Nontrivial solutions to Serrin's problem in annular domains
Nikola Kamburov Luciano Sciaraffia



	We construct nontrivial smooth bounded domains Ø⊆^n of the form Ø_0 ∖Ø_1, bifurcating from annuli, for which there exists a positive solution to the overdetermined boundary value problem 
	
    - u = 1 ,  u > 0      in Ø, 
    
            u = 0 , _ u = const    on Ø_0, 
    
            u = const , _ u = const    on Ø_1,

	where  stands for the inner unit normal to Ø. From results by Reichel <cit.> and later by Sirakov <cit.>, it is known that the condition _ u ≤ 0 on Ø_1 is sufficient for rigidity to hold, namely, the only domains which admit such a solution are annuli and solutions are radially symmetric. Our construction shows that the condition is also necessary. Furthemore, we show that the constructed domains are self-Cheeger.



alpha

Architecting Dependable Learning-enabled Autonomous Systems: A Survey
    
Chih-Hong Cheng1
Dhiraj Gulati1
Rongjie Yan2

    December 30, 2023
=====================================================================






§ INTRODUCTION AND MAIN RESULT


Let Ø⊆^n be a bounded, connected C^2-domain of the form Ø = Ø_0∖Ø_1, where Ø_0 and Ø_1 are bounded domains in ^n, n≥ 2, with Ø_1⋐Ø_0. The present paper is devoted to the overtermined boundary value problem

    - u = 1       in Ø , 
    
        u = 0 , _ u = c_0     on Ø_0 , 
    
        u = a , _ u = c_1     on Ø_1 ,

where  denotes the inner unit normal to Ø and a≥ 0, c_0 and c_1 are real constants. Note that whenever (<ref>) admits a solution u∈ C^2(Ø), then u is strictly positive in Ø due the strong maximum principle, while the Hopf lemma implies that the constant c_0>0. 


Problem (<ref>) arises in classical models in the theory of elasticity, fluid mechanics and electrostatics – see <cit.> for a discussion of applications. The special case in which a=0 and c_0=c_1 was treated by Serrin in his seminal 1971 paper <cit.>. He showed that a strong property of rigidity was forced upon any solution u∈ C^2(Ø) and upon the shape of the domain Ø supporting it: namely, Ø has to be a ball (Ø_1 = ∅) and u has to be radially symmetric and monotonically decreasing along the radius. Serrin's proof is based on the moving planes method,  pioneered earlier by Alexandrov <cit.> in a geometric context, and has ever since been a powerful tool for establishing symmetry of positive solutions to second-order elliptic problems. An important artifact of the method is that proving radial symmetry comes hand in hand with proving the monotonicity of solutions in the radial direction.

 

Reichel <cit.> adapted Serrin's method to analyze (<ref>) in the case when 
    u|_Ø_1 = a>0   and   u_ν|_Ø_1=c_1≤ 0.
 
Under the additional assumption that 0<u<a in Ø, he showed that u has to be radially symmetric and the domain Ø – a standard annulus. Several years later, Sirakov <cit.> removed the extra assumption and proved a more general rigidity theorem, allowing for separate constant Dirichlet conditions u|__i = a_i>0 and separate constant Neumann conditions u_ν|__i=c_i≤ 0 to be imposed on each connected component _i of the inner boundary Ø_1, i=1,…, l. The assumption of non-positivity of each Neumann condition u_ν|__i=c_i≤ 0 is crucial for the moving planes method to run and yield the radial symmetry and monotonicity of solutions. Furthemore, just like Serrin's result, the rigidity theorems by Reichel and Sirakov apply to a more general class of second order elliptic equations of which (<ref>) in an important special case. In n=2 dimensions, a symmetry result for (<ref>) was obtained earlier by Willms, Gladwell and Siegel <cit.> under some additional boundary curvature assumptions. See also <cit.> for a complex analytic approach to (<ref>) when n=2.

In this paper we focus on a case, in which the Neumann condition on the inner boundary is positive:

    u|_Ø_0 = 0,    u|_Ø_1 = a>0   and   u_ν|_Ø_0= u_ν|_Ø_1=c>0.






 
 
 

We immediately notice that over annuli Ø, (<ref>) now possesses radial solutions that are not monotone along the radius – see Lemma <ref> for the formula of a one-parameter family of such examples. Their presence hints that proving radial symmetry for solutions of (<ref>) would be out of the scope of the moving planes method. Thus, one is led to conjecture that, under (<ref>), radial rigidity does not hold for solutions of (<ref>).

Our main result confirms that this is indeed the case. 


    There exist smooth bounded annular domains of the form Ø=Ø_0 ∖Ø_1 ⊆^n, which are different from standard annuli, and positive constants a and c, for which the overdetermined problem (<ref>) admits a solution u ∈ C^∞(Ø) satisfying (<ref>).


We construct these nontrivial annular domains and their corresponding solutions by the means of the Crandall-Rabinowitz Bifurcation Theorem <cit.>. In this manner, we actually obtain a smooth branch of domains and solutions (in fact, a whole sequence of distinct branches) bifurcating from the trivial branch of standard annuli admitting the radial, non-monotone solutions of (<ref>), mentioned above. This is more precisely stated in the body of Theorem <ref> in Section <ref>, of which Theorem <ref> is an immediate corollary.

The overdetermined problem (<ref>) with boundary conditions (<ref>) has a connection to the so-called Cheeger problem: given a bounded domain Ø⊆^n, find 

    h_1(Ø):=inf{P(E)/|E|: E⊆Ø}
 
where |E| is the Lebesgue measure of E and P(E) denotes the perimeter functional (see <cit.>). The constant h_1(Ø) is known as the Cheeger constant for the domain Ø, and a subset E⊆Ø, for which the infimum in (<ref>) is attained, is called a Cheeger set for Ø. See the surveys <cit.> for an overview of the Cheeger problem and a discussion of applications. It turns out that the domains Ø constructed in Theorem <ref> are precisely their own Cheeger sets. Such domains are called self-Cheeger.

 Let Ø be any one of the smooth annular domains in Theorem <ref> that admits a solution u∈ C^∞(Ø) of (<ref>)-(<ref>). Then

    h_1(Ø) =P(Ø)/|Ø| =1/c

and Ø is the unique minimizer of the Cheeger problem (<ref>).

In this manner, Corollary <ref> establishes the existence of non-radially symmetric domains that are self-Cheeger. 


Constructions of non-trivial solutions to overdetermined elliptic problems have been prominent in the literature in recent years. Many of them have been driven by a famous conjecture of Berestycki, Caffarelli and Nirenberg <cit.>, according to which, if f is a Lipschitz function and Ø⊆^n is an unbounded smooth domain, such that ^n ∖Ø is connected, then the overdetermined problem

    u + f(u) = 0 ,  u > 0      in Ø, 
    
        u = 0 , _ u = const    on Ø,

admits a positive bounded solution if and only if Ø is a half space, a cylinder or the complement of a ball. First,  Sicbaldi <cit.> constructed domain counterexamples Ø to the conjecture when n≥ 3 and f(u)=ł u, which bifurcate from cylinders B^n-1× for appropriate ł>0. Then  Ros, Ruiz and Sicbaldi  <cit.> constructed a different set of counterexamples for all dimensions n≥ 2 and f(u) = u^p-u, p>1, that bifurcate from the complement of a ball ^n∖B_ł^n. The main tool behind the two results is a bifurcation theorem by Krasnoselski (see <cit.>) that is based on topological degree theory and which yields a sequence of domains Ø, rather than a smooth branch. Schlenk and Sicbaldi <cit.> managed to strengthen the construction in <cit.> through the use of the Crandall-Rabinowitz Bifurcation Theorem to obtain a smooth branch of perturbed cylinders Ø. A similar approach leading to perturbed generalized cylinders was pursued by Fall, Minlend and Weth <cit.> in the case of our interest, f≡ 1. The bifurcation method has also been successful in finding nontrivial solutions in versions of (<ref>) set in Riemannian manifolds <cit.>.  For other methods of solution construction in overdetermined elliptic problems, we refer to <cit.> among others.

Our approach to Theorem <ref> is aligned with that of Schlenk and Sicbaldi in <cit.> and Fall, Minlend and Weth in <cit.>: we translate the problem to a non-linear, non-local operator equation in appropriate function spaces and we derive the necessary spectral and Fredholm tranversality properties of the linearized operators in order to implement the Crandall-Rabinowitz Bifurcation Theorem. However, unlike the quoted results above where symmetry considerations allow the authors to perturb all connected boundary components of the underlying domains in the same symmetric fashion, we are bound by the geometry of the standard annulus  to deform its two non-symmetric boundary components differently. Thus, the function spaces that we work in are necessarily product spaces of two factors that correspond to the two separate connected components of Ø_ł. This is a new feature, and as far as we are aware, our treatment is the first one in this line of results that deals with it. 

