Asymptotic behavior for the principal eigenvalue of a reinforcement problem
Toshiaki Yachimura
math.SP 35J20, 49R05

Quark-hadron phase structure of QCD matter from SU(4) Polyakov linear sigma model
    Abdel MagiedAbdel Aal DIAB1,2a.diab@eng.mti.edu.egAbdel NasserTAWFIK1,2a.tawfik@eng.mti.edu.eg
   
    
        

    December 30, 2023
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In this paper, we consider the asymptotic behavior for the principal eigenvalue of an elliptic operator with piecewise constant coefficients. This problem was first studied by Friedman in 1980. 
We show how the geometric shape of the interface affects the asymptotic behavior for the principal eigenvalue. This is a refinement of the result by Friedman.  




2010 Mathematics Subject classification. 35J20, 49R05


Keywords and phrases: eigenvalue problem, two phase, transmission condition, reinforcement problem, domain perturbation 





§ INTRODUCTION AND MAIN RESULT

In this paper, we study a two-phase eigenvalue problem and we investigate the asymptotic behavior for the principal eigenvalue. First we introduce some notations. Let Ω⊂ℝ^n (n ⩾ 2) be a bounded domain with smooth and connected boundary Γ. For sufficiently small > 0, put 

    Σ_ = { x ∈ℝ^n    |    x = ξ + τν_Γ(ξ)    for   ξ∈Γ, 0 < τ < },   Ω_ = Ω∪Σ_∪Γ,

where ν_Γ denotes the outward unit normal vector to the Γ, see Figure 1. 
We consider the two-phase eigenvalue problem on Ω_ as follows:

    -div(Q_∇Φ) = λΦ   in  Ω_, 
    Φ = 0     on ∂Ω_,

where Q_ = Q_(x) ( x ∈Ω_) is a piecewise constant function given by 

    Q_(x) = 
    1,      x ∈Ω, 
    ,      x ∈Σ_,

where = α and α is a positive parameter. 

We consider the problem (<ref>) in a weak sense, namely, λ∈ℂ is an eigenvalue of (<ref>) if there exists Φ∈ H^1_0(Ω_) such that Φ≢0 and 
for any φ∈ H^1_0(Ω), 

    ∫_Ω∇Φ·∇φ  dx + ∫_Σ_∇Φ·∇φ  dx = λ∫_Ω_Φφ  dx.

By a standard argument of self-adjoint operators, the eigenvalues of (<ref>) are non-negative real numbers and the set of all eigenvalues is discrete. 
Let {λ_k() }_k⩾1 be the eigenvalues satisfying 0 < λ_1() < λ_2() ⩽λ_3() ⩽⋯→ +∞ and {Φ_k,}_k⩾1 be the associated eigenfunctions in (<ref>) which are assumed to be normalized so that 

    ∫_Ω_Φ_k,^2  dx = 1.


Since Q_ = Q_(x) (x ∈Ω_) is a piecewise constant function, we can rewrite (<ref>) as follows:

    - ΔΦ_1 = λΦ_1     in  Ω, 
    
    - ΔΦ_2 = λΦ_2     in  Σ_, 
    Φ_1 = Φ_2     on  Γ, 
    
    ∂Φ_1/∂ν_Γ = ∂Φ_2/∂ν_Γ     on  Γ, 
    Φ_2 = 0      on  Ω_.

Here Φ_1 and Φ_2 are the restriction of the eigenfunction Φ on Ω and Σ_, respectively. The fourth equality in (<ref>) is usually called transmission condition, which can be interpreted as the continuity of the flux through the interface Γ in (<ref>). 


The purpose of this paper is to study the asymptotic behavior for the principal eigenvalue λ_1() as → 0. In particular, our aim is to show how the geometric shape of the interface Γ affects the asymptotic behavior for the principal eigenvalue λ_1().  In what follows, let the principal eigenfunction Φ_1, be denoted by Φ_ for the sake of simple notation. 

The study of two-phase eigenvalue problems arise in the study of the material science of composite media. In particular, the problems dealt with in this paper are called reinforcement problems or coating problems, and are related to vibration frequencies of composite materials or coating of composite materials with thermal insulation. 

