UNIQUE CONTINUATION FOR SPECTRAL FRACTIONAL ELLIPTIC OPERATORS
PU-ZHAO KOW AND JENN-NAN WANG
Abstract. In this paper we consider the eigenvalue problem for a weighted spectral fractional second order elliptic operator in a bounded domain. We show that any eigenvalue is strictly mono- tone with respect to the weight function if the corresponding eigen- function satisfies the unique continuation property from a measur- able set of positive Lebesgue measure.
1. Introduction
In this paper we will investigate the relation between the monotonic- ity of eigenvalues and the unique continuation property for the spectral fractional elliptic operator. To motivate our study, we first briefly state the result for the elliptic operator. Consider the weighted eigenvalue problem in a bounded domain Ω ⊂ Rn with Lipschitz boundary ∂Ω:
(1.1)
( Au = µm(x)u in Ω, u = 0 on ∂Ω,
where A is a second order elliptic operator given by Au = −
n
X
i,j=1
∂j(aij(x)∂iu) + a0(x)u
with a0(x) ≥ 0 and (aij(x)) ∈ L∞(Ω) satisfying aij(x) = aji(x), the ellipticity condition
(1.2) