma _d\cap\partial \mathcal{U}_R\\ \partial _{\nu }{\tilde F } = 0, & \textrm{on} \; \Gamma _n\cap\partial \mathcal{U}_R. \end{cases} (7.7)

    (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

    | u( \Upsilon^{-1}(y))-\beta \alpha ^{\frac{2k_0-1}{2}} F_{k_0}(y)|\leq C |y|^{ \frac{2k_0-1}{2}+ \varrho }, \quad \textrm{for every} \;y\in B_{\rho_0}^+, (7.8)

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

    | u(x)-\beta F_{k_0}(\alpha \Upsilon(x))|\leq c |x|^{\frac{2k_0-1}{2}+\varrho}, \quad \textrm{for every}\;x\in \Upsilon^{-1}(B_{\rho_0}^+), (7.9)

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

    |D F_{k_0}(x)|\leq c |x|^{\frac{2k_0-1}{2}-1}

    and thus

    \begin{align*} |F_{k_0}(\alpha \Upsilon(x))-F_{k_0}(\varphi (x))&|\leq c |x|^{\frac{2k_0-1}{2}-1}|\alpha \Upsilon(x) -\varphi (x)|\\ &\leq c |x|^{\frac{2k_0-1}{2}-1} |x|^2\\ &\leq c |x|^{\frac{2k_0-1}{2}+1}, \end{align*}

    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),

    \lim\limits_{r\to 0}r^{-\frac{2k_0-1}{2}} u(r\cos t, r\sin t) = \beta\alpha^{\frac{2k_0-1}{2}}\cos\left(\tfrac{2k_0-1}{2}t\right). (7.10)

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

    u(r\cos t, r\sin t)\geq \beta\alpha^{1/2}r^{1/2}\cos\left(\tfrac{t}{2}\right) -Cr^{1/2+\varrho},

    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.

    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

    A: = \mathbb{C}\setminus\{x_1\in \mathbb{R}\subset \mathbb{C}:x_1\leq 0\}

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

    \eta(z): = \log r+i\theta \quad \text{for any}\; z = re^{i\theta }\in A, r \gt 0, \theta \in\left(-\pi, \pi\right).

    Let us consider the holomorphic function

    v(z): = e^{2\eta(-iz)}\eta(-iz) \qquad \text{for any } z\in \mathbb{C}\setminus \{ix_2: x_2 \le 0\}

    and the set

    \mathcal Z: = \{z\in \mathbb{C}\setminus \{ix_2: x_2 \le 0\}:\Im(v(z)) = 0\} . (8.1)

    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

    r = \rho(\theta ): = \exp\left[-\bigg(\theta -\frac{\pi}2\bigg)\cot(2\theta ) \right] . (8.2)

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

    \Gamma _+: \begin{cases} x_1(\theta ) = \rho(\theta )\cos\theta \\ x_2(\theta ) = \rho(\theta )\sin\theta \end{cases} \qquad \theta \in (-\sigma, 0) \, . (8.3)

    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

    \lim\limits_{x_1\to 0^+} \frac{h_+(x_1)}{x_1} = 0 \, , \quad \lim\limits_{x_1\to 0^+} h_+'(x_1) = 0 \, . (8.4)

    Then we define the harmonic function

    u(x_1, x_2): = -\Im(v(z)) \qquad \text{ for any } z = x_1+ix_2\in \mathbb{C}\setminus \{iy: y \le 0\} \, . (8.5)

    In polar coordinates the function u reads

    u(r, \theta ) = r^2 \left[(\log r) \sin(2\theta )+\left(\theta -\frac{\pi}2\right) \cos(2\theta )\right] . (8.6)

    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

    u(x_1, x_2) = x_1 x_2\log(x_1^2+x_2^2)+\left[\arctan\left(\frac{x_2}{x_1}\right)+\frac \pi 2\right](x_1^2-x_2^2) \quad \text{for any } x_1 \lt 0 , \, x_2\in \mathbb{R} \, .