This type of the two-phase eigenvalue problem was first studied by Friedman<cit.>. He considered the two-phase eigenvalue problem for the principal eigenvalue of some elliptic operators in the case lim_→ 0 /  = α and lim_→ 0 /  = 0. His method is based on H^2-estimate of the eigenfunction. Rosencrans–Wang<cit.> generalized Friedman's results to all eigenvalues in the case lim_→ 0 /  = 0. They only used H^1-estimate of eigenfunctions which is easily obtained by the variational characterization of the eigenvalues. Regarding other two-phase eigenvalue problems in this direction, we refer to <cit.><cit.><cit.>. 

In this paper, we treat the case lim_→ 0 /  = α and focus on a refinement of Friedman's result. 
Friedman proved the following theorem:

Let λ_1() be the principal eigenvalue of the eigenvalue problem (<ref>). Then we have  

    λ_1()    = μ_1 + o(1)    as   → 0, 
    Φ_   → w_1   weakly   in    H^2(Ω),

where μ_1 is the principal eigenvalue and w_1 is the principal eigenfunction of the following Robin eigenvalue problem:

    - Δ w = μ w      in  Ω, 
    α w +  wν_Γ = 0     on  Γ.


Theorem <ref> implies that the condition lim_→ 0 /  = α affects the boundary condition, which becomes the Robin boundary condition. 

We derive a more precise asymptotic behavior for the principal eigenvalue. In the following we mention the main result of this paper. 

Let λ_1() be the principal eigenvalue of the eigenvalue problem (<ref>). Then we have the asymptotic behavior 

    λ_1() = μ_1 - ∫_Γ( α H + μ_1/3)w^2_1 + o()    as   → 0,

where H is the mean curvature defined as the sum of the principle curvatures of Γ.   

From Theorem <ref>, we see that the effect of the geometric shape of the interface Γ appears in the second term of the asymptotic behavior for the principal eigenvalue. 

The outline of the proof of Theorem <ref> is as follows: first, we derive an upper bound of the principal eigenvalue by using a variational approach which is based on <cit.>. Next, we derive a lower bound of the principal eigenvalue by using the upper bound and the Fourier expansion with respect to eigenfunctions of a Robin eigenvalue problem. Then we have the asymptotic behavior for the principal eigenvalue. Once the asymptotic behavior is obtained, we can use it to show an L^2-estimate for the tangential components of the principal eigenfunction in Σ_. This is necessary to control the behavior of the principal eigenfunction in the thin layer Σ_. By using this estimate, H^2-estimate, and transmission condition, we finally prove Theorem <ref>. 

The following sections are organized as follows: in section <ref>, we give some geometric preliminaries concerning the thin layer Σ_. In section <ref>, we prove the asymptotic behavior for the principal eigenvalue. In section <ref>, based on the results of section <ref>, we prove Theorem <ref>. 



§ GEOMETRIC PRELIMINARIES

We present some geometric preliminaries of thin layer Σ_. Every x ∈Σ_ can be represented by 

    x = ξ + τν_Γ(ξ),   ξ∈Γ,    0 < τ < .

We introduce a local coordinate system (ξ_1, ξ_2, ⋯ , ξ_n-1, ξ_n) = (ξ_1, ξ_2, ⋯ , ξ_n-1, τ) for Γ× (0, ) and let g = ( g_ij(ξ,τ) ) denote the metric tensor associated with it. Then from (<ref>), g_ij(ξ,τ) is given by 

    g_ij(ξ,τ) = 
    g_0,ij(ξ) + τg_0,ij(ξ) + τ^2g_0,ij(ξ)    if   1⩽ i,j ⩽ n-1, 
    
    0    if   i = n, j ≠ n   or   i ≠ n, j = n, 
    
    1    if   i , j = n,

where g_0 = ( g_0, ij(ξ) ) denotes the Riemannian metric associated with the local coordinates (ξ_1, ξ_2, ⋯ , ξ_n-1) and we put

    g_0,ij = (/ξ_i, ν_Γ/ξ_j) + (/ξ_j, ν_Γ/ξ_i),   g_0,ij = (ν_Γ/ξ_i, ν_Γ/ξ_j).