The paper is organized as follows. In the next section we outline the strategy of the construction leading up to the statement of Theorem <ref>, which refines Theorem <ref>, and we show how the latter follows from the former. In Section <ref> we set up the problem as an operator equation F_ł(v⃗)=0⃗, where F_ł: U ⊆ (C^2,(^n-1))^2 → (C^1,(^n-1))^2, and compute a formula for its linearization L_ł:=D_v⃗ F_ł|_v⃗=0⃗ in terms of the Dirichlet-to-Neumann operator for the Laplacian in Ø_ł (Proposition <ref>). In Section <ref> we study the spectrum of L_ł: we find that for each k∈=∪{0}, L_ł has two different eigenvalue branches μ_k,1(ł)<μ_k,2(ł) with associated eigenvectors in the subspaces 𝒴_k ⊕𝒴_k, where 𝒴_k is any spherical harmonic of degree k. In Lemmas <ref> and <ref> we establish key monotonicity properties for μ_k,1(ł) and μ_k,2(ł) in both ł and k, from which we infer that for k≥ 2 the first eigenvalue branch _k,1(ł) is strictly decreasing in ł∈ (0,1), changing sign once, while the second _k,2(ł)>0 is always positive. In Section <ref> we restrict the operators to pairs of G-invariant functions on the sphere for an appropriate group of isometries G so as to ensure that, whenever 0 is an eigenvalue of the restricted L_ł, it is simple. We then verify the relevant Fredholm mapping properties (Proposition <ref>), necessary to apply the Crandall-Rabinowitz Bifurcation Theorem and complete the proof of Theorem <ref>. Finally, in Section <ref> we provide the proof of Corollary <ref>.


 


§ ACKNOWLEDGEMENTS.

The authors would like to thank Dimiter Vassilev for bringing up Serrin's problem in annular domains to their attention. 




§ OUTLINE OF STRATEGY AND REFINEMENT OF THE MAIN THEOREM


Let us first introduce some notation. For any λ∈ (0,1) we denote the standard annulus of inner radius ł and outer radius 1 by

    Ø_ł    := { x ∈^n: ł < |x| < 1 }

and let its two boundary components be

    _1     := { x ∈^n: |x| = 1 } = ^n-1 , 
    _ł    := { x ∈^n: |x| = ł} = ł^n-1 ,

where ^n-1 is the unit sphere in ^n, centered at the origin. 

We will construct the nontrivial solutions u and domains Ø solving (<ref>)-(<ref>) by bifurcating away, at certain critical values of the bifurcation parameter ł, from the branch of non-monotone radial solutions u_ł of (<ref>) defined on the annuli Ø_ł, for which _ν u_ł=c_ł is the same constant on all of Ø_ł. We describe this branch of solutions explicitly in the lemma below. 


    For each ł∈ (0,1), there exist unique positive values a_ł and c_ł given by
    
    a_ł    = 1/2łlogł + 1/4( 1-ł^2 )     if  n=2, 
    1/nłł^n-2-1/n-21+ł/1+ł^n-1 + 1/2n( 1-ł^2 )     if  n ≥ 3, 
    
            c_ł    = 1/n1-ł^n/1+ł^n-1

    such that for Ø = Ø_λ the problem (<ref>) has a unique positive solution u=u_ł with boundary conditions
    
    u = 0  on _1,      u = a_ł on _ł,      u/ = c_ł on Ø_ł.

    The solution is radially symmetric and given by
    
    u_ł(x) =
                1/2łlog |x| + 1/4( 1 -|x|^2 ),     if  n = 2, 
    λ^n-1/n(n-2)1+λ/1+λ^n-1( 1-|x|^2-n) + 1/2n( 1-|x|^2 )     if  n ≥ 3.

    
    
        If u=u(r), where r=|x|, is a radially symmetric solution to (<ref>), then u satisfies the ODE        
    u” + n-1/ru' = -1 ,

        where the prime denotes differentiation with respect to r. Then simple integration yields
        
    u'(r) = C/r^n-1 -r/n.

        Solving -u'(1) = u'(ł) for C, we obtain 
    C = 1+ł/n(1+ł^1-n).
 The formulas (<ref>)-(<ref>) for a_ł, c_ł and u_ł follow by integrating (<ref>) once again and setting u(1)=0. It remains to confirm that a_ł>0 when ł∈(0,1). When n=2, this follows from the observation that 
        
    lim_ł→ 1^-da_ł/dł= 0 = lim_ł→ 1^- a_ł  and  d^2 a_ł/dł^2= (ł^-1 - 1)/2 >0   for ł∈(0,1).
 
For n≥ 3, we can rewrite the expression (<ref>) for a_ł as
        
    a_ł = 1+ł/2n(n-2)(1+ł^n-1)g(ł)   where   g(ł) = (n-2)- nł - (n-2)ł^n +nł^n-1.

       Then the positivity of g(ł), and thus of a_ł, over ł∈ (0,1), follows from the fact that 
        
    lim_ł→ 1^-dg(ł)/dł= 0 = lim_ł→ 1^- g(ł)    and  d^2 g(ł)/dł^2= n(n-1)(n-2)ł^n-3 (1-ł)>0   for ł∈(0,1).

    


We will be perturbing the boundary of each annulus Ø_ł in the direction of the inner unit normal to Ø_ł. For a pair of functions v⃗ = (v_1,v_2) ∈ C^2,(^n-1) × C^2,(^n-1) of sufficiently small C^2,-norm, 0<<1, denote the v⃗-deformation of Ø_ł by: 

    Ø_λ^v⃗    := { x ∈^n : ł + v_1(x/|x|) < |x| < 1-v_2(x/|x|)  },

so that its boundary Ø_ł^v⃗ = _1^v⃗⊔_ł^v⃗, where

    _1^v⃗    := { x ∈^n : |x| = 1-v_2(x/|x|)  } , 
    _λ^v⃗    := { x ∈^n : |x| = ł + v_1(x/|x|)  }.


Our perturbations v⃗ will ultimately be taken to be invariant with respect to the action of a certain subgroup G of the orthogonal group O(n). Recall that a function ψ: Ø→, defined on a G-invariant domain Ø, is called G-invariant if

    ψ = ψ∘ g,     for every  g ∈ G.

The notation for the usual Hölder and Sobolev function spaces, restricted to G-invariant functions, will include a subscript-G, as in C^k,_G, L^2_G, H^k_G, etc. 


We know that for each ł∈ (0,1) and every v⃗∈ (C^2,(^n-1))^2 of appropriately small C^2,-norm, the Dirichlet problem in the perturbed annulus Ø_ł^v⃗
    
    - u = 1     in Ø_ł^v⃗ , 
    
                u = 0     on _1^v⃗ , 
    
                u = a_λ    on _ł^v⃗ ,

with a_ł defined as in (<ref>), has a unique solution u = u_λ(v⃗) ∈ C^2,(Ø_ł^v⃗). Moreover, u_λ(0⃗) = u_ł, the map (v⃗,ł) ↦ u_λ(v⃗) is smooth by standard regularity theory, and if v⃗ is G-invariant, then so are u_λ(v⃗) and _ν u_λ(v⃗), by the uniqueness of solutions. 



We would like to find out when the Dirichlet problem solution u_ł(v⃗) also satisfies a constant Neumann condition on Ø_ł^v⃗. For the purpose, given ł∈ (0,1) and 
U ⊆ C^2,(^n-1) × C^2,(^n-1) – a sufficiently small neighbourhood of 0⃗, we define the operator 

    F_ł   :  U  → C^1,(^n-1) × C^1,(^n-1) , 
    
    	F_ł(v⃗)    :=  1/c_ł( 
        . u_ł(v⃗)/|__ł^v⃗ - c_ł, . u_ł(v⃗)/|__1^v⃗ -c_ł) ,

where we canonically identify functions on Ø_ł^v⃗ with pairs of functions on ^n-1.





Schauder theory implies that (<ref>) is a good definition and the factor of 1/c_ł provides a convenient normalization. Now, the solution u_ł(v⃗) of the Dirichlet problem (<ref>) in Ø_ł^v⃗ solves the full overdetermined problem (<ref>)-(<ref>) if and only if

    F_ł(v⃗) = 0⃗.

Note that F_ł(0⃗) = 0⃗ for every ł∈ (0,1). Our goal is to find a branch of solutions (v⃗, ł) of (<ref>), bifurcating from this trivial branch (0⃗,ł). To achieve it, we will need to understand how the kernel of the linearization L_ł:=D_v⃗F_ł|_v⃗=0⃗ depends on ł. In Proposition <ref> of the next section we will derive a working formula for L_ł:






    L_ł(w_1,w_2) = ( . -_w⃗/|__ł  +  . w_1/c_ł^2u_ł/ r^2|__ł, . -_w⃗/|__1  +  . w_2/c_ł^2u_ł/ r^2|__1) ,

where ϕ_w⃗ is the harmonic function ϕ on Ø_ł with boundary values |__ł(x) = w_1(x/|x|) and  |__1(x) = w_2(x/|x|). 

