Domain perturbations of two-phase eigenvalue
problems and a related inverse problem
著者
谷地村 敏明
学位授与機関
Tohoku University
学位授与番号
11301甲第19347号
Domain perturbations of two-phase
eigenvalue problems and a related inverse
problem
A thesis submitted for the degree of
Doctor of Philosophy
by
Toshiaki Yachimura
System Information Sciences
Graduate School of Information Sciences
Tohoku University
Contents
1 Introduction 3
1.1 Introduction and main results of this thesis . . . 3
1.1.1 Two-phase eigenvalue problems on thin domains . . . 4
1.1.2 Reinforcement problems . . . 7
1.1.3 On a Robin inverse spectral problems related to the reinforce-ment problems . . . 10
2 Basic results for eigenvalues of second-order elliptic operators 14 2.1 Second-order elliptic operators and Green operators . . . 14
2.1.1 Eigenvalues and eigenfunctions of elliptic operators . . . 16
2.1.2 Variational characterization of the eigenvalues of elliptic op-erators . . . 17
2.1.3 Fredholm alternative . . . 18
3 On the two-phase eigenvalue problem on thin domains 19 3.1 Problem setting for the two-phase eigenvalue problem on thin domains 19 3.2 Geometric preliminaries for thin domains . . . 21
3.3 Upper estimate of the eigenvalues . . . 23
3.4 Lower estimate of the eigenvalues . . . 26
3.5 The case where λk is simple . . . 32
3.6 The case where the interface is a sphere . . . 39
4 Asymtotic behavior of the eigenvalues of reinforcement problems 42 4.1 Problem setting for the reinforcement problem . . . 42
4.3 Asymptotic behavior for λ1(ε) . . . 45
4.3.1 Upper bound of λ1(ε) . . . 45
4.3.2 Lower bound of λ1(ε) . . . 46
4.4 Proof of Theorem 1.1.7 . . . 50
5 Robin inverse spectral problems 54 5.1 Introduction . . . 54
5.2 Uniqueness and Differentiability . . . 55
5.2.1 Uniqueness of the inverse eigenvalue problem . . . 55
5.2.2 Differentiability . . . 56
5.2.3 Local Lipschitz stability . . . 64
5.3 Neumann tracking type functional and its properties . . . 65
5.4 Reconstruction algorithm and numerical tests . . . 68
5.5 The FreeFem++ code for the numerical tests . . . 71
Chapter 1
Introduction
This thesis is based on papers [40, 41, 35].
1.1
Introduction and main results of this thesis
An eigenvalue problem for the Laplace operator in a bounded domain Ω ⊂ Rn(n ≥ 2)
−∆u = λu in Ω (1.1)
arises from various problems of mathematical physics, for example, vibration modes of a thin membrane, the long-time asymptotic distribution of particles diffusing on a container, eigenstates of a particle in quantum mechanics.
The eigenvalues of (1.1) strongly depend on the geometry of the domain Ω. The question of how eigenvalues of (1.1) change when perturbing the domain has been studied by many mathematicians since the pioneering work of Courant–Hilbert [12] who proved that the eigenvalues change continuously for smooth domain perturba-tion.
Since this result, domain perturbations of eigenvalue problems for the Laplace operator have been considered in each situation of domains. For instance, various mathematicians have considered cases where a domain partially degenerates (such as the bar part of a dumbbell domain) [21, 5, 22] or where a domain has small holes [33, 37, 30].
On the other hand, domain perturbations of eigenvalue problems for elliptic operators with piecewise constant coefficients have many open questions due to the
discontinuity of the coefficients. In this thesis, we will consider two-phase eigenvalue problems on thin domains and two-phase eigenvalue problems called reinforcement (or thin coating) problems. In particular, we will discuss how the geometric shape of the domain affects the asymptotic behavior of the eigenvalues.
1.1.1
Two-phase eigenvalue problems on thin domains
Let us formulate the two-phase eigenvalue problem on thin domains. Let D ⊂ Rn
(n ≥ 2) be a bounded domain whose boundary Γ is smooth and connected. For sufficiently small ε > 0, we put
Ω−ε = {x ∈ D | d (x, Γ) < ε}, Ω+ε = {x ∈ Rn\ D | d (x, Γ) < ε},
where d (x, Γ) denotes the Euclidean distance from x to Γ. We define
Ωε= Ω−ε ∪ Ω + ε ∪ Γ.
Let ν denote the outward unit normal vector to the boundary ∂Ωε and νΓ the
outward unit normal vector to the interface Γ. We will now consider a two-phase eigenvalue problem on Ωε as follows:
−div (σ∇u) = λu in Ωε,
∂u
∂ν = 0 on ∂Ωε,
(1.2)
where σ = σ(x) (x ∈ Ωε) is the piecewise constant function given by
σ(x) = σ−, x ∈ Ω−ε ∪ Γ, σ+, x ∈ Ω+ε,
and σ−, σ+ are distinct positive constants (i.e. σ−, σ+> 0, σ−6= σ+), see Figure 1.
Physically speaking, the problem is to consider the frequency of a composite medium when two different materials are joined thinly. This problem is also related to the heat diffusion on thin two-phase heat conductors.
Figure 1: Problem setting for thin domains.
In the case of the Laplacian (i.e. σ− = σ+ = 1), there have been many
re-sults concerning various boundary conditions such that Dirichlet, Neumann, and mixed boundary condition. Krejˇciˇr´ık–Raymond–Tuˇsek [27] proved that for Dirich-let boundary condition, the asymptotic behavior of the eigenvalues of the Laplacian is influenced by the eigenvalues of a Schr¨odinger operator with a potential depending on principal curvatures of the hypersurface. Krejˇciˇr´ık [26] and Jimbo–Kurata [23] showed that for Dirichlet–Neumann mixed boundary condition, it is influenced by maximum values of the mean curvature of the hypersurface.
The results of Schatzman [36] are closely related to our research although the geometric situation is different from ours. Schatzman considered the asymptotic be-havior of the eigenvalues of the Laplacian on a thin domain with Neumann boundary condition as the domain degenerates to its boundary and proved that the asymptotic behavior of the eigenvalues is influenced by a geometric quantities such as the co-efficients of the second fundamental form and the mean curvature of the boundary. In our situation, Schatzman’s results give us the following theorem.
Theorem 1.1.1 (Schatzman). Let λk,ε be the k-th eigenvalue of the Laplacian in Ωε
Beltrami operator on Γ. Then we have
λk,ε = λk+ O(ε) as ε → 0.
Moreover if λk is simple, then we have
λk,ε = λk+ O(ε2) as ε → 0.
On the other hand, in the two-phase eigenvalue problem (1.2), the method used in the previous study can not be applied since the coefficients are discontinuous at the interface Γ. We treat this problem by using a variational method and the Fourier expansions with respect to an appropriate orthonormal basis. This idea is based on the approach used in [23]. It works well even if the coefficients are discontinuous.
Now we present one of the main results of this thesis about the asymptotic behavior of the eigenvalues as ε → 0, see [40].
Theorem 1.1.2. Let λk,ε be the k-th eigenvalue of the two-phase eigenvalue problem
(1.2), and let λk be the k-th eigenvalue of the Laplace–Beltrami operator on Γ. Then
we have
λk,ε =
σ−+ σ+
2 λk+ O(ε) as ε → 0. Note that the remainder term O(ε) depends on σ−, σ+, and k.
From Theorem 1.1.2, we see that the influence of the discontinuity of the coeffi-cients on the asymptotic behavior of the eigenvalues appears as the arithmetic mean of coefficients.
If we suppose that λk is simple, we obtain a more precise asymptotic behavior
of λk,ε.
Theorem 1.1.3. Suppose that λk is simple. Then we have
λk,ε = σ−+ σ+ 2 λk+ σ+− σ− 4 Λkε + o(ε) as ε → 0, where Λk = ˆ Γ n−1 X i,j=1 e Gij− Hgij0∂uk ∂ξi ∂uk ∂ξj p G0dξ.
Remark 1.1.4. Here eGij − Hgij
0 denote the components of the matrix (2W −
tr(W )I)g−10 , where W is the Weingarten matrix, I is the identity matrix, and g0−1 is the inverse metric matrix of Γ. Note that the eigenvalues of the Weingarten matrix W are the principal curvatures of Γ and H = tr(W ). As the matrix 2W − tr(W )I is in general neither positive definite nor negative definite, it follows that the term Λk can be either positive or negative.
Note that the remainder term o(ε) depends on σ−, σ+, and k.
The exact meaning of the symbols of Theorem 1.1.3 is explained in Section 3.2. The term Λk depends on the geometric shape of the interface Γ. It consists of
the quantities eGij related to the second fundamental form, the mean curvature H, and the k-th normalized eigenfunction uk on the interface Γ. We mention that a
term similar to Λk appears in Schatzman’s original results [36, Section 11, Theorem
4]. From Theorem 1.1.3, we see that the influence of the geometric shape of the interface Γ appears only in the second term of the asymptotic behavior of eigenvalues. Moreover, we notice that the second term only appears when σ− 6= σ+.
If λk is not simple, it is difficult to get a more precise asymptotic behavior of the
eigenvalues in general. If the interface Γ is a sphere, however, we can still obtain it although the eigenvalues of the sphere are not simple.
Theorem 1.1.5. If Γ is Sn−1(r) (i.e. the n − 1 dimensional sphere with radius
r > 0), then we have λk,ε = σ−+ σ+ 2 λk+ n − 3 4r (σ+− σ−) λkε + o(ε) as ε → 0. (1.3) Note that the remainder term o(ε) depends on σ−, σ+, and k.
1.1.2
Reinforcement problems
Let Ω ⊂ Rn (n ≥ 2) be a bounded domain with smooth and connected boundary Γ. For sufficiently small ε > 0, put
where νΓdenotes the outward unit normal vector to the Γ. We consider the following
two-phase eigenvalue problem on Ωε:
−div (σε∇u) = λu in Ωε,
u = 0 on ∂Ωε,
(1.4)
where σε = σε(x) (x ∈ Ωε) is the piecewise constant function given by
σε(x) = 1, x ∈ Ω, qε, x ∈ Σε,
where qε = αε and α is a positive parameter, (see Figure 2).
Figure 2: Problem setting for reinforcement problems.