Here / ξ_i and / ξ_j are tangent vectors on ξ∈Γ and (· , ·) is the Euclidean inner product. Let (b_ij)_1⩽ i,j ⩽ n-1 denote the coefficients of the second fundamental form on Γ. In the local coordinate, b_ij = (^2 / ξ_iξ_j, ν_Γ).
By the definition of g_0,ij, we have g_0,ij = -2b_ij. 
Also we denote the inverse matrix of (g_ij) by (g^ij) and put G =  (g_ij). By using this local coordinates we can express the norm of the gradient of Φ as follows: 

    ∇_xΦ^2 = ∑^n_i,j = 1 g^ijΦ/ξ_iΦ/ξ_j = ∇_tanΦ^2 + ( Φ/τ)^2,

where ∇_tanΦ^2 = ∑^n-1 _i,j=1 g^ijΦ / ξ_iΦ / ξ_j. Moreover, by (<ref>) we can obtain the following asymptotic formula for √(G): 

    √(G(ξ,τ)) = √(G(ξ,0))(1 - H(ξ) τ) + O(τ^2)    as   τ→ 0,

where H(ξ) is the mean curvature at ξ∈Γ with respect to ν_Γ (defined as the sum of the principle curvatures of Γ). 
The asymptotic formula (<ref>) will play an important role in obtaining the asymptotic behavior for the principal eigenvalue λ_1(). 
For the details about the geometric property of a thin layer, see <cit.><cit.><cit.> and the references given there. 

In the following sections, √(G(ξ, τ)) will be denoted by √(G_τ) for simplicity and C will be used to represent any positive constant independent of . The same letter C will be used to denote different constants.



§ ASYMPTOTIC BEHAVIOR FOR Λ_1()



 §.§ Upper bound of λ_1()

By the min-max principle, 

    λ_1() = inf_u ∈ H^1_0(Ω),   u ≠ 0∫_Ω∇ u^2 dx + ∫_Σ_∇ u^2 dx/∫_Ω_u^2 dx.

We construct a test function in order to estimate the principal eigenvalue λ_1(). We extend the normalized Robin principal eigenfunction w_1 = w_1(x) (x ∈Ω) along ν_Γ to Σ_ by setting w_1(ξ, τ) = w_1(ξ) for every ξ∈Γ. 
Also we put

    ϕ(x) = 
    1   in  Ω, 
    
    1 - τ  in  Σ_.

Taking ũ = w_1ϕ as a test function in (<ref>), we obtain 

    λ_1() ⩽∫_Ω∇ũ^2 dx + ∫_Σ_∇ũ^2 dx/∫_Ωũ^2 dx = ∫_Ω∇ w_1^2 dx + ∫_Σ_∇(w ϕ)^2 dx/∫_Ωw_1^2 dx + ∫_Σ_w_1ϕ^2 dx.


By using the normalization ∫_Ωw_1^2 dx = 1 and ∫_Σ_w_1ϕ^2 dx = O(), we have 

    ∫_Ωw_1^2 dx + ∫_Σ_w_1ϕ^2 dx = 1 + O().
 
Also we have ∇ w_1·∇ϕ = 0 since w_1 and ϕ only depend on ξ and τ in Σ_, respectively. Hence,    

    ∫_Σ_∇(w_1ϕ)^2 dx    = ∫_Σ_( ϕ^2∇ w_1^2 + 2 ∇ w_1·∇ϕ + w_1^2∇ϕ^2) dx 
       = ∫_Σ_ϕ^2∇ w_1^2 dx + ∫_Σ_ w_1^2∇ϕ^2 dx 
       = ∫_Σ_ w_1^2∇ϕ^2 dx + O(^2).

We note that ∇ϕ = ϕ/τν_Γ = - ν_Γ/. By using the asymptotic formula (<ref>), we get 

    ∫_Σ_ w_1^2∇ϕ^2 dx    = α w_1(ξ)^2·-ν_Γ/^2
       = α·1/^2 w_1(ξ)^2 (1 + O(1) τ) 
       = α w_1^2 + O().

Thus, 

    λ_1()    ⩽∫_Ω∇ũ^2 dx + ∫_Σ_∇ũ^2 dx/∫_Ωũ^2 dx
       ⩽∫_Ω∇ w_1^2 dx + α w_1^2 + C/1 - C⩽μ_1 + C.

Therefore we obtain the following upper bound of the principal eigenvalue λ_1():  

    λ_1() ⩽μ_1 + C.




 §.§ Lower bound of λ_1()

Recall the weak form (<ref>): for any φ∈ H^1_0(Ω), 

    ∫_Ω∇Φ_·∇φ  dx + ∫_Σ_∇Φ_·∇φ  dx = λ_1() ∫_Ω_Φ_φ  dx.

First of all, we mention that we can get the following H^1 and H^2-estimates of the principal eigenfunction Φ_ by using the upper bound (<ref>). 