In order to study the kernel of L_ł, we will look more generally at the eigenvalue problem

    L_ł(w⃗) = μ(ł) w⃗ ,     w⃗∈(C^2,(^n-1))^2.

For each spherical harmonic 𝒴_k of degree k∈, the subspace W_k=Span{(𝒴_k, 0), (0, 𝒴_k)} is invariant under L_ł and decomposes into eigenspaces for L_ł|_W_k, associated with two distinct eigenvalues μ_k,1(ł)<μ_k,2(ł), for which we will calculate explicit formulas in Section <ref>. We will study the dependence of these eigenvalues on both ł∈ (0,1) and k∈, focussing on whether they cross 0 as ł varies in (0,1). It turns out that when k=1, μ_1,1(ł) < 0 while μ_1,2(ł) = 0 for all ł, i.e. the eigenspace correponding to μ_1,2 is always in the kernel of L_ł (see Remark <ref>). This part of L_ł comes from the deformations of Ø_ł, generated by translations. What we find for k≥ 2 is that the first eigenvalue branch μ_k,1(ł) is strictly decreasing in ł and that it crosses 0 at a unique ł_k^*∈ (0,1). This is done in Lemma <ref>. Moreover, we establish in Lemma <ref> that both μ_k,1(ł) and μ_k,2(ł) increase strictly in k∈ for fixed ł. This means that for k≥ 2,  the second eigenvalue branch μ_k,2(ł)>0 never crosses 0, while the values of ł_k^*, at which  μ_k,1(ł) is zero, form a strictly increasing k-sequence; in addition, the eigenvalues for k=0, μ_0,1(ł)<μ_0,2(ł) <0 (Proposition <ref>). Therefore, at the critical values ł=ł_k^*, k ≥ 2, the kernel of L_ł_k^* over (C^2,(^n-1))^2 consists of the μ_k,1(ł_k^*)-eigenspace plus the always present component of the μ_1,2≡ 0 eigenspace. 

We would like to point out that the arguments behind proving the key eigenvalue monotonicity properties in Lemmas <ref> and <ref> are of a different nature from the ones used at the analogous phase of the constructions in <cit.>. The first two papers employed an argument based on rescaling and Bessel function identities, while the third utilized a neat characterization of the eigenvalues of the linearized operator in terms of solutions to a mixed boundary value problem for a first-order ODE. We are afforded neither method in our setting because of the lack of symmetry between the two boundary components of Ø_ł. 
We prove the ł-monotonicity of the first branch μ_k,1(ł), k≥ 2, by analyzing the explicit formula for μ_k,1(ł) directly and using some delicate estimates, based on hyperbolic trigonometric identities (see the proof of Lemma <ref>).  Unfortunately, this approach does not extend to the second eigenvalue branch μ_k,2(ł), k≥ 2, which we also believe to be decreasing in ł, based on numerics. Showing the latter is ultimately not necessary, since we prove that μ_k,2(ł)>0  never contributes to the kernel of L_ł. Neither do the eigenvalues branches for k=0, _0,j(ł), j=1,2, which are shown to be strictly negative.  In order to establish the k-monotonicity of μ_k,j(ł), for fixed ł, j=1,2, we treat k as a continuous positive variable and extend the μ_k,j(ł) to be smooth functions in (k,ł)∈ (0,∞)×(0,1), continuous up to k=0 (see Remark <ref>). Even so, trying to prove  _k μ_k,j(ł) >0 directly from the formula for the eigenvalue turns out to be unyielding, and we accomplish it instead by looking at the matrix representation M_ł,k of L_ł|_W_k and showing that _k M_ł,k is positive definite (see the proof of Lemma <ref>).

The Crandall-Rabinowitz Bifurcation Theorem (see the Appendix) requires that the linearized operator has a one-dimensional kernel at bifurcation values. To achieve this, in Section <ref> we will restrict  the operators F_ł and L_ł to G-invariant functions in (C^2,(^n-1))^2. The symmetry group G will be chosen so as to completely eliminate the eigenspaces of L_ł corresponding to k=1 (which contain the μ_1,2=0 component of ker L_ł), and ensure that, whenever μ_k,1(ł) is an eigenvalue of the restricted L_ł for some k ≥  2, it is of multiplicity 1. Additionally, we will choose the group G in a way that will guarantee that the constructed G-invariant domains Ø_ł^v⃗ are not merely translations of the standard annulus. 

More precisely, let -_^n-1 be the Laplace-Beltrami operator on the sphere ^n-1 and let {_i }_i=0^∞ be the sequence of its eigenvalues, i.e.  We shall fix any group G ⩽ O(n) that has the following two properties:

	
  (P1) If T is a translation of ^n and T(^n-1) is a G-invariant set, then T is trivial.

	
	
  (P2)  If {_i_k}_k=0^∞ are the eigenvalues of -_^n-1 when restricted to the G-invariant functions, then _i_k has multiplicity equal to 1, i.e. there exists a unique (up to normalization) G-invariant spherical harmonic of degree i_k, k∈.

Note that i_0=0 and that because of (P1), spherical harmonics of degree 1 are not G-invariant, i.e. i_1≥ 2. An example of a group, satisfying both (P1) and (P2), is G=O(n-1) ×_2. For instance, when n=2, G = _2 ×_2 acts on ^2 by reflections with respect to the two coordinate axes; the eigenvalues of -_^1=-^2_þ, restricted to _2 ×_2-invariant functions, are _i_k = (2k)^2, k∈, and the corresponding eigenfunctions (up to normalization) are cos(2k þ).

Let us denote Y_k := 𝒴_i_k, where 𝒴_i_k is the unique G-invariant spherical harmonic of degree i_k∈, normalized in the L^2-norm, that is,

    _^n-1 Y_k + _i_k Y_k = 0 ,     ∫_^n-1 |Y_k|^2  dS = 1 ,      k ∈ .

Finally, denote by

    ⟨w⃗,z⃗⟩_ł := ł^n-1∫_^n-1 w_1 z_1  dS + ∫_^n-1 w_2 z_2  dS ,     w⃗,z⃗∈ L^2(^n-1) ×  L^2(^n-1) ,

the inner product on L^2(^n-1)×  L^2(^n-1) induced by the standard inner product on L^2(Ø_ł) under the natural identification 

    [                                w⃗                                 ↔                                 ψ;                   (w_1(x),w_2(x))                                 = (ψ(λ x), ψ(x))   for all  x∈^n-1, ]

and let

    w⃗_ł := √(⟨w⃗,w⃗⟩_ł)

be the induced L^2-norm. We point out that L_ł is formally self-adjoint with respect to ⟨·, ·⟩_ł (see Remark <ref>).

As a result of restricting to pairs of G-invariant functions in C^2,(^n-1), the linearized operator L_ł:(C^2,_G(^n-1))^2 → (C^1,_G(^n-1))^2 will now possess a one-dimensional kernel at each critical value ł_k:=ł_i_k^*, k∈ –  spanned by an element of the form z⃗_k=(a_kY_k, b_kY_k) – and its image will be the closed subspace of co-dimension 1 orthogonal to z⃗_k with respect to the inner product (<ref>). Moreover, because of the strict ł-monotonicity of μ_i_k, 1(ł), the tranversality condition 
    _ł L_ł|_ł=ł_k(z⃗_k) = μ_i_k, 1'(ł_k) z⃗_k ∉im  L_ł_k
 is going to hold. Invoking the Crandall-Rabinowitz Bifurcation Theorem <ref>, we reach at the statement of the refinement to Theorem <ref>.


 Let n≥ 2, ∈ (0,1), and let G and Y_k, k ∈, be as above. There is a strictly increasing sequence {λ_k }_k=1^∞ of positive real numbers with lim_k→∞λ_k = 1 and a sequence {z⃗_k }_k=1^∞ of non-zero elements of the form z⃗_k = (a_kY_k,b_kY_k) with z⃗_k_ł_k=1, such that for each k ∈, there exists > 0 and a smooth curve 
    
    [                     (-,)                        → ( C^2,(^n-1) )^2 × (0,1);                        s                        ↦             (w⃗(s),ł(s)) ]

    satisfying w⃗(0) = 0, ł(0) = ł_k, such that for v⃗(s) ∈ C^2,(^n-1) × C^,(^n-1) defined by
    
    v⃗(s) = s(z⃗_k + w⃗(s)),

    the overdetermined problem
    
    - u = 1      in Ø_ł(s)^v⃗(s) , 
    
                u = 0     on _1^v⃗(s) , 
    
                u = >0      on _ł(s)^v⃗(s) ,
    _ν u = const >0     on Ø_ł^v⃗(s),

	admits a positive solution u ∈ C^2,(Ø_ł(s)^v⃗(s)). Moreover, for every s ∈ (-,) the two components of w⃗(s) = (w_1(s),w_2(s)) are G-invariant functions that satisfy
    
    ⟨w⃗(s),z⃗_k ⟩_ł_k = 0.