    From direct computation we obtain

    \frac{\partial u}{\partial x_1}(x_1, x_2) = x_2\log(x_1^2+x_2^2)+x_2 +2\left[\arctan\left(\frac{x_2}{x_1}\right)+\frac \pi 2\right]x_1 \, , \\ \frac{\partial u}{\partial x_2}(x_1, x_2) = x_1\log(x_1^2+x_2^2)+x_1 -2\left[\arctan\left(\frac{x_2}{x_1}\right)+\frac \pi 2\right]x_2 \, .

    We now define

    H_1(x_1, x_2) = \frac{2\left[\arctan\left(\frac{x_2}{x_1}\right)+\frac \pi 2\right]x_1}{\log(x_1^2+x_2^2)} \quad \text{and} \quad H_2(x_1, x_2) = \frac{2\left[\arctan\left(\frac{x_2}{x_1}\right)+\frac \pi 2\right]x_2}{\log(x_1^2+x_2^2)}

    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

    H_1(x_1, x_2): = 3H_1(-x_1, x_2)-2H_1(-2x_1, x_2) \quad \text{and} \quad H_2(x_1, x_2): = 3H_2(-x_1, x_2)-2H_2(-2x_1, x_2)

    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

    V_1(x_1, x_2): = \begin{cases} x_2+\frac{x_2}{\log(x_1^2+x_2^2)}+H_1(x_1, x_2), & \quad \text{if } (x_1, x_2)\neq (0, 0), \\ 0, & \quad \text{if } (x_1, x_2) = (0, 0) \, , \end{cases} \\ V_2(x_1, x_2): = \begin{cases} x_1+\frac{x_1}{\log(x_1^2+x_2^2)}-H_2(x_1, x_2), & \quad \text{if } (x_1, x_2)\neq (0, 0), \\ 0, & \quad \text{if } (x_1, x_2) = (0, 0) \, . \end{cases}

    One may verify that V_1, V_2\in C^1(B_{1/2}). Moreover we have

    \frac{\partial V_1}{\partial x_1}(0, 0) = 0 \, , \quad \frac{\partial V_1}{\partial x_2}(0, 0) = 1 \, , \quad \frac{\partial V_2}{\partial x_1}(0, 0) = 1 \, , \quad \frac{\partial V_2}{\partial x_2}(0, 0) = 0 \, .

    Then we consider the dynamical system

    \begin{cases} x_1'(t) = V_1(x_1(t), x_2(t)) \\ x_2'(t) = V_2(x_1(t), x_2(t)) \, . \end{cases} (8.37)

    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

    \Omega = \{(x_1, x_2)\in B_R:x_2 \gt h(x_1)\}.

    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

    \frac{1}2 \log \big(x_1^2+(h_+(x_1))^2\big) \tan\left[2\arctan\left(\frac{h_+(x_1)}{x_1}\right)\right] +\arctan\left(\frac{h_+(x_1)}{x_1}\right)-\frac{\pi}2 = 0 . (8.8)

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

    \tan \left[2\arctan\left(\frac{h_+(x_1)}{x_1}\right)\right] = -\frac{2\arctan\big(\frac{h_+(x_1)}{x_1}\big)-\pi}{\log \big(x_1^2+(h_+(x_1))^2\big)} = \frac{\pi}2 \frac{1}{\log x_1}+o\left(\frac{1}{\log x_1}\right) . (8.9)

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

    \big(x_1+h_+(x_1)h_+'(x_1)\big) \, \tan\left[2\arctan\left(\frac{h_+(x_1)}{x_1}\right)\right] \\ \quad \quad \quad \quad \quad +\left\{ 1+\frac{\log \big(x_1^2+(h_+(x_1))^2\big)} {\cos^2\left[2\arctan\left(\frac{h_+(x_1)}{x_1}\right)\right]} \right\} \big(x_1h_+'(x_1)-h_+(x_1) \big) = 0 (8.10)

    and hence (8.4) and (8.9) yield

    x_1h_+'(x_1)-h_+(x_1)\sim -\frac{\pi}4 \, \frac{x_1}{\log^2 x_1} \qquad \text{as } x_1\to 0^+ . (8.11)

    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

    F(z): = r^{\frac 43} \, e^{i\frac 43 \theta } \quad \text{for any } z = re^{i\theta } \, , \ r \gt 0\, , \theta \in \left(-\frac{\pi}2, \frac{3\pi}2\right) .