The principal eigenfunction Φ_ satisfies 


    ∫_Ω∇Φ_^2  dx + ∫_Σ_∇Φ_^2  dx ⩽ C, 
       ∫_ΩD^2Φ_^2  dx + ∫_Σ_D^2Φ_^2  dx ⩽ C

for a positive constant C independent of . 


It is easy to show the estimate (<ref>) by taking φ = Φ_ in (<ref>) and using the upper bound (<ref>). 
The H^2-estimate (<ref>) derives from a boundary estimate on Γ, which was first established by Brezis–Caffarelli–Friedman<cit.> in the case of two-phase elliptic equations. Friedman<cit.> proved the H^2-estimate (<ref>) by using a similar method. Thus we omit this proof.


We take any ζ∈ C^1(Ω). Let us extend ζ along ν_Γ to Σ_ by ζ(ξ,τ) = ζ(ξ) for every ξ∈Γ. We take φ = ζϕ as a test function in (<ref>), then we have









    ∫_Ω∇Φ_·∇ζ  dx + ∫_Σ_ϕ∇Φ_·∇ζ  dx + ∫_Σ_ζ∇Φ_·∇ϕ  dx 
                                    = λ_1() ∫_ΩΦ_ϕζ  dx + λ_1() ∫_Σ_Φ_ϕζ  dx.

The second term on the left-hand side and the second term on the right-hand side are O(). Indeed, for any ζ∈ C^1(Ω), by using the H^1-estimate (<ref>) we have 

    ∫_Σ_ϕ∇Φ_·∇ζ  dx   ⩽∫_Σ_^1/2∇Φ_·^1/2∇ζ dx 
       ⩽( ∫_Σ_∇Φ_^2 dx )^1/2( ∫_Σ_∇ζ^2 dx )^1/2
       ⩽ C.

By using the upper bound of λ_1(), we also have

    λ_1() ∫_Σ_Φ_ϕζ  dx   ⩽ C ∫_Σ_Φ_ζ dx 
       ⩽ C ( ∫_Σ_Φ_^2 dx )^1/2( ∫_Σ_ζ^2 dx )^1/2.

Now we need to estimate ∫_Σ_Φ_^2 dx. By the Dirichlet boundary condition on Ω_, we get

    Φ_(ξ,τ) = - ∫^_τΦ_τ ds.

This identity implies that

    Φ_^2⩽( ∫^_0Φ_τ ds )^2⩽∫^_0Φ_τ^2 ds.

Thus we have 

    ∫_Σ_Φ_^2 dx ⩽∫_Σ_( ∫^_0Φ_τ^2 ds ) dx ⩽ C ( ∫_Σ_∇Φ_^2 dx ).

Therefore we obtain the following estimate: 

    λ_1() ∫_Σ_Φ_ϕζ  dx⩽ C.


Note that ∇ϕ = ϕ/τν_Γ = - ν_Γ/ and, by using the asymptotic formula (<ref>), we have 

    ∫_Σ_ζ∇Φ_·∇ϕ  dx    = ∫_Σ_ζ∇Φ_·( - ν_Γ/)   dx 
    
       =  - αζΦ_/τ
       = - αζΦ_/τ(1 + O(1) τ) 
       = αΦ_ζ + O().


Therefore, we obtain

    ∫_Ω∇Φ_·∇ζ dx + αΦ_ζ = λ_1() ∫_ΩΦ_ζ dx + O().


We consider the Fourier expansions of Φ_ with respect to the orthonormal basis given by the eigenfunctions of the following Robin eigenvalue problem:  

    - Δ w = μ w      in  Ω, 
    α w +  wν_Γ = 0     on  Γ.

Let {μ_k}_k⩾1 be the eigenvalues corresponding to the problem (<ref>) ordered so that they satisfy 0 < μ_1⩽μ_2⩽μ_3⩽⋯→ +∞ and { w_k}_k⩾1 be the associated eigenfunctions which are assumed to be normalized so that 

    ∫_Ωw_k^2 dx = 1.

Then Φ_ admits the following the Fourier expansions in H^1(Ω): 

    Φ_ = ∑_k ⩾ 1 c_k() w_k,    c_k = ∫_ΩΦ_ w_k dx.


Taking ζ = c_1 w_1 in (<ref>) and using the orthogonality of the Robin eigenfunctions { w_k}_k⩾1, we have 

    (c_1)^2μ = λ_1() (c_1)^2 + O().

From the estimate (<ref>), it will be sufficient to show the following lemma to get the lower bound of λ_1(). 