Let us show how Theorem <ref> entails Theorem <ref>.

Fix any k∈ and ∈ (0,1). We need only explain why for s≠ 0 the C^2, domains Ø_ł(s)^v⃗(s), constructed in Theorem <ref>, are different from a standard annulus, and why they are actually smooth. 

Since the functions v_1(s), v_2(s), are G-invariant, the corresponding domains Ø_ł(s)^v⃗(s) and solutions u of (<ref>) are also G-invariant. We point out that the orthogonality condition (<ref>) implies that for s≠ 0, the non-zero v⃗(s) is also non-constant, which means that at least one of the boundary components _r^v⃗(s), r=ł(s),1, is different from a central dilation of ^n-1 with respect to the origin. In addition, Property (P1) of G prevents Ø_ł(s)^v⃗(s) from being an affine transformation of the annulus Ø_ł that involves a non-trivial translation. All this guarantees the nontriviality of Ø_ł(s)^v⃗(s). 

The domains Ø=Ø_ł(s)^v⃗(s) are constructed to be of class C^2,, but by the classical regularity result of Kinderlehrer and Nirenberg <cit.>, the boundary of a C^2,-domain Ø, supporting a solution u∈ C^2,(Ø) to (<ref>), gets upgraded to a smooth one (in fact, to an analytic one). The solution u itself is in C^∞(Ø). 







§ REFORMULATING THE PROBLEM AND DERIVING ITS LINEARIZATION


Let us first recast the operator F_ł, defined in (<ref>), by pulling back the Dirichlet problem (<ref>) from Ø_ł^v⃗ to the annulus Ø_ł, where we shall use polar coordinates

    (0,∞) ×^n-1≅^n ∖{0}  under   (r,þ) ↦ x = rþ
 
to describe the geometry. In this way, Ø_ł≅ (ł,1) ×^n-1, its boundary components _ł≅{ł}×^n-1, _1 ≅{ 1 }×^n-1 are two copies of ^n-1, and we  naturally get the identification of functions (<ref>). 

For any v⃗ = (v_1,v_2) ∈ U ⊆ (C^2,(^n-1))^2 of sufficiently small norm, we consider the diffeomorphism : Ø_ł→Ø_ł^v⃗ defined in polar coordinates by
	
    (r,þ) = ((1 + η_1(r) v_1(þ) + η_2(r) v_2(þ))r, þ) ,

where η_j, j=1,2, are smooth functions satisfying
	
    η_1(r) =  1/ł    if  r ≤ł +  0     if  r ≥ł +2 ,     η_2(r) =  -1     if  r ≥ 1-   0     if  r ≤ 1 -2 ,

for some small enough >̣0.  We set on Ø_ł the pull-back metric g = ^*g_0, where g_0 is the Euclidean metric on Ø_ł^v⃗. Near Ø_ł, the metric g equals
	
    g = (1 + η_j v_j)^2dr^2 + 2r η_j   (1 + η_j v_j)drdv_j + r^2η_j^2 dv_j^2 + r^2(1 + η_j v_j)^2 g_^n-1 ,

where g_^n-1 is the standard metric on ^n-1. Since  is an isometry between (Ø_ł, g) and (Ø_ł^v⃗, g_0), u_ł(v⃗) is the solution of the Dirichlet problem (<ref>) in Ø_ł^v⃗ if and only if u_ł^*(v⃗) := ^*u_ł(v⃗) is the solution of
	
    -_g u = 1     in Ø_ł , 
    
                u = 0     on _1 , 
    
                u = a_ł    on _ł .

By Schauder theory, u_ł^*(v⃗) ∈ C^2,(Ø_ł) and it depends smoothly on v⃗.


Let ^* denote the inner unit normal field to Ø_ł with respect to the metric g. We have _*^* =, and to find the expression for _ u_ł(v⃗) in these coordinates, we need to compute _^*u_ł^*(v⃗) = g(∇_g u_ł^*(v⃗), ^*). In the new coordinates, the operator F_ł: U → (C^2,(^n-1)^2 thus becomes 
	
    F_ł(v⃗) := 1/c_ł( . u_ł^*(v⃗)/^*|__ł - . u_ł/|__ł , . u_ł^*(v⃗)/^*|__1 -. u_ł/|__1).





Then F_ł(0⃗) = 0⃗ for all ł∈ (0,1), since u_ł^*(0⃗) = u_ł, and F_ł(v⃗) = 0⃗ if and only if _^* u_ł^*(v⃗)=c_ł is constant on Ø_ł. The latter is equivalent to u_ł(v⃗) solving the overdetermined problem (<ref>)-(<ref>) in Ø_ł^v⃗.

In the following proposition we will compute the linearization at v⃗=0⃗ of the operator F_ł(v⃗), reformulated as in (<ref>). Recall that we use the identification (<ref>) of functions on Ø_ł with a pair of functions on ^n-1. 


    The smooth operator F_ł, defined in (<ref>), has a linearization at v⃗ = 0⃗, 
 L_λ := D_v⃗ F_λ|_v⃗=0⃗ : (C^2,(^n-1))^2 → (C^1,(^n-1))^2 given by 
    
    L_λ(w_1,w_2) = (  ._w⃗/|__ł + w_1/c_ł.^2 u_ł/ r^2|__ł, 
            ._w⃗/|__1 + w_2/c_ł.^2 u_ł/ r^2|__1) ,

    where for w⃗=(w_1, w_2) ∈ (C^2,(^n-1))^2, _w⃗ denotes the harmonic function ∈ C^2,(Ø_ł) with boundary values 
    |__ł(x) = w_1(x/|x|) and |__1(x) = w_2(x/|x|).
    
        As F_ł is a smooth operator, its linearization at 0⃗ is given by the directional derivative
        
    L_λ(w⃗) = lim_t→0F_λ(tw⃗)/t .

        Write v⃗=(v_1, v_2)=t(w_1, w_2) for small t and consider the diffeomorphism = _t defined in (<ref>) and the induced metric g = g_t on Ø_ł. Let u_ł^*(v⃗) = u_t be the solution of the Dirichlet problem (<ref>) in Ø_ł, which smoothly depends on the parameter t. Since u_ł is a radial function and can be extended by (<ref>) to solve -_g_0 u_ł =1 in the whole of ^n ∖{ 0 }, we have that u_ł^* := _t^*u_ł is well defined and solves
        
    -_g_t u = 1    in   Ø_ł.
         
        Expanding  u_ł^* = u_ł^*(r,þ) in a neighbourhood of Ø_ł to first order in t, we obtain
        
    u_ł^*(r,þ) = u_ł((1+tη_jw_j)r,þ) = u_ł(r) + tr η_j(r) w_j(þ)  u_ł/ r + O(t^2),

        where η_j, j=1,2, are the functions defined in (<ref>).
        
         
        Let ψ_t := u_t - u_λ^*. Then ψ_t ∈ C^2,(Ø_ł) is a solution of
        
    _g_tψ_t = 0      in Ø_ł , 
    ψ_t = -u_λ^*      on _1 , 
    ψ_t = a_λ - u_λ^*     on _ł ,

        which depends smoothly on t, with ψ_0 = 0. Setting ψ̇ := d/dtψ_t |_t=0, we can differentiate (<ref>) at t=0 to obtain
        
    ψ̇ = 0      in Ø_ł , 
    ψ̇ = (_ru_λ) w_2  = -c_ł w_2    on _1 , 
    ψ̇ = -(_ru_λ) w_1 = -c_ł w_1    on _ł ,

        so that 
        
    ψ̇ =  -c_łϕ_w⃗.

        Now, given that ψ_t = t ψ̇ + O(t^2), we have in a neighbourhood of Ø_λ
        
    u_t = u_λ + t ( ψ̇ + rη_j w_j  u_ł/ r) + O(t^2).

        
        Recall that ^* = _t denotes the inner unit normal field to Ø_ł with respect to the metric g_t. In the following we compute __tu_t to first order in t. As u_t is constant on each boundary component, it follows that __t u_t = |∇_g_t u_t| on Ø_ł. Using formula (<ref>) for g_t in a neighbourhood of Ø_ł, we find
        
    g_t^-1 = ( 1 - 2tη_j w_j + O(t^2) ) dr^2 + O(t) drdw_j + ( 1/r^2 + O(t) ) dþ^2.