Such two-phase eigenvalue problems are called reinforcement (or thin coating) problems, and are related to the vibration frequencies of composite materials or coating of composite materials with thermal insulation.
Reinforcement problems were first studied by Friedman [14]. He considered a two-phase eigenvalue problem for the principal eigenvalue of some elliptic operators in the case limε→0qε/ε = α and limε→0qε/ε = 0. His method is based on H2-estimate
and the comparison principle for the two-phase eigenvalue problem. Rosencrans– Wang [34] generalized Friedman’s results to all eigenvalues in the case limε→0qε/ε =
0. Their methods are based on H1-estimate of the eigenfunctions and the variational
characterization of the eigenvalues. Regarding other two-phase eigenvalue problems in this direction, we refer to [31, 15, 22].
In this thesis, we consider the case limε→0qε/ε = α and focus on a refinement of
Friedman’s result. Friedman considered more general setting, but in our situation, Friedman’s result reads as follows.
Theorem 1.1.6 (Friedman). Let λ1(ε) be the principal eigenvalue of the eigenvalue
problem (1.4). Then we have
λ1(ε) = µ1+ o(1) as ε → 0,
uε → w1 weakly in H2(Ω),
where µ1 is the principal eigenvalue and w1 is the principal eigenfunction of the
following Robin eigenvalue problem: −∆w = µw in Ω, αw + ∂w ∂νΓ = 0 on Γ. (1.5)
Theorem 1.1.6 implies that the condition limε→0qε/ε = α affects the boundary
condition, which becomes the Robin boundary condition.
We derive a more precise asymptotic behavior for the principal eigenvalue of (1.4), see [41].
Theorem 1.1.7. Let λ1(ε) be the principal eigenvalue of the eigenvalue problem
(1.4). Then we have the asymptotic behavior
λ1(ε) = µ1 − ε ˆ Γ αH +µ1 3 w12pG0dξ + o(ε) as ε → 0,
where H is the mean curvature defined as the sum of the principle curvatures of Γ.
From Theorem 1.1.7, 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.
1.1.3
On a Robin inverse spectral problems related to the
reinforcement problems
In the previous subsection, the thin layer Σε had a uniform thickness. However,
Friedman’s original results considered a thin layer Σεwith varying thicknesses. That
is, Friedman considered the case where the thin layer Σε is defined by
Σε=x ∈ Rn | x = ξ + tp(ξ)νΓ(ξ) for ξ ∈ Γ, 0 < t < ε ,
where p is a given positive smooth function on Γ. Then the following result holds: Theorem 1.1.8 (Friedman). Let λ1(ε) be the principal eigenvalue of the eigenvalue
problem (5.1). Then we have
λ1(ε) = µ1+ o(1) as ε → 0,
uε → w1 weakly in H2(Ω),
where µ1 is the principal eigenvalue and w1 is the principal eigenfunction of the
following Robin eigenvalue problem: −∆w = µw in Ω, αw + p∂w ∂νΓ = 0 on Γ.
From Theorem 1.1.8, the thickness of the thin layer p appears in the boundary condition.
Let us consider an inverse eigenvalue problem related to the reinforcement prob-lems above. Let Ω ⊂ Rn (n ≥ 2) be a bounded domain with smooth boundary, and γ, ΓD be disjoint nonempty closed subsets of the boundary ∂Ω such that ∂Ω = ΓD∪γ.
Let h ∈ C0(γ) and h > 0. Let us consider the following Robin eigenvalue problem:
−∆u = λu in Ω, u = 0 on ΓD, hu + ∂νu = 0 on γ, (1.6)
where ν is the outward unit normal vector of ∂Ω. In what follows, we only consider the principal eigenvalue and eigenfunction. Moreover, we assume that the principal
eigenfunction is positive and it is normalized by ˆ Ω u(h) 2 dx = 1.
Our aim is to study an inverse problem of the Robin eigenvalue problem (1.6). We consider the following inverse problem:
Inverse problem 1.1.9. Recover the unknown Robin coefficient h defined in the inaccessible part γ of the boundary ∂Ω from the principal eigenvalue λ(h) and the Neumann data ∂νu(h)|ΓD on the accessible part ΓD.
From the viewpoint of the reinforcement problems, the Robin coefficient h is equal to α/p. That is, the Robin coefficient h represents a quantity of the thickness effect. The inverse problem 1.1.9 can be interpreted as the question of whether the quantity can be determined when the principal eigenvalue and the Neumann data on the accessible part are given.
We remark that the setting of the inverse problem is similar to the detection problem of internal corrosion. Let us consider the following problem:
−∆u = 0 in Ω, u = 0 on ΓD, hu + ∂νu = 0 on γ.
The detection problem of internal corrosion is to recover the unknown Robin co-efficient h defined in the inaccessible part γ of the boundary ∂Ω from the Neu-mann data ∂νu(h)|ΓD on the accessible part ΓD. Physically speaking, the
Neu-mann data ∂νu(h)|ΓD is the current and the Robin coefficient h is the corrosion.
There are many results concerning uniqueness, stability, and reconstruction algo-rithm for this inverse problem. For the details about the inverse problem, see [19, 9, 1, 10, 8, 11, 7, 24, 4, 18, 38] and the references therein.
To my knowledge, there are a few results concerning the Robin inverse eigenvalue problem 1.1.9. The papers [3, 2] dealt with the Robin inverse eigenvalue problem when the support of the Robin coefficient is sufficiently small and gave a non-iterative algorithm for detecting the Robin coefficient from the measurements of an eigenvalue and the Neumann data on the accessible part of the boundary.
Generally speaking, inverse problems are ill-posed problems, that is, a given problem either has no solutions in the desired class, or it has many (two or more) solutions, or the solution is unstable (i.e. arbitrarily small errors in the measurement data may lead to large errors in the solutions). Therefore, the main issues of inverse problems are those of showing uniqueness, stability, and giving a reconstruction algorithm.
Our purpose is twofold. First, we study uniqueness and local stability for the inverse problem 1.1.9, see [35].
Theorem 1.1.10. Let Ω ⊂ Rn (n ≥ 2) be a bounded domain with smooth boundary
∂Ω and γ, ΓD be disjoint nonempty closed subsets of ∂Ω such that ∂Ω = ΓD ∪ γ.
Let (λ(hj), uj) be the solution of the Robin eigenvalue problems (1.6),
correspond-ing to the Robin coefficients hj, with hj ∈ C0(γ) and hj > 0 for j = 1, 2. If
(λ(h1), ∂νu1|ΓD) = (λ(h2), ∂νu2|ΓD), then we have h1 = h2.
Theorem 1.1.10 can be easily obtained by Holmgren’s unique continuation theo-rem. Local stability is also proved by using Fr´echet differentiability for the principal eigenfunction u(h) of (1.6) as follows.
Theorem 1.1.11. Let A be the admissible set of Robin coefficients defined by A = h ∈ C1(γ) : h(x) > 0 . Let h, ξ ∈A . Then lim ε→0 k∂νu(h + εξ) − ∂νu(h)kL2(Γ D) |ε| + λ(h + εξ) − λ(h) |ε| ! > 0.
Second, we deal with the numerical computation. Let us define a Neumann tracking type functional
F (h) = 1 2 ˆ ΓD (∂νu(h) − g)2ds + 1 2 λ(h) − λ 2 , (1.7)
where (λ, g) are the given spectral data. By the Fr´echet derivative of the func-tional (1.7) and the gradient descent method, we can solve the inverse problem 1.1.9 numerically.
This thesis is organized as follows. In Chapter 2, we provide basic results for eigenvalue problems of second-order elliptic operators. In Chapter 3, we consider the two-phase eigenvalue problems on thin domains (1.2) and prove Theorems 1.1.2, 1.1.3, and 1.1.5. In Chapter 4, we consider the reinforcement problem (1.4) and prove Theorem 1.1.7. The main tools used to prove these theorems are the min-max principle and a suitable Fourier expansions. Finally, the Robin inverse eigenvalue problem 1.1.9 is analyzed in Chapter 5.
Chapter 2
Basic results for eigenvalues of
second-order elliptic operators
In this chapter, we recall the basic results for eigenvalues of second-order elliptic operators. A Typical example is the Laplace operator, but we state the general results for second-order elliptic operators. For the general references in this chapter, we refer to [12, 17, 13].
2.1
Second-order elliptic operators and Green
op-erators
First, let us consider the following Dirichlet boundary-value problem: Lu = f in Ω, u = 0 on ∂Ω, (2.1)
where L is a second-order elliptic operator defined by
Lu := − n X i,j=1 ∂ ∂xi aij(x) ∂u ∂xj ! (2.2)
and f ∈ L2(Ω). Moreover, we assume that a
ij ∈ L∞(Ω) (i, j = 1, . . . , n) and the
uniform ellipticity condition, that is, there exists a constant θ > 0 such that
n
X
i,j=1
for all x ∈ Ω and all ξ ∈ Rn (n ≥ 1), where |ξ|2 = (ξ2
1 + · · · + ξ2n)1/2 denotes the
Euclidean norm of the vector ξ. We also assume the symmetry condition
aij = aji (i, j = 1, . . . , n). (2.4)
Definition 2.1.1 (Weak solution). We say that u ∈ H1
0(Ω) is a weak solution of
the boundary-value problem (2.1) if
n X i,j=1 ˆ Ω aij(x) ∂u ∂xi ∂ψ ∂xj dx = ˆ Ω f (x)ψ(x) dx (2.5) for all ψ ∈ H1 0(Ω).
Existence and uniqueness of the weak solution of (2.1) are shown by Lax–Milgram theorem. Let G be the linear operator defined by
G : L2(Ω) −→ H01(Ω) ⊂ L2(Ω), (2.6) f 7→ u solution of (2.5).
The operator (2.6) is sometimes called Green operator of (2.5).
Let us examine the spectral properties of the Green operator (2.6). It is closely related to the eigenvalues and eigenfunctions of the elliptic operator (2.2).
Proposition 2.1.2. The Green operator G is a self-adjoint, compact and positive operator in L2(Ω), that is, G satisfies the following three conditions:
1. (positive): for all f ∈ L2(Ω), (Gf, f )
L2(Ω) ≥ 0,
2. (self-adjoint): for all f, g ∈ L2(Ω), (Gf, g)
L2(Ω)= (f, Gg)L2(Ω),
3. (compact): For any bounded sequence {fk}∞k=1 ⊂ L2(Ω), there exists some
subsequence {fkj}
∞
j=1 such that {Gfkj}
∞
j=1 converges in L2(Ω).