The following estimate holds:

    c_1() = 1 + o(1)    as   → 0.



From Lemma <ref>, we obtain the H^1-boundedness of the principal eigenfunction Φ_ in Ω. 



Applying Rellich's Theorem, after passing to a subsequence, there exists Φ∈ H^1(Ω) such that Φ_→Φ strongly in L^2(Ω) and weakly in H^1(Ω). Moreover, for some nonnegative value λ we also have λ_1() →λ and λ⩽μ_1.  If we let → 0 in (<ref>), then 

    ∫_Ω∇Φ·∇ζ dx + αΦζ = λ∫_ΩΦζ dx.
 
Thus λ is a Robin eigenvalue and Φ is the corresponding Robin eigenfunction. It implies that μ_1⩽λ. 
Therefore we obtain λ = μ_1. Since μ_1 is the principal eigenvalue, we have Φ = ± w_1. Also, since Φ_ is chosen to be positive function, we get Φ = w_1. By using the fact that Φ_ converges to Φ strongly in L^2(Ω), we get the estimate c_1 = 1 + o(1) as → 0.  

From (<ref>) and Lemma <ref> we have

    λ_1() ⩾μ_1 - C.





Combining the upper bound (<ref>) with the lower bound (<ref>), we obtain

    λ_1() = μ_1 + O()    as   → 0.




§ PROOF OF THEOREM <REF>


First of all, we show the L^2 estimate for the tangential components of ∇Φ_. 

The following estimate holds:

    ∫_Σ_∇_tanΦ_^2 dx = O()    as   → 0.



By using (<ref>) we have 

    λ_1()    = ∫_Ω∇Φ_^2 dx + ∫_Σ_∇Φ_^2 dx 
       = ∫_Ω∇Φ_^2 dx + ∫_Σ_∇_tanΦ_^2 dx + ∫_Σ_Φ_/ν_Γ^2 dx.

Now we estimate ∫_Σ_Φ_/ν_Γ^2 dx. By using the estimate (<ref>), we obtain

    αΦ_^2⩽∫_Σ_Φ_/ν_Γ^2 dx + C.

Therefore, 

    λ_1()    = ∫_Ω∇Φ_^2 dx + ∫_Σ_∇_tanΦ_^2 dx + ∫_Σ_Φ_/ν_Γ^2 dx 
       ⩾∫_Ω∇Φ_^2 dx + ∫_Σ_∇_tanΦ_^2 dx + αΦ_^2 - C
       ⩾μ_1 + ∫_Σ_∇_tanΦ_^2 dx - C.

From the asymptotic behavior (<ref>), we get the following estimate: 

    ∫_Σ_∇_tanΦ_^2 dx = O()    as   → 0.


The estimate in the following lemma will be useful to examine the behavior for Φ_ in Σ_. Recall that Φ_1, and Φ_2, denote the restriction of Φ_ on Ω and Σ_, respectively.  

The following estimate holds: 

    Φ_1, + 1/αΦ_1,/ν_Γ^2_L^2(Γ) = O()    as   → 0.



For any k ∈ (0,), 

    Φ_2,/τ(ξ,k) - Φ_2,/τ(ξ,0) = ∫^k_0^2Φ/τ^2 ds.

By integrating from 0 to τ we have 

    Φ_2,(ξ, τ) = Φ_2,(ξ,0) + τΦ_2,/τ(ξ,0) + ∫^τ_0 dk ∫^k_0^2Φ_2,/τ^2 ds.

Due to the Dirichlet boundary condition on Ω_, we obtain 

    Φ_2,(ξ,0) + Φ_2,/ν_Γ(ξ,0) = - ∫^_0∫^k_0^2Φ_2,/τ^2 ds dk.

Integrating on Γ and using transmission condition, we have 

    Φ_1,(ξ,0) + 1/αΦ_1,/ν_Γ(ξ,0)^2⩽ C ( ∫_Σ_D^2Φ_2,^2 dx ).

Therefore from Lemma <ref> we obtain 

    Φ_1, + 1/αΦ_1,/ν_Γ^2_L^2(Γ) = O()    as   → 0.


























We are now going to derive a more precise asymptotic behavior for the principal eigenvalue λ_1(). Recall that

    ∫_Ω∇Φ_·∇ζ  dx + ∫_Σ_ϕ∇Φ_·∇ζ  dx + ∫_Σ_ζ∇Φ_·∇ϕ  dx 
                                    = λ_1() ∫_ΩΦ_ζ  dx + λ_1() ∫_Σ_Φ_ϕζ  dx.