        Now, |∇_g_t u_t|^2 = g_t^rr(_r u_t)^2 on Ø_ł and after differentiating (<ref>) with respect to r, we obtain
        
    |∇_g_t u_t| =  u_λ/ + t ( ψ̇/ + w_j ^2 u_ł/ r^2) + O(t^2)   on  Ø_ł.

        The formula (<ref>) for L_ł hence follows from (<ref>) and (<ref>).
        
    






§ SPECTRUM OF THE LINEARIZED OPERATOR


In this section we give an account of the spectral properties of the linearized operator L_ł, which we derived in Proposition <ref>.

Recall that a function 𝒴∈ C^∞(^n-1) is a spherical harmonic of degree k ∈ if it is an eigenfunction of the Laplace-Beltrami operator -_^n-1 on ^n-1, that is,

    _^n-1𝒴 + _k 𝒴 = 0 ,

where _k := k(k+n-2) is the corresponding eigenvalue. We first observe that the subspace W generated by { (𝒴,0), (0,𝒴) } is invariant under L_ł and we shall derive a matrix representation of L_ł|_W with respect to a certain convenient basis.


    Let 𝒴∈ C^∞(^n-1) be a spherical harmonic of degree k∈ and unit L^2(^n-1) norm, and let W be the subspace of C^∞(^n-1) × C^∞(^n-1) spanned by { (𝒴,0), (0,𝒴) }. 
Then W is invariant under the action of L_ł. Moreover, if 

    e⃗_1 := (ł^1-n/2𝒴,0),   e⃗_2 := (0,𝒴),
     
then ℬ={e⃗_1, e⃗_2} is an orthonormal basis for W with respect to the inner product ⟨·, ·⟩_ł on L^2(^n-1)× L^2(^n-1), defined in (<ref>), and the matrix representing the restriction L_ł|_W with respect to the basis ℬ is given by

    M_ł,k = ( [ 1/ł(k+n-2)ł^2-n-k + kł^k/ł^2-n-k - ł^k - n-1/ł                    ł^1-n/22-n-2k/ł^2-n-k - ł^k;                    ł^1-n/22-n-2k/ł^2-n-k - ł^k    kł^2-n-k + (k+n-2)ł^k/ł^2-n-k - ł^k + (n-1) ]) - 1/c_ł𝕀,      k≥ 1,

while for k=0, 

    M_ł,0   = ( [ 1/ł(n-2)ł^2-n/ł^2-n - 1 - n-1/ł            ł^1-n/22-n/ł^2-n - 1;            ł^1-n/22-n/ł^2-n - 1           n-2/ł^2-n - 1 + (n-1) ]) - 1/c_ł𝕀  when   n≥ 3, 
    
           M_ł,0   = ( [ -1/ł1/logł - 1/ł     ł^-1/21/logł;     ł^-1/21/logł     - 1/logł + 1 ]) - 1/c_ł𝕀  when   n=2.


    
        It is easy to verify that ℬ is an orthonormal basis for W with respect to the inner product ⟨·, ·⟩_ł. Let w⃗ = a e⃗_1 + b e⃗_2. For n≥ 3 and k∈, as well as for n=2, k∈,  the function _w⃗∈ C^∞(Ø_ł), defined in polar coordinates r ∈ [ł,1], þ∈^n-1 by 
        
    _w⃗(r,þ) = (aA(r) + bB(r)) 𝒴(þ) ,
 
        where
        
    A(r) = ł^1-n/2r^2-n-k-r^k/ł^2-n-k-ł^k ,      B(r) =  ł^2-n-kr^k-ł^k r^2-n-k/ł^2-n-k-ł^k ,

        is harmonic in Ø_ł and satisfies 
    _w⃗(ł,þ) = a  ł^1-n/2𝒴(þ),   _w⃗(1,þ) = b 𝒴 (þ).

        Formula (<ref>) for L_ł hence gives
        
    L_ł(a e⃗_1 + b e⃗_2) = ( -(aA'(ł)+bB'(ł)) 𝒴 +_r^2 u_ł(ł)/c_ł ał^1-n/2𝒴 , (aA'(1)+bB'(1)) 𝒴 + _r^2 u_ł(1)/c_ł b𝒴) 
       =  ( -(aA'(ł)+bB'(ł))𝒴 - n-1/ł ał^1-n/2𝒴 , (aA'(1)+bB'(1)) 𝒴 + (n-1) b𝒴) - 1/c_ł(a e⃗_1 + b e⃗_2) ,

where in the last equality we used the fact that _r^2 u_ł = -1 - n-1/r_r u_ł. Therefore, W is an invariant subspace of L_ł. Plugging in (<ref>) in (<ref>), we easily derive formulas (<ref>) and (<ref>) for the matrix representation M_ł,k of L_ł|_W with respect to the basis ℬ.

When n=2, and k=0, the substitute for (<ref>) is 

    A(r) = ł^-1/2log r/logł,      B(r) = -log r - logł/logł

and we easily check again the invariance of W under L_ł and derive (<ref>).
    


  
Note that the matrix M_ł,k in Lemma <ref> is symmetric. This is not surprising taking into account the fact that the operator L_ł is formally self-adjoint with respect to the inner product ⟨·,·⟩_ł over the space V:=(C^2,(^n-1))^2. Indeed, for w⃗_⃗1⃗, w⃗_⃗2⃗∈ V, identified as functions on Ø_ł under (<ref>), let ϕ_w⃗_⃗1⃗, ϕ_w⃗_⃗2⃗ denote their corresponding harmonic extensions to Ø_ł. Using formula (<ref>) for L_ł and the definition (<ref>) of the inner product ⟨·,·⟩_ł, an application of Green's formula and harmonicity yield

    ⟨ L_łw⃗_⃗1⃗, w⃗_⃗2⃗⟩_ł    = ∫_Ø_ł(-ϕ_w⃗_⃗1⃗/ν + ^2 u_ł/ r^2ϕ_w⃗_⃗1⃗/c_ł)ϕ_w⃗_⃗2⃗  dS 
        = ∫_Ø_łϕ_w⃗_⃗1⃗(-ϕ_w⃗_⃗2⃗/ν + ^2 u_ł/ r^2ϕ_w⃗_⃗2⃗/c_ł)   dS = ⟨w⃗_⃗1⃗, L_łw⃗_⃗2⃗⟩_ł.




Note that the matrices M_ł, k, ł∈ (0,1), k∈, derived in Lemma <ref>, fit in a two-parameter family of symmetric matrices 

    ℳ={M_ł,k: (ł, k)∈ (0,1)× [0,∞)}.

where we define M_ł, k for non-integral k∈ [0,∞) by the analytic formula in equation (<ref>). In that way, the family ℳ is analytic in both (ł, k)∈ (0,1)× (0,∞). Moreover, we can see that ℳ is continuous up to (ł, k)∈ (0,1)×{0} after easily checking

    lim_k→ 0^+ M_ł,k = M_ł,0,

where M_ł, 0 is given by (<ref>) when n=2 and by (<ref>) when n=3. 

The symmetric matrices of ℳ are never multiples of the identity (the off-diagonal entries are non-zero), whence every M_ł,k∈ℳ has two distinct real eigenvalues 
    _k,1(ł)<_k,2(ł)
 and each is a smooth function of  (ł, k) ∈ (0,1)× (0,∞), continuous up to (ł, k)∈ (0,1)×{0}. Furthermore, since any eigenvector of M_ł,k∈ℳ has two non-zero entries, we can define v_j(ł,k)∈^2 to be the unique eigenvector of M_ł,k, associated with _k,j(ł), j=1,2, that has unit Euclidean norm and positive first entry. Clearly, the eigenvector v_j(ł,k), j=1,2, depends smoothly in (ł, k) ∈ (0,1)× (0,∞) and is contininuous up to (ł, k)∈ (0,1)×{0}, as well.



Before we continue, it will be convenient to recast the matrices M_ł,k∈ℳ in new notation that will greatly facilitate the computations when we analyze the behaviour of its eigenvalues _k,j(ł), j=1,2. For that purpose, first define the matrix

    M̃_ł,k := M_ł, k + 1/c_ł𝕀 ,      M_ł,k∈ℳ

whose eigenspaces are the same as those of M_ł,k and whose eigenvalues _k,j(ł) are shifts of _k,j(ł) by 1/c_ł:

    _k,j(ł) = _k,j(ł) - 1/c_ł ,      j=1,2 .

For a given k ∈ (0,∞) we shall denote

    := n/2+k-1,   ∈ (0,∞)  and   e^ø:=ł^-,   ø∈ (0,∞).