Proof. Positivity and self-adjointness of the Green operator G can be easily seen by the uniform ellipticity condition (2.3) and the symmetry condition (2.4) of the elliptic operator (2.2). Moreover, we can check that (Gf, f )L2(Ω) > 0 if f 6= 0.
Compactness follows by the following version of the Rellich theorem [17, Chapter 1, Theorem 1.1.1]:
Theorem 2.1.3 (Rellich theorem). Let Ω ⊂ Rn be a bounded domain. Then the
embedding H1
0(Ω) ,→ L2(Ω) is compact. Moreover, if Ω is a bounded open set with
Lipschitz boundary, then the embedding H1
0(Ω) ,→ L2(Ω) is compact.
2.1.1
Eigenvalues and eigenfunctions of elliptic operators
Compact operators in Hilbert spaces have good properties. As an example, we can show that self-adjoint, compact and positive operators have real eigenvalues and an orthonormal basis consisting of eigenfunctions, just like finite-dimensional matrices. In fact, the following theorem holds [13, Appendix D, Theorem 7].
Theorem 2.1.4. Let H be a separable Hilbert space of infinite dimension and let T be a self-adjoint, compact and positive operator. Then there exists a sequence of real nonnegative eigenvalues {νk}∞k≥1 converging to 0 and a sequence of eigenvectors
{xk}∞k≥1 defining a Hilbert basis of H such that T xk = νkxk for all k ≥ 1.
Applying Theorem 2.1.4 to the Green operator G, we can find a Hilbert basis {uk}∞k≥1 of L
2(Ω) and a real nonnegative eigenvalues {ν
k}∞k≥1 such that Guk = νkuk.
Moreover, the eigenvalues {νk}∞k≥1 are positive since νkkukk 2
L2(Ω) = (Guk, uk)L2(Ω) >
0 due to the strict positivity of G. By the definition of the Green operator, uk ∈
H1 0(Ω) satisfies νk n X i,j=1 ˆ Ω aij(x) ∂uk ∂xi ∂ψ ∂xj dx = ˆ Ω ukψ dx.
Thus Luk = 1/νk uk. In summary, the following theorem holds:
Theorem 2.1.5. There exists a sequence of positive eigenvalues satisfying 0 < λ1 ≤
λ2 ≤ . . . → +∞ and a sequence of corresponding eigenfunctions u1, u2, · · · satisfying
Luk= λkuk in Ω, u = 0 on ∂Ω. (2.7)
Remark 2.1.6. By Theorem 2.1.5, the eigenfunctions {uk}k≥1 of (2.7) satisfy Ψ = ∞ X k=1 αkuk, αk= ˆ Ω Ψukdx (2.8) in L2(Ω), for all Ψ ∈ H1
0(Ω). We note that the expansion (2.8) converges not only
in L2-norm but also in H01-norm. Moreover, we can prove by the strong maximum principle that the first (so-called principal) eigenvalue λ1 is simple (that is, its
mul-tiplicity is one) and the first eigenfunction u1 has a constact sign on Ω. For the
proof, see [13, Section 6.5, Theorem 2].
2.1.2
Variational characterization of the eigenvalues of
el-liptic operators
The eigenvalues of the elliptic operator (2.7) can be characterized by variational method known as min-max principle. Let us introduce the following functional called a Rayleigh quotient:
R(u) = n X i,j=1 ˆ Ω aij(x) ∂u ∂xi ∂u ∂xj dx ˆ Ω |u|2dx . (2.9)
Then, the following theorem holds.
Theorem 2.1.7. For any natural number k ≥ 1, λk = sup
E⊂L2(Ω),dim E≤k−1
inf{R(u) | u ∈ H01(Ω), u ⊥ E, u 6= 0}. (2.10)
In particular, for the principal eigenvalue λ1, we obtain
λ1 = inf{R(u) | u ∈ H01(Ω), u 6= 0}. (2.11)
In (2.11), the infimum is achieved by the principal eigenfunction u1.
The variational characterizations (2.10) and (2.11) are very powerful and useful tool to estimate the eigenvalues. The strategy to estimate eigenvalues throughout this thesis is based on the min-max Theorem 2.1.7. First, by substituting an appro-priate function into (2.10), we can obtain an upper bound for the eigenvalues. Next,
we consider the Fourier expansion with respect to the eigenfunctions of a suitable eigenvalue problem. By using the upper bound, we can get estimates of the Fourier coefficients. Then we can derive the asymptotic behavior of the eigenvalues. We will find that this strategy works well in Chapters 3, 4, and 5.
2.1.3
Fredholm alternative
By the definition of eigenvalues, L − λI has a non-trivial kernel, where I is the identity operator. However, we sometimes need to solve an equation of the form (L − λI)u = f . Similarly to the well known property of finite-dimensional matrices, the following theorem holds [28, Theorem 7.41.7].
Theorem 2.1.8 (Fredholm alternative). Let λ be an eigenvalue of the elliptic op-erator (2.7) and let us consider the following problem
Lu − λu = f in Ω, u = 0 on ∂Ω, (2.12)
where f ∈ L2(Ω). Then the problem (2.12) has a solution u ∈ H1
0(Ω) if and only if
(f, w)L2(Ω) = 0 for all eigenfunctions w corresponding to λ.
Theorem 2.1.8 will be used in the Chapter 5 for a mixed boundary condition. We remark that previous theorems in this chapter also hold for the Neumann and Robin boundary condition by replacing H01(Ω) with H1(Ω).
Chapter 3
On the two-phase eigenvalue
problem on thin domains
In this chapter, we study the two-phase eigenvalue problem on thin domains (1.2). We analyze the asymptotic behavior of each eigenvalue as the domain degenerates to a certain hypersurface being the set of discontinuities of the coefficients.
3.1
Problem setting for the two-phase eigenvalue
problem on thin domains
Let us recall the problem setting for the two-phase eigenvalue problem on thin domains. Let D ⊂ Rn (n ≥ 2) be a bounded domain whose boundary Γ is smooth
and connected. For sufficiently small ε > 0, we put
Ω−ε = {x ∈ D | d (x, Γ) < ε}, Ω+ε = {x ∈ Rn\ D | d (x, Γ) < ε},
where d (x, Γ) denotes the Euclidean distance from x to Γ. We define
Ωε= Ω−ε ∪ Ω+ε ∪ Γ.
Let ν denote the outward unit normal vector to the boundary ∂Ωε and νΓ the
two-phase eigenvalue problem:
−div (σ∇u) = λu in Ωε,
∂u
∂ν = 0 on ∂Ωε,
(3.1)
where σ = σ(x) (x ∈ Ωε) is the piecewise constant function given by
σ(x) = σ−, x ∈ Ω−ε ∪ Γ, σ+, x ∈ Ω+ε,
and σ−, σ+ are distinct positive constants (i.e. σ−, σ+> 0, σ−6= σ+).
We consider the problem (3.1) in a weak sense, namely, λ ∈ C is an eigenvalue of (3.1) if there exists a function u ∈ H1(Ω
ε) such that u 6≡ 0 and
ˆ
Ωε
σ∇u · ∇ψdx = λ ˆ
Ωε
uψdx for any ψ ∈ H1(Ωε). (3.2)
By the same argument of Theorem 2.1.5, the eigenvalues of (3.1) are non-negative real numbers and the set of all eigenvalues is discrete. Let {λk,ε}k≥1 be the
eigen-values satisfying 0 = λ1,ε< λ2,ε ≤ λ3,ε≤ · · · → +∞ and {uk,ε}k≥1 be the associated
eigenfunctions in (3.1).
Since σ = σ(x) (x ∈ Ωε) is a piecewise constant function, by integration by parts,
we can rewrite (3.2) as follows: −σ±∆u±= λu± in Ω±ε, u−= u+ on Γ, σ− ∂u− ∂νΓ = σ+ ∂u+ ∂νΓ on Γ, ∂u± ∂ν = 0 on ∂Ω ± ε \ Γ. (3.3)
Here u− and u+ are the restriction of u on Ω−ε and Ω+ε, respectively. The third
equality of (3.3) is usually called transmission condition, and can be interpreted as the continuity of the flux through the interface Γ. The purpose of this chapter is to consider the asymptotic behavior of the eigenvalues {λk,ε}k≥1 as ε → 0 when
the domain Ωε degenerates to the interface Γ. In particular, our aims are to show
affect the asymptotic behavior of eigenvalues as the domain Ωε degenerates to the
interface Γ.
By the min-max Theorem 2.1.7, the k-th eigenvalue λk,ε can be characterized
directly as follows: for any natural number k ≥ 1,
λk,ε = sup
E⊂L2(Ω(ε)),dim E≤k−1
inf{Rε(u) | u ∈ H1(Ωε), u ⊥ E, u 6= 0}, (3.4) where Rε(u) is Rε(u) = ˆ Ωε σ|∇u|2dx ˆ Ωε |u|2dx . (3.5)
According to the min-max principle (3.4), it is sufficient to estimate the Rayleigh quotient (3.5) in order to estimate the k-th eigenvalue λk,ε. However, it is not easy
to estimate the Rayleigh quotient above because Ω−ε and Ω+ε are perturbed as ε → 0. Thus we will fix the domains by a suitable coordinate transformation.
3.2
Geometric preliminaries for thin domains
Since the interface Γ is an n−1 dimensional compact manifold in Rn, we can take the
union of a finite number of local patches in Γ, each of which has local coordinates (ξ1, ξ2, · · · , ξn−1). Note that we regard a point ξ ∈ Γ as its corresponding local
coordinates (ξ1, ξ2, · · · , ξn−1) through a local coordinate map. Every x ∈ Ωε in the
neighborhood of the interface Γ is then represented by
x = ξ + tνΓ(ξ), ξ ∈ Γ, |t| < ε. (3.6)
We introduce local coordinates (ξ1, ξ2, · · · , ξn−1, t) for Γ × (−ε, ε). Let g = gij(ξ, t)
denote the Riemannian metric associated to the local coordinates. By (3.6), gij(ξ, t)
is given by gij(ξ, t) = g0,ij(ξ) + teg0,ij(ξ) + t 2 b g0,ij(ξ) if 1 ≤ i, j ≤ n − 1, 0 if i = n, j 6= n or i 6= n, j = n, 1 if i, j = n, (3.7)
where g0 = g0,ij(ξ) denotes the Riemannian metric associated to the local
coordi-nates (ξ1, ξ2, · · · , ξn−1) and we write
e g0,ij = ∂ ∂ξi ,∂νΓ ∂ξj ! + ∂ ∂ξj ,∂νΓ ∂ξi ! , bg0,ij = ∂νΓ ∂ξi ,∂νΓ ∂ξj ! .