Noting that ζ only depends on ξ in Σ_ and using Cauchy–Schwarz's inequality, we obtain 

    ∫_Σ_ϕ∇Φ_·∇ζ  dx   = ∫_Σ_ϕ∇_tanΦ_·∇_tanζ  dx
       ⩽( ∫_Σ_∇_tanΦ_^2 dx)^1/2( ∫_Σ_∇_tanζ^2 dx)^1/2 = O(^3/2).

Moreover, 

    ∫_Σ_ζ∇Φ_·∇ϕ  dx    = ∫_Σ_ζ∇Φ_·( - ν_Γ/)   dx 
       =  - αζΦ_/τ
       = - αζΦ_/τ(1 - H(ξ) τ)  + O(^2) 
       = αΦ_ζ + α H(ξ) ζτΦ_/τ + O(^2).

Using integration by parts and (<ref>), we have








    α H(ξ) ζτΦ_/τ = - α  H Φ_ζ + O(^2).

Thus we obtain 

    ∫_Σ_ζ∇Φ_·∇ϕ  dx = αΦ_ζ - α  H Φ_ζ + O(^2).

Furthermore, 

    ∫_Σ_Φ_ϕζ  dx    = Φ_ϕζ
       = ζ( Φ_(ξ,0) + τΦ_/τ(ξ,0) 
    + ∫^τ_0 dk ∫^k_0^2Φ_/τ^2 ds ) 
                   ×( 1 - τ/) ×( 1 - H(ξ) τ + O(τ^2) ) .









By direct computation, we have 

    ∫^_0( 1 - τ/) dτ = /2,   ∫^_0τ( 1 - τ/) dτ = ^2/6,   ∫^_0τ^2( 1 - τ/) dτ = ^3/12.

Thus we obtain 

    ζ( Φ_(ξ,0) + τΦ_/τ(ξ,0) 
    + ∫^τ_0 dk ∫^k_0^2Φ_/τ^2 ds ) 
                   ×( 1 - τ/) ×( 1 - H(ξ) τ + O(τ^2) ) 
       = /2ζΦ_2,(ξ,0)  + ^2/6ζΦ_2,/ν_Γ(ξ,0)  + O(^2) 
       = /2ζΦ_1,(ξ,0)  + /6αζΦ_1,/ν_Γ(ξ,0)  + O(^2),

where we used the transmission condition. 






From Lemma <ref> we have 

    /2ζΦ_1,(ξ,0)  + /6αζΦ_1,/ν_Γ(ξ,0)  + O(^2) 
       = /2ζΦ_1,(ξ,0)  - /6ζΦ_1,(ξ,0)  + O(^3/2) 
       = /3ζΦ_1,(ξ,0)  + O(^3/2).

Therefore

    ∫_Σ_Φ_ϕζ  dx = /3ζΦ_1,(ξ,0)  + O(^3/2).

By using the above and the asymptotic behavior (<ref>), we obtain the following estimate: 

    λ_1() ∫_Σ_Φ_ϕζ  dx = ( μ_1/3ζΦ_1,(ξ,0) ) + O(^3/2).

Combining (<ref>), (<ref>), and (<ref>) we have 

    ∫_Ω∇Φ_·∇ζ  dx + αΦ_ζ - α H Φ_ζ
                                    = λ_1() ∫_ΩΦ_ζ  dx + ( μ_1/3Φ_ζ) + O(^3/2).

By using the fact that Φ_ - w_1_L^2(Γ)→ 0 as → 0, we obtain 

    Φ_ζ =  w_1ζ + o(1)    as   → 0.

Taking ζ = c_1 w_1, we get 

    (c_1)^2μ - α H c_1 w_1^2 = λ_1() (c_1)^2 + ( μ_1/3 c_1 w_1^2) + o().

Dividing by (c_1)^2 and using Lemma <ref>, we finally obtain the following more precise asymptotic behavior for λ_1(): 

    λ_1() = μ_1 - ( α H +μ_1/3) w_1^2 + o()    as   → 0.

The proof of Theorem <ref> is complete. 



Acknowledgments. 

The author would like to thank Professor Shigeru Sakaguchi (Tohoku University) for many stimulating discussions. Also the author would like to thank Lorenzo Cavallina (Tohoku University) for his warm encouragement. 

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