	Let k ∈ (0,∞) be given and let ,ø be as in (<ref>). The matrix M̃_ł,k defined in (<ref>) takes the form
	
    M̃_ł,k = ( [  1/ł(ø - n/2) -/√(ł)1/sinhø; -/√(ł)1/sinhø       ø + n/2 ])

	and its eigenvalues are given by
	
    _k,j(ł) = C ∓√(C^2 - 4ł D)/2ł ,      j=1,2,

	where
	
    C = (ł+1)ø + n/2(ł-1) ,       D = ^2 - n^2/4 = (n+k-1)(k-1) .

	
	
		These are straightforward computations. First note that
		
    (k+n-2)ł^1-n/2-k + kł^n/2+k-1    = (2-k)ł^- + kł^
        = ( n/2-1 )(ł^- - ł^) + (ł^- + ł^)

		and, using the expresion in (<ref>), we find the (1,1)-entry of M̃_ł,k to be
		
    1/ł((k+n-2)ł^1-n/2-k + kł^n/2+k-1/ł^1-n/2-k - ł^n/2+k-1 - (n-1) )    =1/ł( ( n/2-1 ) + ł^- + ł^/ł^- - ł^ -(n-1) ) 
        = 1/ł(ø - n/2).

		In a similar fashion we compute the (2,2)-entry of M̃_ł,k. Also
		
    ł^1-n/22-n-2k/ł^2-n-k-ł^k = -2/√(ł)1/ł^--ł^ = -/√(ł)1/sinhø .

		This establishes equation (<ref>). The characteristic equation for M̃_ł,k then computes to
		
    ł^2 - {(ł+1)ø +n/2(ł-1) } + {^2 - n^2/4} = 0 ,

		
		from which we derive formulas (<ref>)-(<ref>) for its eigenvalues.
	




	Note that when k=1, we have = n/2 and e^ø = ł^-n/2, so that C and D in (<ref>) evaluate to
	
    C=n/2((ł + 1)ł^-n/2 + ł^n/2/ł^-n/2 - ł^n/2 + ł -1 ) = n ł + ł^n/1-ł^n = ł/c_ł,      D=0.

	Hence, (<ref>) gives us _1,1(ł) = 0 and _1,2(ł)=1/c_ł, and the eigenvalues of the matrix M_ł,1  are then given by 
	
    _1,1(ł) = -1/c_ł ,     _1,2(ł) = 0.
 
	We observe that, over subspaces W=Span{ (𝒴,0), (0,𝒴) }, where 𝒴 is a spherical harmonic of degree 1, the linearization L_ł|_W has a kernel of dimension 1 for every ł∈ (0,1). This kernel precisely corresponds to deformations of the standard annulus Ø_ł generated by translations. 
	
	



In the following key sequence of lemmas we will examine the behaviour of the eigenvalues μ_k,1(ł) and μ_k,2(ł) in both k∈ and ł∈ (0,1). See Figure <ref> below for a plot of these branches for k=0,1,2,3, in dimension n=3. 




First, we will establish the first branch μ_k,1(ł) is strictly decreasing in ł∈ (0,1) for any given k ≥ 2. The proof of the next lemma is based on a delicate use of hyperbolic trigonometric identities. 

	For k∈, let M_ł,k be the matrix defined in (<ref>) and let _k,1: (0,1) → denote its first eigenvalue. For every k ≥ 2 the following are satisfied:
	
		
  * lim_ł→0_k,1(ł) = k-1, lim_ł→1_k,1(ł) = -∞;
		
  * _k,1'(ł) < 0, and so _k,1(ł) is strictly decreasing in ł.
	

	
		Fix k ∈, k>1. We shall first prove that '_k,1(ł) < 0, where ' denotes differentiating with respect to ł. As 1/c_ł=n1+ł^n-1/1-ł^n is strictly increasing in ł,  by (<ref>) it suffices to show that '_k,1(ł) ≤ 0. 
        Since by (<ref>)
        
    _k,1(ł) = 2D/C + √(C^2-4ł D)

        and given that D=(n+k-1)(k-1) is positive and constant in ł, we only need to show that
        
    /ł( C + √(C^2-4ł D)) =  C'(√(C^2-4ł D)+C)-2D/√(C^2-4ł D) > 0.

        Using the identity ^2ø - 1/sinh^2ø = 1 successively, we get 
        
    C^2 - 4ł D     = {(1+ł)ø - n/2(1-ł) }^2 - 4ł{^2 - n^2/4}
        = ^2(1+ł)^2 ( 1 + 1/sinh^2 ø) - n(1+ł)(1-ł)ø + n^2/4(1-ł)^2 - 4ł( ^2 - n^2/4) 
        = ^2(1-ł)^2 - n(1+ł)(1-ł)ø + n^2/4(1+ł)^2 + ^2(1+ł)^2/sinh^2ø
        = ^2(1-ł)^2 ( 1 + 1/sinh^2ø) - n(1+ł)(1-ł)ø + n^2/4(1+ł)^2 + 4ł^2/sinh^2ø
        = {(1-ł) ø - n/2(1+ł) }^2 + 4ł^2/sinh^2 ø ,
        
        which gives the estimate 
        
    √(C^2-4ł D) > (1-ł)ø - n/2(1+ł).

         On the other hand, using the fact that ø' = -/ł,
         
    C' = ø + 1+ł/ł^2/sinh^2 ø + n/2 > ø + ^2/sinh^2 ø + n/2,

        so that (<ref>) and (<ref>) yield
        
    C'(√(C^2-4ł D) + C) - 2D     > ( ø + ^2/sinh^2 ø + n/2) ( 2ø - n ) - ( 2^2 -n^2/2) 
        = 2^2 ( 1 + 1/sinh^2ø) + 2^3ø/sinh^2ø - n^2/sinh^2ø - 2^2 
        = 2^2/sinh^2 ø( 1 + ø - n/2) 
        > 2^2/sinh^2 ø( 1+-n/2) = 2^2k/sinh^2 ø > 0,

        where in the penultimate inequality we used ø >1. This confirms (<ref>) and completes the proof of the strict monotonicity of _k,1(ł) in ł.
         
         To derive the limiting behaviour of _k,1(ł) as ł→ 0, we first note that from the definition (<ref>), we have lim_ł→0ø = ∞, so that (<ref>) gives lim_ł→0 C =  - n/2 = k-1, and since lim_ł→0 1/c_ł = n, we calculate
         
    lim_ł→0_k,1(ł) = lim_ł→0(2D/C + √(C^2-4ł D) - 1/c_ł) =  D/k-1 - n = k-1 .

         As to the limiting behaviour of  _k,1(ł) as ł→ 1, the fact that _k,1(ł) is decreasing in ł and lim_ł→1 1/c_ł = ∞  yields 
    lim_ł→ 1_k,1(ł) = -∞.

	


Next, we will prove that, for fixed ł, both _k,1(ł) and _k,2(ł) increase with k. As the explicit formulas (<ref>)-(<ref>) for the eigenvalues turn out to be unyielding, we accomplish this instead by treating k as a continuous variable and showing that _k M_ł,k is positive definite. 

	For fixed ł∈ (0,1) and j=1,2 the sequence {_k,j(ł) }_k=0^∞ is strictly increasing. 
	
		
		We shall treat k∈ [0,∞) as a continuous variable, following the discussion in Remark <ref>. First, we restrict ourselves to k>0 and fix ł. Recall that in the remark we defined v = v_j(ł,k)∈^2 to be the unique eigenvector of M_ł,k associated with eigenvalue _k,j(ł), j=1,2, which has unit Euclidean norm and positive first entry. Then _k v ∈^2 is orthogonal to v and since M_ł,k is symmetric, we have
        
    ⟨ M_ł,k(_k v),v ⟩ = ⟨_k v, M_ł,k v ⟩ =_k,j⟨_k v, v ⟩ =0.

         Differentiating the identity _k,j=⟨ M_ł,kv,v ⟩ with respect to k and using (<ref>), we obtain
        
    _k _k,j = ⟨ (_k M_ł,k)v,v ⟩ + ⟨ M_ł,k(_k v),v ⟩ + ⟨ M_ł,k v,_k v ⟩ = ⟨_k M_ł,kv,v ⟩.

        Therefore, we will have the desired _k _k,j>0 once we show that the symmetric matrix _k M_ł,k is positive definite. Using / k = 1 and ø/ k =ø /, we compute from (<ref>)
        
    _k M_ł,k = _k M̃_ł,k = ([               1/ł( ø - ø/sinh^2 ø) 1/√(ł)( øcoshø/sinh^2 ø - 1/sinhø); 1/√(ł)( øcoshø/sinh^2 ø - 1/sinhø)                    ø - ø/sinh^2 ø. ]).