Here ∂/∂ξi and ∂/∂ξj are tangent vectors on ξ ∈ Γ and (·, ·) is the Euclidean inner
product. Let (bij)1≤i,j≤n−1 denote the coefficients of the second fundamental form
of Γ. In the local coordinates, we have bij = ∂2/∂ξi∂ξj, νΓ. By the definition of
e g0,ij, we have ∂ ∂ξi ,∂νΓ ∂ξj ! = − ∂ 2 ∂ξi∂ξj , νΓ ! = −bij. (3.8) Therefore we obtain e g0,ij = −2bij. (3.9)
Let (g0ij) be the inverse matrix of (g0,ij) and let G0 = det(g0,ij). Similarly, let
(gij) denote the inverse matrix of (g
ij) and let G = det(gij). Then, by (3.7), we can
obtain the asymptotic formulas for the inverse metric tensor gij and the Jacobian
√ G as follows: gij(ξ, t) = g0ij(ξ) + t eGij(ξ) + O(t2) as t → 0, (3.10) p G(ξ, t) =pG0 1 − tH(ξ) + O(t2) as t → 0, (3.11) where eGij(ξ) = 2Pn−1
k,l=1g0ik(ξ)bkl(ξ)g0lj(ξ), and H(ξ) is the mean curvature of Γ at
ξ ∈ Γ with respect to νΓ (defined as the sum of the principle curvatures of Γ). The
asymptotic formulas (3.10) and (3.11), which are given by Schatzman’s paper [36, Section 10], will play an important role to obtain the asymptotic behavior of the eigenvalues. For the details about the geometric properties of thin domains, see [23, 36] and the references given there.
By using the local coordinates (ξ1, ξ2, · · · , ξn−1, t), the squared norm of the
gra-dient of u in (3.5) is |∇u|2 = |∇ gu|2+ ∂u ∂t 2 , where |∇gu|2 = n−1 X i,j=1 gij∂u ∂ξi ∂u ∂ξj .
Similarly, we define |∇g0u| 2 = n−1 X i,j=1 g0ij∂u ∂ξi ∂u ∂ξj .
In terms of the local coordinates (ξ1, ξ2, · · · , ξn−1, t), the Rayleigh quotient reads
Rε(u) = ˆ ε −ε ˆ Γ σ |∇gu|2+ ∂u ∂t 2! p G(ξ, t)dξdt ˆ ε −ε ˆ Γ |u|2p G(ξ, t)dξdt . (3.12)
By introducing the variable τ ∈ (−1, 1) by t = ετ and transforming euε(ξ, τ ) =
u(ξ, ετ ), we rewrite the min-max principle and the Rayleigh quotient as follows:
λk,ε = sup
E⊂L2(Γ×(−1,1)),dim E≤k−1
inf{Rε(ueε) |ueε ∈ H 1 (Γ × (−1, 1)),ueε ⊥ε E}, where Rε(euε) = ˆ 1 −1 ˆ Γ σ |∇gueε| 2+ 1 ε2 ∂euε ∂τ 2! p G(ξ, ετ )dξdτ ˆ 1 −1 ˆ Γ |ueε|2 p G(ξ, ετ )dξdτ (3.13)
and euε⊥ε E means that for any Ψ ∈ E,
ˆ 1 −1 ˆ Γ e uε(ξ, τ )Ψ(ξ, τ ) p G(ξ, ετ )dξdτ = 0.
We will estimate the eigenvalue λk,εby using the Rayleigh quotient (3.13). In the
following sections, we will write pG(ξ, ετ ) as √Gε for simplicity and let C denote
any positive constant independent of ε. The same letter C will be used to denote different constants.
3.3
Upper estimate of the eigenvalues
Let k ≥ 1 and E ⊂ L2(Γ × (−1, 1)) be any subspace such that dim E ≤ k − 1. We
define Lk = Span 1 √ 2u1(ξ), 1 √ 2u2(ξ), . . . 1 √ 2uk(ξ) ,
where uk is the normalized eigenfunction associated with the k-th eigenvalue λk of
the Laplace–Beltrami operator on Γ. Since dim E ≤ k − 1 and dim Lk = k, there
exist constants {cp(ε)}kp=1 such that
e Ψε(ξ, τ ) = k X p=1 cp(ε) 1 √ 2up(ξ), Ψeε ⊥εE.
We substitute eΨε into the Rayleigh quotient (3.13) as a test function. Then we have
inf{Rε(ueε) | euε∈ H 1(Γ × (−1, 1)), e uε⊥ε E} ≤ Rε( eΨε) = σ− ˆ 0 −1 ˆ Γ |∇gΨeε|2 p Gεdξdτ + σ+ ˆ ε 0 ˆ Γ |∇gΨeε|2 p Gεdξdτ ˆ 1 −1 ˆ Γ | eΨε|2 p Gεdξdτ =: N1(ε) + N2(ε) M (ε) , where N1(ε) = σ− ˆ 0 −1 ˆ Γ |∇gΨeε|2 p Gεdξdτ, N2(ε) = σ+ ˆ ε 0 ˆ Γ |∇gΨeε|2 p Gεdξdτ, M (ε) = ˆ 1 −1 ˆ Γ | eΨε|2 p Gεdξdτ.
By requiring the normalization,
M (ε) = ˆ 1 −1 ˆ Γ | eΨε|2 p Gεdξdτ = 1. We obtain k X p=1 cp(ε)2 = 1 + O(ε). (3.14)
By using the asymptotic formulas for the inverse metric tensor (3.10) and the Jaco-bian (3.11), we calculate both terms N1(ε) and N2(ε).
N1(ε) = σ− ˆ 0 −1 ˆ Γ |∇gΨeε|2 p Gεdξdτ = σ− 2 k X p,q=1 cp(ε)cq(ε) ˆ 0 −1 ˆ Γ ∇g0up· ∇g0uq p G0dξdτ + O(ε) = σ− 2 k X p=1 λpcp(ε)2+ O(ε). Similary, we have N2(ε) = σ+ ˆ ε 0 ˆ Γ |∇gΨeε|2 p Gεdξdτ = σ+ 2 k X p=1 λpcp(ε)2 + O(ε). Thus we have N1(ε) + N2(ε) M (ε) = σ− 2 k X p=1 λpcp(ε)2+ σ+ 2 k X p=1 λpcp(ε)2+ Cε = σ−+ σ+ 2 k X p=1 λpcp(ε)2+ Cε ≤ σ−+ σ+ 2 λk k X p=1 cp(ε)2+ Cε = σ−+ σ+ 2 λk+ Cε.
Here we used the monotonicity of the eigenvalues {λk}k≥1 and (3.14). Therefore we
obtain the following upper estimate of the eigenvalue λk,ε.
λk,ε ≤
σ−+ σ+
3.4
Lower estimate of the eigenvalues
For any ψ ∈ H1(Γ × (−1, 1)), we consider the weak form of (3.1) in the local
coordinates: ˆ 1 −1 ˆ Γ σ ∇geuk,ε · ∇gψ + 1 ε2 ∂euk,ε ∂τ ∂ψ ∂τ p Gεdξdτ = λk,ε ˆ 1 −1 ˆ Γ e uk,εψ p Gεdξdτ, (3.16) where λk,ε is the k-th eigenvalue andeuk,ε is the k-th eigenfunction associated to λk,ε. We normalize euk,ε as follows: ˆ 1 −1 ˆ Γ |uek,ε|2 p Gεdξdτ = 1. (3.17)
If we take ψ =euk,ε, then we have
λk,ε = ˆ 1 −1 ˆ Γ σ |∇geuk,ε| 2 + 1 ε2 ∂euk,ε ∂τ 2! p Gεdξdτ. (3.18)
The main idea in order to get a lower estimate of eigenvalues λk,ε is to take a test
function that projects euk,ε onto the eigenspace of λk. For that reason, we consider
the Fourier expansions of euk,ε. Let up (p ≥ 1) denote the p-th normalized
eigen-function of Laplace–Beltrami operator on Γ and φl (l ≥ 1) be the l-th normalized
eigenfunction of the following eigenvalue problem: −d
2φ
dτ2(τ ) = µφ(τ ) (τ ∈ (−1, 1)), φ 0
(−1) = 0, φ0(1) = 0. (3.19) Notice that the eigenvalue problem (3.19) can be solved explicitly as follows:
µl = (l − 1)2 4 π 2 (l ≥ 1), φ l= 1 √ 2 (l = 1), cos(l − 1 2 πτ ) (l ≥ 2, l odd), sin(l − 1 2 πτ ) (l ≥ 2, l even). (3.20) Then, sinceupφl p≥1,l≥1is an orthonormal basis of L 2 Γ × (−1, 1), the k-th
eigen-function euk,ε admits the following Fourier expansions:
e uk,ε(ξ, τ ) = ∞ X p,l=1 αkp,l(ε)up(ξ)φl(τ ), (3.21) αp,lk (ε) = ˆ 1 −1 ˆ Γe uk,ε(ξ, τ )up(ξ)φl(τ ) p G0dξdτ. (3.22)
First of all, we will get some estimates for the Fourier coefficients αkp,l(ε). If we substitute the Fourier expansions (3.21) into (3.18), then we can obtain some estimates for the Fourier coefficients αp,lk (ε) by using the upper bound of λk,ε.
Lemma 3.4.1. The following estimates hold:
∞ X p=1,l=2 αp,lk (ε)2 = O(ε2) as ε → 0, (3.23) ∞ X p=1 αp,1k (ε)2 = 1 + O(ε) as ε → 0. (3.24)
Proof. By using the upper bound of λk,ε, we have
ˆ 1 −1 ˆ Γ ∂uek,ε ∂τ 2 p G0dξdτ ≤ Cε2. (3.25)
We substitute the Fourier expansions (3.21) into (3.25) to obtain
∞
X
p=1,l=2
µl(αp,lk )
2 ≤ Cε2.
Note that µl ≥ µ2 = π2/4 for l ≥ 2. Thus we get (3.23). Moreover, by the
normalization (3.17), we obtain
∞
X
p,l=1
(αp,lk )2 = 1 + O(ε). (3.26)
Combining the estimate (3.23) with (3.26) yields (3.24).
Next, we will consider the lower estimate of the eigenvalue λk,ε. As a test function
of (3.16), we take ψ = k(j+1)−1 X p=k(j) αp,1k upφ1, (3.27)
where {k(j)}∞j=1 is the increasing sequence of natural numbers defined by
k(1) = 1, k(j + 1) = min{k ∈ N | λk > λk(j)}. (3.28)
We note that for any k, there exists a unique j such that k(j) ≤ k < k(j + 1). By the definition (3.28), the multiplicity of λk is given by k(j + 1) − k(j). Thus the test
function (3.27) taken as above is the projection of the eigenspace of λk. By using
the asymptotic behavior of the inverse metric tensor (3.10) and the Jacobian (3.11) and also using the orthonormality of eigenfunction up and φl, we infer that the right
hand side of (3.16) is λk,ε ˆ 1 −1 ˆ Γ e uk,εψ p Gεdξdτ = λk,ε k(j+1)−1 X p=k(j) (αkp,1)2+ O(ε). (3.29)
Moreover, the left hand side of (3.16) is ˆ 1 −1 ˆ Γ σ ∇geuk,ε· ∇gψ + 1 ε2 ∂euk,ε ∂τ ∂ψ ∂τ p Gεdξdτ = k(j+1)−1 X p=k(j) ∞ X l=1 αp,lk αp,1k λp σ− ˆ 0 −1 φlφ1dτ + σ+ ˆ 1 0 φlφ1dτ ! + O(ε) = σ−+ σ+ 2 λk k(j+1)−1 X p=k(j) (αp,1k )2+ σ+√− σ− 2 λk k(j+1)−1 X p=k(j) ∞ X l=2 αp,lk αp,1k ˆ 1 0 φldτ + O(ε),
where we used λp = λk (k(j) ≤ p ≤ k(j + 1) − 1) and
´1 −1φldτ = 0 (l ≥ 2). Let us define I1 = σ+− σ− √ 2 λk k(j+1)−1 X p=k(j) ∞ X l=2 αp,lk αp,1k ˆ 1 0 φldτ.
Now we will estimate the term I1.
I1 = σ+− σ− √ 2 λk k(j+1)−1 X p=k(j) ∞ X l=2 αp,lk αp,1k ˆ 1 0 φldτ = σ+√− σ− 2 λk k(j+1)−1 X p=k(j) ∞ X m=1 αp,2mk αp,1k ˆ 1 0 φ2mdτ + αp,2m+1k α p,1 k ˆ 1 0 φ2m+1dτ ! .
By (3.20), the following integrals can be calculated explicitly for m ≥ 1: ˆ 1 0 φ2mdτ = 2 (2m − 1)π, ˆ 1 0 φ2m+1dτ = 0. Then we have I1 = σ+− σ− √ 2 λk k(j+1)−1 X p=k(j) ∞ X m=1 αkp,2mαp,1k 2 (2m − 1)π.
Thus we get the following estimate: |I1| ≤ √ 2|σ+− σ−| π λk k(j+1)−1 X p=k(j) ∞ X m=1 |αp,2mk ||αp,1k | 1 2m − 1 ≤ √ 2|σ+− σ−| π λk k(j+1)−1 X p=k(j) ∞ X m=1 ε 2 · (αp,1k )2 (2m − 1)2 + 1 2ε · (α p,2m k ) 2 ! ≤ |σ+√− σ−| 2π λkε k(j+1)−1 X p=k(j) (αp,1k )2· π 2 8 + |σ+− σ−| √ 2πε λk ∞ X p=1 ∞ X l=2 (αp,lk )2,
where we used Cauchy’s inequality and the identity P∞
m=11/(2m − 1)
2 = π2/8.
Moreover, by using the estimate for the Fourier coefficients (3.23) we have
|I1| ≤ Cλkε
k(j+1)−1
X
p=k(j)
(αp,1k )2+ Cλkε.
Thus we get the following estimate for I1,
I1 ≥ −Cλkε
k(j+1)−1
X
p=k(j)
(αp,1k )2− Cλkε.
Therefore, we obtain the estimate of the left hand side of (3.16).
σ−+ σ+ 2 λk k(j+1)−1 X p=k(j) (αp,1k )2+ I1− Cε ≥ σ−+ σ+ 2 λk k(j+1)−1 X p=k(j) (αp,1k )2 1 − 2C σ−+ σ+ ε − Cλkε − Cε. (3.30)
Combining the estimate of the right hand side of (3.29) with that of the left hand side of (3.30) yields λk,ε k(j+1)−1 X p=k(j) (αp,1k )2 ≥ σ−+ σ+ 2 λk k(j+1)−1 X p=k(j) (αp,1k )2 1 − 2C σ−+ σ+ ε − Cλkε − Cε. (3.31) From the above estimate, it will be necessary to show the following lemma to obtain the lower estimate of eigenvalues.
Lemma 3.4.2. The following estimate holds:
k(j+1)−1
X
p=k(j)
αp,1k (ε)2 = 1 + o(1) as ε → 0. (3.32)
Proof. We will study the limit ofeuk,εfor ε → 0. Using the upper estimate of λk,ε, we
have the boundedness of {uek,ε}ε>0 ⊂ H1(Γ × (−1, 1)). Applying Rellich’s Theorem
2.1.3, we can take a subsequence {εp}∞p=1, a nonnegative value bλk, and a function
b uk∈ H1(Γ × (−1, 1)) such that lim p→∞λk,εp = bλk, lim p→∞ e uk,εp−buk L2(Γ×(−1,1)) = 0, e uk,εp *buk weakly in H 1(Γ × (−1, 1)). (3.33)
By using the weak lower semicontinuity of the H1-norm and the estimate (3.25),
we show that ubk(ξ, τ ) does not depend on the variable τ . Therefore we can write
b
uk(ξ, τ ) = buk(ξ). If we take ψ(ξ, τ ) = bψ(ξ) ( bψ ∈ H
1(Γ)) as a test function in (3.16)
and let p → ∞, then by (3.33) we obtain ˆ Γ ∇g0ubk· ∇g0ψb p G0dξ = eλk ˆ Γb ukψb p G0dξ for any bψ ∈ H1(Γ), (3.34)
where we here put
e λk =
2bλk
σ−+ σ+
.
For any test function bψ ∈ H1(Γ), (3.34) holds. Thus eλk is an eingenvalue of the
Laplace–Beltrami operator on Γ andubk is the corresponding eigenfunction. By the
upper bound of λk,ε, we get
b λk ≤ σ−+ σ+ 2 λk. Therefore we obtain e λk≤ λk. (3.35)
Also by using the orthonormality of uek,ε, for each k, k0 ≥ 1 we have
ˆ 1 −1 ˆ Γb ukbuk0 p G0dξdτ = δ(k, k0).
Thus we get ˆ Γ b ukubk0 p G0dξ = δ(k, k0) 2 . (3.36)
Using the orthonormality condition (3.36), we have
e
λk≥ λk. (3.37)
Therefore, from (3.35) and (3.37), we obtain
e
λk= λk.
This implies that eλk is the k-th eigenvalue and buk is the corresponding k-th eigen-function. Thus the eigenspace of λk contains buk. We can express buk as
b uk(ξ) = k(j+1)−1 X p=k(j) cpup(ξ), (3.38) where {cp} k(j+1)−1
p=k(j) are suitable constants. By (3.36), we get k(j+1)−1 X p=k(j) c2p = 1 2. Thus we have e uk,ε−buk 2 L2(Γ×(−1,1)) = ∞ X p,l=1 αkp,lupul− k(j+1)−1 X p=k(j) cpup 2 L2(Γ×(−1,1)) = ∞ X p,l=1 (αp,lk )2− 2 k(j+1)−1 X p=k(j) αp,1k √2cp+ 2 k(j+1)−1 X p=k(j) c2p = 2 1 − k(j+1)−1 X p=k(j) αp,1k √2cp + O(ε). From (3.33), since ||uek,ε−ubk|| 2 L2(Γ×(−1,1))→ 0 as ε → 0, we obtain k(j+1)−1 X p=k(j) αp,1k √2cp = 1 + o(1). (3.39)
By using Cauchy–Schwarz’s inequality for (3.39) and the estimate (3.24), we have 1 + o(1) = k(j+1)−1 X p=k(j) αp,1k √2cp ≤ k(j+1)−1 X p=k(j) (αp,1k )2 1/2 k(j+1)−1 X p=k(j) (√2cp)2 1/2 = k(j+1)−1 X p=k(j) (αp,1k )2 1/2 ≤ ∞ X p=1 (αp,1k )2 1/2 = 1 + O(ε). Therefore we obtain k(j+1)−1 X p=k(j) (αp,1k )2 = 1 + o(1).
From the estimate (3.31) and Lemma 3.4.2, we have
λk,ε ≥ σ−+ σ+ 2 λk 1 − 2C σ−+ σ+ ε − (Cλkε + Cε) / k(j+1)−1 X p=k(j) (αp,1k )2 = σ−+ σ+ 2 λk 1 − 2C σ−+ σ+ ε − Cλkε + Cε 1 + o(1) ≥ σ−+ σ+ 2 λk− Cε.
Thus we obtain the following lower estimate for the eigenvalue λk,ε:
λk,ε ≥
σ−+ σ+
2 λk− Cε. (3.40)
Combining the upper estimate (3.15) with the lower estimate (3.40), we obtain the complete proof of Theorem 1.1.2.