    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

    \begin{align*} \int_{\widetilde\Gamma _-} \frac{\partial \widetilde u}{\partial\nu_{\widetilde \Gamma _-}}\widetilde \varphi\, ds & = \int_{\widetilde\Omega } \nabla\widetilde u(y)\nabla \widetilde \varphi(y)\, dy = \int_{\widetilde\Omega } [\nabla u(F^{-1}(y))(DF(F^{-1}(y)))^{-1}] \nabla \widetilde \varphi(y)\, dy \\[7pt] & = \int_{\Omega } \big[\nabla u(x)(DF(x))^{-1}\big] \nabla \widetilde \varphi(F(x))\, |\mathop{\rm det}(DF(x))| \, dx \\[7pt] & = \int_{\Omega } \big[\nabla u(x)(DF(x))^{-1}\big] \big[\nabla\varphi(x)(DF(x))^{-1}\big]\, |\mathop{\rm det}(DF(x))| \, dx \\[7pt] & = \int_{\Omega } \nabla u(x)\nabla\varphi(x) \, dx = \int_{\Gamma _-} \frac{\partial u}{\partial \nu_{\Gamma _-}}\varphi \, ds = 0 \end{align*}

    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

    r = \widetilde\rho(\theta ): = \exp\left[-\left(\theta -\frac{2\pi}3\right)\cot\left(\frac{3\theta }2\right)\right] \, . (8.12)

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

    \frac{1}2 \log \big(x_1^2+(\widetilde h_+(x_1))^2\big) \tan\left[\frac 32\arctan\left(\frac{\widetilde h_+(x_1)}{x_1}\right)\right] +\arctan\left(\frac{\widetilde h_+(x_1)}{x_1}\right)-\frac{2\pi}3 = 0 \, . (8.13)

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

    \lim\limits_{x_1\to 0} \frac{\widetilde h_+(x_1)}{x_1} = 0 \, , \quad \lim\limits_{x_1\to 0^+} \widetilde h_+'(x_1) = 0 \, . (8.14)

    By (8.14) we have

    \tan \left[\frac 32\arctan\left(\frac{\widetilde h_+(x_1)}{x_1}\right)\right] = -\frac{2\arctan\big(\frac{\widetilde h_+(x_1)}{x_1}\big)-\frac{4\pi}3}{\log \big(x_1^2+(\widetilde h_+(x_1))^2\big)} = \frac{2\pi}3 \frac{1}{\log x_1}+o\left(\frac{1}{\log x_1}\right) \quad \text{as } x_1\to 0^+ \, . (8.15)

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

    \big(x_1+\widetilde h_+(x_1)\widetilde h_+'(x_1)\big) \, \tan\left[\frac 32\arctan\left(\frac{\widetilde h_+(x_1)}{x_1}\right)\right] \\ \quad \quad \quad \quad \quad +\left\{ 1+\frac{3\log \big(x_1^2+(\widetilde h_+(x_1))^2\big)} {4\cos^2\left[\frac 32\arctan\left(\frac{\widetilde h_+(x_1)}{x_1}\right)\right]} \right\} \big(x_1\widetilde h_+'(x_1)-\widetilde h_+(x_1) \big) = 0. (8.16)

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

    x_1\widetilde h_+'(x_1)-\widetilde h_+(x_1)\sim -\frac{4\pi}9 \, \frac{x_1}{\log^2 x_1} \qquad \text{as } x_1\to 0^+ . (8.17)

    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 }).

    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".

    All authors declare no conflicts of interest in this paper.

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