        We see that its determinant
        
    (_k M_ł,k)     = 1/ł{( ø - ø/sinh^2 ø)^2 - ( øcoshø/sinh^2 ø - 1/sinhø)^2 }
        = 1/łsinh^4 ø (sinh^2 ø - ø^2) (cosh^2 ø -1) > 0,

        as sinhø > ø and coshø >1 for ø∈ (0,∞). Furthermore, the (2,2)-entry of _k M_ł,k satisfies
        
    ø - ø/sinh^2 ø = coshøsinhø - ø/sinh^2 ø >sinhø - ø/sinh^2 ø> 0 .

        Therefore, by Sylvester's criterion the matrix  _k M_ł,k is positive definite for k>0. 
        Since according to Remark <ref>, _k,j(ł) is continuous in k∈ [0,∞) for fixed ł, we can conclude 
    _k+1,j(ł) > _k,j(ł)   for all   k∈,    j=1,2.
 
	


In the final lemma of this section we derive the asymptotics of  _k,1(ł) and _k,2(ł) as k→∞.

	For fixed ł∈ (0,1) the sequences {_k,j(ł) }_k=0^∞, j=1,2, have the asymptotics 
    lim_k→∞_k,1(ł)/k = 1 ,     lim_k→∞_k,2(ł)/k = 1/ł .

	
		From the definition of C and D in (<ref>) and the fact that lim_k→∞ø = 1, we calculate 
		
    lim_k→∞C/k = 1 + ł ,    lim_k→∞D/k^2 = 1 .

		Hence, using equations (<ref>) and (<ref>), we obtain
		
    lim_k→∞_k,j(ł)/k   = lim_k→∞(_k,j(ł)/k - 1/c_ł k) = lim_k→∞1/2ł( C/k∓√(C^2/k^2 - 4ł D/k^2)) 
        = (1+ł) ∓√((1+ł)^2 - 4ł)/2ł = {[   1 j=1; 1/ł j=2 ]..

	


As a corollary to the lemmas above, we state the following proposition.
 Let k∈ and let μ_k,1(ł) and μ_k,2(ł) be the eigenvalues of the matrix M_ł,k defined in Lemma <ref>. The following statements are satisfied: 


  * both eigenvalues for k=0 are negative

    μ_0,1(ł) < μ_0,2(ł) < 0   for all  ł∈ (0,1);



  * for k=1, the eigenvalues are equal to
	
    _1,1(ł) = -1/c_ł ,     _1,2(ł) = 0;
 


  * for every k≥ 2, the second eigenvalue

    μ_k,2(ł) > 0   for all  ł∈ (0,1);



  * for every k≥ 2, there exists a unique value ł_k^*∈ (0,1) such that the first eigenvalue

    μ_k,1(ł^*_k)=0.

Moreover, the sequence {ł_k^*}_k=2^∞ is strictly increasing with lim_k→∞ł_k^* =1.



In (<ref>) we calculated that μ_1,2≡ 0, hence by Lemma <ref> we have that for k≥ 2

    μ_k,2 > μ_1,2≡ 0 > μ_0,2 > μ_0,1.

so that we show both (<ref>) and (<ref>). Equation (<ref>) is (<ref>) reproduced here for the sake of completeness. Only the last bullet point remains to be established.

According to Lemma <ref>, for k≥ 2 the first branch μ_k,1(ł) is strictly decreasing in ł and

    lim_ł→ 0μ_k,1(ł)= k-1 > 0   while  lim_ł→ 1μ_k,1(ł) = -∞.

Thus, when k≥ 2, μ_k,1(ł) has a unique zero ł=ł_k^* in (0,1), where it changes sign from positive to negative. Since the k-monotonicity Lemma <ref> implies that  

    μ_k+1,1(ł_k^*)>μ_k,1(ł_k^*) = 0

we must have ł_k+1^*>ł_k^*,  so that the sequence of zeros {ł_k^*}_k=2^∞ is strictly increasing. Denote its limit by l=lim_k→∞ł_k^*. Obviously, ł_k^*≤ l≤ 1 for all k≥ 2. If it were the case that l<1, we would have by the asymptotic behaviour of μ_k,1(ł), established in Lemma <ref>, that for any large enough k, μ_k,1(l)/k>1/2. But then the zero ł_k^* of μ_k,1(ł) would have to be greater than l, which is a contradiction. Hence, l=1.







§ THE PROOF OF THEOREM <REF>


We now turn to the proof of Theorem <ref>. Following the discussion given in Section <ref>, it will be necessary to specialize to functions that are invariant under the action of a subgroup G of the orthogonal group O(n) satisfying (P1)-(P2), stated in Section <ref>. Recall that C_G^k,(^n-1) denotes the Hölder space of G-invariant functions. 

We begin by observing that the operator F_ł defined in (<ref>) restricts to the G-invariant function spaces (C_G^k,(^n-1))^2 and, therefore, so does its linearization L_ł.


	The nonlinear operator F_ł defined in (<ref>) and its linearization L_ł=D_v⃗F_ł(0⃗) have well defined restrictions
	
    [               F_ł  :                    U                    → ( C_G^1,(^n-1) )^2 ,;               L_ł  :   ( C_G^2,(^n-1) )^2                    → ( C_G^1,(^n-1) )^2 , ]

	where U ⊆ (C_G^2,(^n-1))^2 is a sufficiently small neighbourhood of 0⃗. 
	
	
		We just have to explain why F_ł(v⃗) ∈ (C_G^1,(^n-1))^2 if v⃗∈ U ⊆ (C_G^2,(^n-1))^2. Clearly, if v⃗ is G-invariant, then so is the pull-back metric g=g(v⃗) = ^* g_0 on Ø_ł, where  is the diffeomorphism defined in (<ref>). Hence, by unique solvability, the solution u^*_ł(v⃗)∈ C^2,(Ø_ł) of the Dirichlet problem (<ref>) is also G-invariant, and we confirm that F_ł(v⃗) belongs to (C^1,_G(^n-1))^2, indeed.
	



Recall that properties (P1)-(P2) of G say that the G-invariant spherical harmonics are only the ones of degree {i_k}_k∈, with i_0 =0 and i_1≥ 2, and for each k∈, they form a one-dimensional subspace – spanned by the unique G-invariant spherical harmonic Y_k of degree i_k and unit L^2(^n-1) norm. For each k∈, let W_k= Span{(Y_k, 0), (0,Y_k)}, let ℬ_k={e⃗_1, e⃗_2 } be the orthonormal basis for W_k, defined in (<ref>), and let M_ł, i_k be the matrix of L_ł|_W_k with respect to ℬ_k. Also, recall that in Remark <ref> we chose the eigenvector

    v_j(ł, i_k)=(a_k,j, b_k,j) ,    k∈,   j=1,2,   where   a_k,j>0   and    a_k,j^2 + b_k,j^2 = 1,

to span the eigenspace of M_ł, i_k, associated with  μ_i_k, j(ł). The corresponding eigenvector of L_ł is

    z⃗_k,j:= a_k,je⃗_1 + b_k,je⃗_2 ,      k∈,    j=1,2,   and its norm  z⃗_k,j_ł = 1.



    The sequence of eigenvectors {z⃗_k,j(ł)}_k∈, j=1,2 of L_ł forms an orthonormal basis for the Hilbert space L_G^2(^n-1) × L_G^2(^n-1),  endowed with the inner product ⟨·, ·⟩_ł defined in (<ref>), which is equivalent to the usual one.


Since i_1≥ 2, Proposition <ref> says that the eigenvalues _i_k,1(ł), k∈, cross 0 at values ł_k := ł_i_k^* ∈ (0,1), with ł_k↗ 1, while the eigenvalues _i_k,2(ł)>0. In addition, the eigenvalues _i_0,1(ł) and _i_0,2(ł) are strictly negative for all ł∈ (0,1).  Theorem <ref> will follow after a direct application of the Crandall-Rabinowitz Theorem (see Appendix, Theorem <ref>) to the smooth family of nonlinear operators F_ł: U → (C_G^1,(^n-1))^2 from Lemma <ref>, and the following proposition puts us exactly in the framework of that theorem. In order to simplify notation, we will denote  
    z⃗_k := z⃗_k,1  for   k∈,
 
where z⃗_k,1 is defined in (<ref>).


    For every k ∈, the linear operator L_ł_k: (C_G^2,(^n-1))^2 → (C_G^1,(^n-1))^2 in Lemma <ref> has kernel of dimension 1 spanned by z⃗_k, closed image of co-dimension 1 given by
    
    L_ł_k = {w⃗∈( C_G^1,(^n-1) )^2 : ⟨w⃗, z⃗_k ⟩_ł_k = 0 } ,

    and satisfies
    
    . /ł L_ł|_ł=ł_k (z⃗_k) ∉ L_ł_k .