3.5
The case where λ
kis simple
If we suppose that the eigenvalue λk is simple, we obtain a more precise asymptotic
behavior of λk,ε. In this section, we prove Theorem 1.1.3. First of all, we need to
Lemma 3.5.1. The following estimate holds:
∞
X
p=1,l=2
αp,lk (ε)2 = o(ε2) as ε → 0. (3.41)
Proof. We recall that the functionubk, expressed by (3.38), satisfies ||uek,ε−ubk||
2
L2(Γ×(−1,1))→
0 as ε → 0 and also (3.34). In (3.34), we take the same test function as in (3.16). Then we obtain ˆ Γ ∇g0ubk· ∇g0ψ p G0dξ = λk ˆ Γb ukψ p G0dξ. (3.42) By (3.42), we get σ− ˆ 0 −1 ˆ Γ ∇g0buk· ∇g0ψ p G0dξdτ + σ+ ˆ 1 0 ˆ Γ ∇g0buk· ∇g0ψ p G0dξdτ = σ−+ σ+ 2 λk ˆ 1 −1 ˆ Γ b ukψ p G0dξdτ +σ−− σ+ 2 λk ˆ Γ ˆ 0 −1 ψdτ − ˆ 1 0 ψdτ ! b uk p G0dξ. (3.43)
Subtracting (3.43) from (3.16) and taking ψ =euk,ε−buk, we obtain ˆ 1 −1 ˆ Γ σ ∇g0(uek,ε−ubk) 2 + 1 ε2 ∂ e uk,ε ∂τ 2! p G0dξdτ = λk,ε − σ−+ σ+ 2 λk ˆ 1 −1 ˆ Γ e uk,ε euk,ε−buk pG0dξdτ + σ−+ σ+ 2 λk ˆ 1 −1 ˆ Γ uek,ε−ubk 2p G0dξdτ +σ+− σ− 2 λk ˆ Γ ˆ 0 −1e uk,εdτ − ˆ 1 0 e uk,εdτ ! b uk p G0dξ + O(ε) =: S1+ S2+ S3+ O(ε), where S1 = λk,ε− σ−+ σ+ 2 λk ˆ 1 −1 ˆ Γ e uk,ε uek,ε−ubk pG0dξdτ, (3.44) S2 = σ−+ σ+ 2 λk ˆ 1 −1 ˆ Γ e uk,ε−ubk 2p G0dξdτ, (3.45) S3 = σ+− σ− 2 λk ˆ Γ ˆ 0 −1e uk,εdτ − ˆ 1 0 e uk,εdτ ! b uk p G0dξ. (3.46)
From Theorem 1.1.2 and the fact that ||uek,ε−ubk||
2
L2(Γ×(−1,1))→ 0 as ε → 0, we have
S1 = o(ε) and S2 = o(1). Furthermore, we claim that S3 = O(ε). Indeed, let
V (ξ) = ˆ 0 −1e uk,εdτ − ˆ 1 0 e uk,εdτ. Then, we get V (ξ) = ˆ 0 −1e uk,εdτ − ˆ 1 0 e uk,εdτ ≤ ˆ 1 −1 ˆ τ 0 ∂uek,ε ∂τ (ξ, s)ds dτ ≤ 2 ˆ 1 −1 ∂uek,ε ∂τ (ξ, s) 2 ds !1/2 . By (3.25), ˆ Γ V (ξ)2pG0dξ ≤ 4 ˆ 1 −1 ˆ Γ ∂euk,ε ∂τ (ξ, s) 2 p G0dξds ≤ Cε2.
Thus we can estimate S3 as follows:
|S3| = σ+− σ− 2 λk ˆ Γ V (ξ)buk p G0dξ ≤ C ˆ Γ V (ξ)2pG0dξ 1/2ˆ Γb u2kpG0dξ 1/2 = √C 2 ˆ Γ V (ξ)2pG0dξ 1/2 ≤ Cε.
Therefore, we obtain S3 = O(ε) as claimed.
By the estimate above, we have ˆ 1 −1 ˆ Γ σ ∇g0(euk,ε−buk) 2 + 1 ε2 ∂ e uk,ε ∂τ 2! p G0dξdτ = S1+ S2+ S3+ O(ε) = o(1). (3.47) From (3.47), we obtain ˆ 1 −1 ˆ Γ ∇g0(uek,ε−ubk) 2p G0dξdτ = o(1), (3.48) ˆ 1 −1 ˆ Γ ∂ e uk,ε ∂τ 2 p G0dξdτ = o(ε2). (3.49)
The estimate (3.49) implies the estimate for the Fourier coefficients we wanted. This completes the proof of Lemma 3.5.1.
Remark 3.5.2. Lemma 3.5.1 also holds if λk is not simple. This estimate will be
used for the proof of Theorem 1.1.5.
Set V = {(p, l) ∈ N2| (p, l) 6= (k, 1)}. Then we can also obtain an estimate for Fourier coefficients as follows by using the assumption that λk is simple.
Lemma 3.5.3. If λk is simple, then we obtain the following estimate for the Fourier
coefficients of euk,ε:
X
(p,l)∈V
αp,lk (ε)2λp = o(1) as ε → 0. (3.50)
Proof. Since λk is simple, the eigenspace of λk is one dimensional. Thus by (3.38),
we have b uk= ± 1 √ 2uk = ±ukφ1. (3.51) Without loss of generality, we may assume buk= ukφ1. Then we have
ˆ 1 −1 ˆ Γ ∇g0(uek,ε− ukφ1) 2p G0dξdτ = X (p1,l1)∈V X (p2,l2)∈V αp1,l1 k α p2,l2 k ˆ Γ ∇g0up1 · ∇g0up2 p G0dξ × ˆ 1 −1 φl1φl2dτ ! + 2 X (p,l)∈V αp,lk (αk,1k − 1) ˆ Γ ∇g0up· ∇g0uk p G0dξ × ˆ 1 −1 φlφ1dτ ! + (αk,1k − 1)2 ˆ Γ ∇g0uk· ∇g0uk p G0dξ × ˆ 1 −1 φ21dτ ! = X (p,l)∈V (αp,lk )2λp+ (αk,1k − 1) 2λ k.
By (3.48), the left hand side of the above goes to 0 as ε → 0. Therefore we obtain (3.50).
of (3.16) is λk,ε ˆ 1 −1 ˆ Γ e uk,εψ p Gεdξdτ = λk,ε(αk,1k )2− ε λk,ε √ 2 ∞ X p=1 ∞ X l=2 αp,lk αk,1k ˆ Γ Hupuk p G0dξ × ˆ 1 −1 τ φldτ ! + O(ε2) = λk,ε(αk,1k ) 2− I 2ε + O(ε2), where I2 = λk,ε √ 2 ∞ X p=1 ∞ X l=2 αp,lk αkk,1 ˆ Γ Hupuk p G0dξ × ˆ 1 −1 τ φldτ ! .
We will estimate I2. Set
fl = ˆ 1 −1 τ φldτ and Fl= ∞ X p=1 αp,lk up.
Since the eigenfunction of ul is explicitly given by (3.20), by a straightforward
com-putation we get ∞ X l=1 |fl| 2 < ∞. (3.52)
Furthermore, since H = H(ξ) is continuous on Γ and Γ is compact, we have
|I2| ≤ C ∞ X l=2 |αk,1k | |fl| ˆ Γ HFluk p G0dξ ≤ C ∞ X l=2 |αk,1k | |fl|kFlkL2(Γ). Here kFlkL2(Γ) = P∞ p=1(α p,l k )2 1/2
. Thus by using Lemma 3.5.1, we have
|I2| ≤ C ∞ X l=2 |αk,1k | |fl| ∞ X p=1 (αp,lk )2 1/2 ≤ C ∞ X l=2 (αk,1k )2|fl|2 1/2 ∞ X p=1,l=2 (αkp,l)2 1/2 = o(ε).
Therefore, the right hand side of (3.16) is
λk,ε(αk,1k )
2− ε × o(ε) + O(ε2) = λ
k,ε(αk,1k )
2+ O(ε2).
Moreover, by using the asymptotic formulas for the inverse metric tensor (3.10) and the Jacobian (3.11), the left hand side of (3.16) becomes
ˆ 1 −1 ˆ Γ σ ∇geuk,ε· ∇gψ + 1 ε2 ∂uek,ε ∂τ ∂ψ ∂τ p Gεdξdτ = σ−+ σ+ 2 λk(α k,1 k ) 2 +σ+√− σ− 2 λk ∞ X l=2 αk,lk αk,1k ˆ 1 0 φldτ +σ+− σ− 4 ε(α k,1 k ) 2 ˆ Γ n−1 X i,j=1 e Gij − Hg0ij∂uk ∂ξi ∂uk ∂ξj p G0dξ + ε X (p,l)∈V αkp,lαk,1k ˆ Γ n−1 X i,j=1 e Gij − Hg0ij∂up ∂ξi ∂uk ∂ξj p G0dξ × √σ− 2 ˆ 0 −1 τ φldτ + σ+ √ 2 ˆ 1 0 τ φldτ ! + O(ε2).
In order to prove Theorem 1.1.3, it suffices to show that the second and fourth terms in the above are o(ε). Indeed, if these were true, then by using the estimate for the Fourier coefficients (αk,1k )2 = 1 + o(1), we could prove Theorem 1.1.3. We can easily show that the second term is o(ε) by Lemma 3.5.1. Thus if we prove that the fourth term is (ε), then it will complete the proof. From now on, we will estimate the fourth term. Set
hl = σ− √ 2 ˆ 0 −1 τ φldτ + σ+ √ 2 ˆ 1 0 τ φldτ and I3 = X (p,l)∈V αp,lk αk,1k ˆ Γ n−1 X i,j=1 e Gij − Hg0ij∂up ∂ξi ∂uk ∂ξj p G0dξ hl.
In the same way (3.52), we get
∞ X l=1 |hl| 2 < ∞.