    
    
        For a proof of (<ref>), see <cit.>, as it follows almost verbatim. For the sake of completeness, we shall provide some of the details. Our first observation is that the Sobolev space H^s(^n-1) can be characterized as the subspace of funtions  v ∈ L^2(^n-1) such that
        
    ∑_j=0^∞ (1+j^2)^s  P_j (v) _L^2^2 < ∞ ,

        where P_j denotes the L^2-orthogonal projection on the subspace generated by the spherical harmonics of degree j, and we denote H_G^s(^n-1) := H^s(^n-1) ∩ L_G^2(^n-1). As stated in Remark <ref>, the sequence {z⃗_k,j} is an orthonormal basis for (L_G^2(^n-1))^2 with inner product ⟨·, ·⟩_ł, and so we can define the map  (H_G^2(^n-1))^2 → (H_G^1(^n-1))^2
        
    w⃗ = ∑_ℓ=0^∞ ( a_ℓ,1z⃗_ℓ,1 + a_ℓ,2z⃗_ℓ,2 ) ↦∑_ℓ=0^∞ ( a_ℓ,1_i_ℓ,1(ł) z⃗_ℓ,1 + a_ℓ,2_i_ℓ,2(ł) z⃗_ℓ,2 ) .

         Due to the asymptotic behavior of the sequences {_m,j(ł) }_m=1^∞ proved in Lemma <ref>, we can see that (<ref>) defines a continuous mapping. Since it agrees with L_ł on finite linear combinations of {z⃗_k,j}, which are dense both in (C_G^2,(^n-1))^2 and (H_G^2(^n-1))^2, (<ref>) defines an extension of L_ł. Moreover, for ł=ł_k the map
        
    w⃗ = ∑_ℓ=0^∞ ( b_ℓ,1z⃗_ℓ,1 + b_ℓ,2z⃗_ℓ,2 ) ↦∑_ℓ=0 
    ℓ≠ k^∞( b_ℓ,1/_i_ℓ,1(ł_k)z⃗_ℓ,1 + b_ℓ,2/_i_ℓ,2(ł_k)z⃗_ℓ,2) + b_k,2/_i_k,2(ł_k)z⃗_k,2

        is a right inverse for L_ł_k, which is also continuous by Lemma <ref>. Thus, L_ł_k defines an isomorphism between the spaces
        
    𝔛_k     := {v⃗∈( H_G^2(^n-1) )^2 : ⟨v⃗, z⃗_k ⟩_ł_k = 0 } ,  
    𝔜_k     := {v⃗∈( H_G^1(^n-1) )^2 : ⟨v⃗, z⃗_k ⟩_ł_k = 0 } .

        It follows that L_ł_k: 𝔛_k ∩ (C_G^2,(^n-1))^2 →𝔜_k ∩ (C_G^1,(^n-1))^2 is a well defined, injective mapping. It only remains to prove its surjectivity. For that purpose, let y⃗∈𝔜_k ∩ (C_G^1,(^n-1))^2. Then there exists w⃗∈𝔛_k such that L_ł_k(w⃗) = y⃗. The latter means that the weak solution ∈ H^2(Ø_ł) to
        
    = 0    in Ø_ł , 
     = w_1    on _ł , 
     = w_2    on _1 ,

        satisfies
                
    -/ + w_1/c_ł_r^2 u_ł/ r^2    = y_1   on _ł , 
    
                -/ + w_2/c_ł_r^2 u_ł/ r^2    = y_2    on _1 .

        
        
        From here  one argues that ∈ W^2,p(Ø_ł) for every p ∈ (1,∞) so that by Sobolev embedding ∈ C^1,(Ø_ł), for all 0<<1. But then ϕ is also weak solution to  the Neumann problem
              
    = 0    in Ø_ł , 
    / =w_1/c_ł_r^2 u_ł/ r^2- y_1    on _ł , 
    / = w_2/c_ł_r^2 u_ł/ r^2 - y_2    on _1 ,

  with Neumann conditions in C^1,. Hence, ϕ∈ C^2,(Ø_ł),    
         which implies w⃗∈ (C_G^2,(^n-1))^2.
        
        Therefore, L_ł_k: 𝔛_k ∩ (C_G^2,(^n-1))^2 →𝔜_k ∩ (C_G^1,(^n-1))^2 is also an isomorphism. This readily implies equality (<ref>) and that L_ł_k is spanned by z⃗_k. 
        The tranversality condition (<ref>) follows from the fact that
        
    /ł L_ł|_ł=ł_k (z⃗_k) = μ_i_k,1'(ł_k) z⃗_k .

        which by Lemma <ref> is a non-zero scalar multiple of z⃗_k.
    



Let I_k⊆ (0,1) be a small interval around each critical value ł_k↗ 1. Let U_k⊆ (C_G^2,(^n-1))^2 be an appropriately small neighbourhood of 0⃗, such that for all ł∈ I_k, F_ł is well defined on U_k  via (<ref>). Then the operator 
    
    F: U_k × I_k →  Y:=(C_G^1,(^n-1))^2,      F(v⃗, ł):= F_ł(v⃗) ,
 
is in C^∞(U_k × I_k , Y), and by Proposition <ref>, we can apply the Crandall-Rabinowitz Bifurcation Theorem to get a smooth curve
    
    [                     (-,)                        → ( C_G^2,(^n-1) )^2 × I_k;                        s                        ↦             (w⃗(s),ł(s)) ]

    such that
    
        
  * w⃗(0) =0⃗, ł(0) = ł_k, and ⟨w⃗(s), z⃗_k ⟩_ł_k =0; 
        
  * F_ł(s)(v⃗(s)) = 0, where v⃗(s) = s(z⃗_k + w⃗(s)).
    
    Then, for every s ∈ (-,), the solution u_ł(s)(v⃗(s)) ∈ C_G^2,(Ø_ł(s)^v⃗(s))  to the Dirichlet problem (<ref>) also solves the overdetermined problem (<ref>). 




§ PROOF OF COROLLARY <REF>


Let Ø be any one of the domains constucted in Theorem <ref> and let u∈ C^∞(Ø) be the solution of the corresponding overdetermined problem

    - u = 1       in Ø , 
    
        u = 0     on Ø_0 , 
    
        u = a     on Ø_1 , 
    _ u = c     on Ø.

for some constants a>0 and c>0.


First, let us show that 

    |∇ u| < c   in  Ø.

Indeed, since - u =1, 

    Δ |∇ u|^2 = 2 |D^2 u|^2 + 2 ∇ u ·∇ ( u) =  2 |D^2 u|^2 >0,

so that the function |∇ u|^2 is subharmonic in Ø and, by the strict maximum principle 

    |∇ u|^2 (x) < sup_Ø|∇ u|^2 = c^2   for all   x∈Ø.

Now let E⊆Ø be any subset of finite perimeter and let ^* E⊆ E be its reduced boundary – where one can define a measure-theoretic inner unit normal ν_E. By De Giorgi's theorem (see <cit.>), the (n-1)-Haudorff dimensional measure H^n-1(^* E) = P(E) and we can apply the version of the Divergence Theorem to obtain

    |E| = ∫_E (- u)   dx = ∫_^* E∇ u ·ν_E   d H^n-1≤ c H^n-1(^* E) = c P(E),

where we used (<ref>) in the inequality above. Hence, P(E)/E ≥ 1/c and equality holds if and only if E=Ø.





§ APPENDIX


We give here a version of the Crandall-Rabinowitz Theorem, equivalent to the one stated on <cit.>, which is the principal tool behind Theorem <ref>. For a proof of the theorem and applications, we refer the reader to <cit.>.


    Let X and Y be Banach spaces, and let U ⊂ X and I ⊂ be open sets, such that 0 ∈ U. Let F ∈ C^2 (U × I , Y) and assume
    
    
        
  * F(0,ł) = 0 for all ł∈ I;
        
  * _x F(0,ł_0) is a dimension 1 subspace and _x F(0,ł_0) is a closed co-dimension 1 subspace for some ł_0 ∈ I;
        
  * _ł_x F(0,ł_0)(x_0) ∉_x F(0,ł_0), where x_0 ∈ X spans _x F(0,ł_0).
    
    
    Write X = X̂⊕ x_0. Then there exists a C^1 curve
    
    (-,) →X̂× ,      s ↦ (x(s),ł(s))

    such that
    
    
        
  * x(0) = 0 and ł(0) = ł_0;
        
  * s(x_0+x(s)) ∈ U and ł(s) ∈ I;
        
  * F(s(x_0+x(s)),ł(s)) = 0.
    
    
    Moreover, there is a neighbourhood of (0,ł_0) such that { (s(x_0+x(s)),ł(s)) : s ∈ (-,) } are the only solutions bifurcating from { (0,ł) : ł∈ I }.