We separate I3 into two parts as follows: I3 = ∞ X l=2 αk,lk αk,1k ˆ Γ n−1 X i,j=1 e Gij − Hg0ij∂uk ∂ξi ∂uk ∂ξj p G0dξ hl + ∞ X l=1 X p6=k αp,lk αk,1k ˆ Γ n−1 X i,j=1 e Gij − Hg0ij∂up ∂ξi ∂uk ∂ξj p G0dξ hl =: A1+ A2. (3.53)
At first, we estimate A1. Since the term eGij− Hgij0 is continuous on Γ and Γ is
compact, we can show that there exists a constant C > 0 such that n−1 X i,j=1 e Gij − Hg0ij∂uk ∂ξi ∂uk ∂ξj ≤ C|∇g0uk| 2. Thus we obtain ˆ Γ n−1 X i,j=1 e Gij − Hg0ij∂uk ∂ξi ∂uk ∂ξj p G0dξ ≤ Cλk. Therefore, |A1| ≤ Cλk ∞ X l=2 |αk,lk ||αk,1k ||hl| ≤ Cλk ∞ X l=2 (αkk,l)2 1/2 ∞ X l=2 (αk,1k )2|hl|2 1/2 ≤ Cλk ∞ X p=1,l=2 (αp,lk )2 1/2 (αk,1k )2 ∞ X l=2 |hl|2 1/2 = o(ε),
where we used Lemma 3.5.1. Next, we estimate A2. Let
Fl=
X
p6=k
Then, |A2| ≤ ∞ X l=1 αk,1k ˆ Γ n−1 X i,j=1 e Gij− Hgij0∂Fl ∂ξi ∂uk ∂ξj p G0dξ hl ≤ C ∞ X l=1 |αk,1k ||hl| ∇g0Fl L2(Γ) ∇g0uk L2(Γ). Here ∇g0Fl L2(Γ) = P p6=k(α p,l k )2λp 1/2 and ∇g0uk L2(Γ)= λ 1/2 k . We get |A2| ≤ C ∞ X l=1 |αk,1k ||hl|λ 1/2 k X p6=k (αp,lk )2λp 1/2 ≤ C ∞ X l=1 (αk,1k )2λk|hl|2 1/2 ∞ X l=1 X p6=k (αp,lk )2λp 1/2 ≤ C (αk,1k )2λk ∞ X l=1 |hl|2 1/2 X (p,l)∈V (αkp,l)2λp 1/2 = o(1).
Therefore we can obtain the estimate |I3| ≤ |A1| + |A2| = o(ε) + o(1) = o(1). This
implies that the fourth term is o(ε), which proves Theorem 1.1.3.
3.6
The case where the interface is a sphere
If λk is not simple, it is difficult to get a more precise asymptotic behavior of the
eigenvalues in general. If the interface Γ is a sphere, however, we can obtain it al-though the eigenvalues of a sphere are not simple. In this section, we prove Theorem 1.1.5.
The idea of the proof is to take ψ = Pk(j+1)−1
p=k(j) α p,1
k upφ1 as the test function in
(3.16) and use a property of the coefficients of the second fundamental form of the sphere. Now, let Sn−1(r) denote n − 1 dimensional sphere with radius r > 0. First
of all, we will consider the coefficients of the second fundamental form of Sn−1(r).
Proof. Since Γ = Sn−1(r), the outward normal vector ν Γ to Γ is represented by ξ/r. By (3.8), we have bij = − ∂ ∂ξi ,∂νΓ ∂ξj ! = −1 r ∂ ∂ξi , ∂ ∂ξj ! = −1 rg0,ij. (3.54) Next, we take ψ = Pk(j+1)−1 p=k(j) α p,1
k upφ1 as a test function in (3.16). Then, the
right hand side of (3.16) is
λk,ε ˆ 1 −1 ˆ Γ e uk,εψ p Gεdξdτ = λk,ε k(j+1) X p=k(j) (αp,1k )2− I4ε + O(ε2), where I4 = λk,ε √ 2 ∞ X p1=1 k(j+1) X p2=k(j) ∞ X l=2 αp1,l k α p2,1 k fl ˆ Γ Hup1up2 p G0dξ .
Since Γ = Sn−1(r), the mean curvature H (defined as the sum of the principal
curvatures) equals −(n − 1)/r. Thus we have
I4 = − n − 1 r λk,ε √ 2 ∞ X p1=1 k(j+1) X p2=k(j) ∞ X l=2 αp1,l k α p2,1 k fl ˆ Γ up1up2 p G0dξ = −n − 1 r λ√k,ε 2 k(j+1) X p=k(j) ∞ X l=2 αp,lk αkp,1fl. Therefore, |I4| ≤ C k(j+1) X p=k(j) ∞ X l=2 (αp,lk )2 1/2 k(j+1) X p=k(j) ∞ X l=2 (αp,1k )2fl2 1/2 ≤ C ∞ X p=1 ∞ X l=2 (αkp,l)2 1/2 k(j+1) X p=k(j) (αp,1k )2 ∞ X l=2 fl2 1/2 = o(ε).
Furthermore, the left hand side of (3.16) is ˆ 1 −1 ˆ Γ σ ∇guek,ε· ∇gψ + 1 ε2 ∂uek,ε ∂τ ∂ψ ∂τ p Gεdξdτ = σ−+ σ+ 2 λk k(j+1) X p=k(j) (αp,1k )2+σ+√− σ− 2 λk k(j+1) X p=k(j) ∞ X l=2 αp,lk αp,1k ˆ 1 0 φldτ + ε ∞ X p1,l=1 k(j+1) X p2=k(j) αp1,l k α p2,1 k hl ˆ Γ n−1 X i,j=1 e Gij − Hg0ij∂up1 ∂ξi ∂up2 ∂ξj p G0dξ + O(ε 2 ).
The second term can be treated in the same way as I4: we can show that it is o(ε).
Moreover, by the definition of the term eGij and Lemma 3.6.1, we have
e Gij− Hgij0 = 2 n−1 X k,l=1 gik0 bklglj0 − Hg ij 0 = −2 r n−1 X k,l=1 g0ikg0,klg0lj+ n − 1 r g ij 0 = −2 rg ij 0 + n − 1 r g ij 0 = n − 3 r g ij 0 . Thus, ∞ X p1,l=1 k(j+1) X p2=k(j) αp1,l k α p2,1 k hl ˆ Γ n−1 X i,j=1 e Gij − Hg0ij∂up1 ∂ξi ∂up2 ∂ξj p G0dξ = n − 3 r ∞ X p1,l=1 k(j+1) X p2=k(j) αp1,l k α p2,1 k hl ˆ Γ n−1 X i,j=1 g0ij∂up1 ∂ξi ∂up2 ∂ξj p G0dξ = n − 3 r λk k(j+1) X p=k(j) ∞ X l=1 αp,lk αp,1k hl= n − 3 4r (σ+− σ−) λk k(j+1) X p=k(j) (αkp,1)2+ o(1).
Hence the third term is n − 3 4r (σ+− σ−) λkε k(j+1) X p=k(j) (αkp,1)2 + o(ε).
Chapter 4
Asymtotic behavior of the
eigenvalues of reinforcement
problems
In this chapter, we study the reinforcement problem (1.4) and investigate the asymp-totic behavior for the principal eigenvalue.
4.1
Problem setting for the reinforcement
prob-lem
Let us recall the problem setting for the reinforcement problem (1.4). Let Ω ⊂ Rn (n ≥ 2) be a bounded domain with smooth and connected boundary Γ. For sufficiently small ε > 0, put
Σε=x ∈ Rn | x = ξ + tνΓ(ξ) for ξ ∈ Γ, 0 < t < ε , Ωε = Ω ∪ Σε∪ Γ,
where νΓ denotes the outward unit normal vector to Γ. We consider the following
two-phase eigenvalue problem on Ωε:
−div (σε∇u) = λu in Ωε,
u = 0 on ∂Ωε,
where σε = σε(x) (x ∈ Ωε) is the piecewise constant function given by σε(x) = 1, x ∈ Ω, qε, x ∈ Σε,
where qε = αε and α is a positive parameter.
We consider the problem (4.1) in a weak sense, namely, we say that λ ∈ C is an eigenvalue of (5.1) if there exists a function u ∈ H01(Ωε) with u 6≡ 0 and such that
for any ψ ∈ H1 0(Ω), ˆ Ω ∇u · ∇ψ dx + qε ˆ Σε ∇u · ∇ψ dx = λ ˆ Ωε uψ dx. (4.2)
By Theorem 2.1.5, the eigenvalues of (4.1) are positive real numbers and the set of all eigenvalues is discrete. Let {λk(ε)}k≥1 be the eigenvalues satisfying 0 < λ1(ε) <
λ2(ε) ≤ λ3(ε) ≤ · · · → +∞ and {uk,ε}k≥1 be the associated eigenfunctions in (4.1)
which are assumed to be normalized so that ˆ Ωε uk,ε 2 dx = 1.
Since σε = σε(x) (x ∈ Ωε) is a piecewise constant function, by integration by
parts, we can rewrite (4.2) as follows: −∆u1 = λu1 in Ω, −qε∆u2 = λu2 in Σε, u1 = u2 on Γ, ∂u1 ∂νΓ = qε ∂u2 ∂νΓ on Γ, u2 = 0 on ∂Ωε. (4.3)
Here u1 and u2 are the restriction of the eigenfunction u on Ω and Σε, respectively.
The fourth equality in (4.3), the so-called transmission condition, plays an important role in this problem.
The purpose of this chapter is to study the asymptotic behavior for the principal eigenvalue λ1(ε) as ε → 0. In particular, our aim is to show how the geometric
shape of the interface Γ affects the asymptotic behavior for the principal eigenvalue λ1(ε).
In what follows, let the principal eigenfunction u1,εbe denoted by uε for the sake
of simple notation.
4.2
Geometric preliminaries for the thin layer
We present some geometric preliminaries of thin layer Σε in the same manner of
Section 3.2. Every x ∈ Σε can be represented by
x = ξ + tνΓ(ξ), ξ ∈ Γ, 0 < t < ε. (4.4)
We introduce a local coordinate system (ξ1, ξ2, · · · , ξn−1, ξn) = (ξ1, ξ2, · · · , ξn−1, t)
for Γ × (0, ε) and let g = gij(ξ, t) denote the metric tensor associated with it.
Then from (4.4), gij(ξ, t) is given by
gij(ξ, t) = g0,ij(ξ) + teg0,ij(ξ) + t 2 b g0,ij(ξ) if 1 ≤ i, j ≤ n − 1, 0 if i = n, j 6= n or i 6= n, j = n, 1 if i, j = n, (4.5)
where g0 = g0,ij(ξ) denotes the Riemannian metric associated with the local
co-ordinates (ξ1, ξ2, · · · , ξn−1) and we put
e g0,ij = ∂ ∂ξi ,∂νΓ ∂ξj ! + ∂ ∂ξj ,∂νΓ ∂ξi ! , bg0,ij = ∂νΓ ∂ξi ,∂νΓ ∂ξj ! .
Here ∂/∂ξi and ∂/∂ξj are tangent vectors on ξ ∈ Γ and (·, ·) is the Euclidean inner
product. Let (bij)1≤i,j≤n−1 denote the coefficients of the second fundamental form
on Γ. In this local coordinates, bij = ∂2/∂ξi∂ξj, νΓ. By the definition of ge0,ij, we have eg0,ij = −2bij. Also, let the inverse matrix of (gij) be denoted by (gij) and
put G = det(gij). By using this local coordinates, we can express the norm of the
gradient of u as follows: |∇xu| 2 = n X i,j=1 gij∂u ∂ξi ∂u ∂ξj = |∇tanu| 2 + ∂u ∂t 2 , (4.6)
where |∇tanu|2 = Pn−1i,j=1gij∂u/∂ξi∂u/∂ξj. We recall the following asymptotic
for-mula for √G: p
where H(ξ) is the mean curvature at ξ ∈ Γ with respect to νΓ (defined as the sum of
the principal curvatures of Γ). Similar to Chapter 3, the asymptotic formula (4.7) will play an important role in obtaining the asymptotic behavior for the principal eigenvalue λ1(ε).
In what follows, pG(ξ, t) will be denoted by √Gt for simplicity and C will be
used to represent any positive constant independent of ε. The same letter C will be used to denote different constants.
4.3
Asymptotic behavior for λ
1(ε)
4.3.1
Upper bound of λ
1(ε)
By Theorem 2.1.7, we have λ1(ε) = inf u∈H1 0(Ω), u6=0 ˆ Ω |∇u|2dx + qε ˆ Σε |∇u|2dx ˆ Ωε |u|2dx . (4.8)We construct a test function in order to estimate the principal eigenvalue λ1(ε). We
extend the normalized Robin principal eigenfunction w1 = w1(x) (x ∈ Ω) in (1.5)
along νΓ to Σε by setting w1(ξ, t) = w1(ξ) for every ξ ∈ Γ. Also we put
φ(x) = 1 in Ω, 1 − t ε in Σε. Taking ˜u = w1φ as a test function in (4.8), we obtain
λ1(ε) ≤ ˆ Ω |∇˜u|2dx + qε ˆ Σε |∇˜u|2dx ˆ Ω |˜u|2dx = ˆ Ω |∇w1|2dx + qε ˆ Σε ∇(wφ) 2 dx ˆ Ω |w1|2dx + ˆ Σε |w1φ|2dx .
By using the normalization ´Ω|w1| 2 dx = 1 and ´Σ ε|w1φ| 2 dx = O(ε), we have ˆ Ω |w1|2dx + ˆ Σε |w1φ|2dx = 1 + O(ε).
Also we have ∇w1·∇φ = 0 since w1 and φ only depend on ξ and t in Σε, respectively. Hence, qε ˆ Σε ∇(w1φ) 2 dx = qε ˆ Σε φ2|∇w1|2+ 2∇w1· ∇φ + w12|∇φ| 2 dx = qε ˆ Σε φ2|∇w1|2dx + qε ˆ Σε w21|∇φ|2dx = qε ˆ Σε w21|∇φ|2dx + O(ε2).
We note that ∇φ = ∂φ∂tνΓ = −νΓ/ε. By using the asymptotic formula (4.7), we get
qε ˆ Σε w21|∇φ|2dx = αε ˆ ε 0 ˆ Γ w1(ξ)2· −νΓ ε 2 p Gtdξdt = αε · 1 ε2 ˆ ε 0 ˆ Γ w1(ξ)2(1 + O(1)t) p G0dξdt = α ˆ Γ w21pG0dξ + O(ε). Thus, λ1(ε) ≤ ˆ Ω |∇˜u|2dx + qε ˆ Σε |∇˜u|2dx ˆ Ω |˜u|2dx ≤ ˆ Ω |∇w1|2dx + α ˆ Γ w12pG0dξ + Cε 1 − Cε ≤ µ1+ Cε.
Therefore, we obtain the following upper bound of the principal eigenvalue λ1(ε):
λ1(ε) ≤ µ1+ Cε. (4.9)
4.3.2
Lower bound of λ
1(ε)
Let us recall the weak form (4.2): for any ψ ∈ H01(Ω), ˆ Ω ∇uε· ∇ψ dx + qε ˆ Σε ∇uε· ∇ψ dx = λ1(ε) ˆ Ωε uεψ dx.
First of all, we mention that we can get the following H1 and H2-estimate of the
first established by Brezis–Caffarelli–Friedman [6] in the case of two-phase elliptic equations by using standard elliptic regularity theory. Friedman [14] proved them by using a similar method.
Lemma 4.3.1. The principal eigenfunction uε satisfies
ˆ Ω |∇uε|2 dx + qε ˆ Σε |∇uε|2 dx ≤ C, (4.10) ˆ Ω D2uε 2 dx + qε ˆ Σε D2uε 2 dx ≤ C (4.11)
for a positive constant C independent of ε.
Let us construct a suitable test function. We take any ζ ∈ C1(Ω). Let us extend
ζ along νΓ to Σε by ζ(ξ, t) = ζ(ξ) for every ξ ∈ Γ. We take ψ = ζφ as a test function
in (4.2), then we have ˆ Ω ∇uε· ∇ζ dx + qε ˆ Σε φ∇uε· ∇ζ dx + qε ˆ Σε ζ∇uε· ∇φ dx = λ1(ε) ˆ Ω uεφζ dx + λ1(ε) ˆ Σε uεφζ dx.
The second term on the left-hand side and the second term on the right-hand side are O(ε). Indeed, for any ζ ∈ C1(Ω), by using the H1-estimate (4.10) we have
qε ˆ Σε φ∇uε· ∇ζ dx ≤ ˆ Σε q 1/2 ε ∇uε · q 1/2 ε ∇ζ dx ≤ qε ˆ Σε |∇uε|2dx !1/2 qε ˆ Σε |∇ζ|2dx !1/2 ≤ Cε.
By using the upper bound of λ1(ε), we also have
λ1(ε) ˆ Σε uεφζ dx ≤ C ˆ Σε |uε||ζ| dx ≤ C ˆ Σε |uε| 2 dx !1/2 ˆ Σε |ζ|2dx !1/2 .
Now we need to estimate ´Σ
ε|uε|
2
dx. By the Dirichlet boundary condition on ∂Ωε,
we get uε(ξ, t) = − ˆ ε t ∂uε ∂t ds. (4.12)
This identity implies that
|uε|2 ≤ ˆ ε 0 ∂uε ∂t ds !2 ≤ ε ˆ ε 0 ∂uε ∂t 2 ds. (4.13) Thus we have ˆ Σε |uε|2dx ≤ ε ˆ Σε ˆ ε 0 ∂uε ∂t 2 ds ! dx ≤ Cε qε ˆ Σε |∇uε|2dx ! . (4.14)
Therefore we obtain the following estimate: λ1(ε) ˆ Σε uεφζ dx ≤ Cε.
Note that ∇φ = ∂φ∂tνΓ = −νΓ/ε and, by using the asymptotic formula (4.7), we have
qε ˆ Σε ζ∇uε· ∇φ dx = qε ˆ Σε ζ∇uε· −νΓ ε dx = −α ˆ ε 0 ˆ Γ ζ∂uε ∂t p Gtdξdt = −α ˆ ε 0 ˆ Γ ζ∂uε ∂t 1 + O(1)t pG0dξdt = α ˆ Γ uεζ p G0dξ + O(ε). Therefore, we obtain ˆ Ω ∇uε· ∇ζdx + α ˆ Γ uεζ p G0dξ = λ1(ε) ˆ Ω uεζdx + O(ε). (4.15)
We consider the Fourier expansions of uε with respect to the orthonormal basis
given by the eigenfunctions of the following Robin eigenvalue problem: −∆w = µw in Ω, αw + ∂w ∂νΓ = 0 on Γ. (4.16)
Let {µk}k≥1 be the eigenvalues corresponding to problem (4.16), ordered so that
they satisfy 0 < µ1 ≤ µ2 ≤ µ3 ≤ · · · → +∞, and {wk}k≥1 be the associated
eigenfunctions which are assumed to be normalized so that ˆ
Ω
|wk|2dx = 1.
Then uε admits the following Fourier expansions in H1(Ω) (see Remark 2.1.6):
uε = X k≥1 ck(ε)wk, ck = ˆ Ω uεwkdx. (4.17)
Taking ζ = c1w1in (4.15) and using the orthogonality of the Robin eigenfunctions
{wk}k≥1, we have
(c1)2µ = λ1(ε)(c1)2+ O(ε). (4.18)
From the estimate (4.18), it will be sufficient to show the following lemma to get the lower bound of λ1(ε). It can be proven in the same way of Lemma 3.4.2.
Lemma 4.3.2. The following estimate holds:
c1(ε) = 1 + o(1) as ε → 0. (4.19)
Proof. From Lemma 4.3.1, we obtain the uniform H1-boundedness of the principal eigenfunction uε in Ω. Applying Rellich’s Theorem 2.1.3, after passing to a
subse-quence, there exists bu ∈ H1(Ω) such that u
ε → u strongly in Lb
2(Ω) and weakly in
H1(Ω). Moreover, for some nonnegative value bλ we also have λ
1(ε) → bλ and bλ ≤ µ1. If we let ε → 0 in (4.15), then ˆ Ω ∇bu · ∇ζdx + α ˆ Γ b uζpG0dξ = bλ ˆ Ω b uζdx. (4.20)
Thus bλ is a Robin eigenvalue andu is the corresponding Robin eigenfunction. It im-b plies that µ1 ≤ bλ. Therefore we obtain bλ = µ1. Since µ1 is the principal eigenvalue,
we have that either bu = w1 or bu = −w1. Also, since uε is chosen to be positive function, we getu = wb 1. By using the fact that uε converges tou strongly in Lb
2(Ω),
we get the estimate c1 = 1 + o(1) as ε → 0.
From (4.18) and Lemma 4.3.2, we have
λ1(ε) ≥ µ1− Cε. (4.21)
Combining the upper bound (4.9) with the lower bound (4.21), we obtain
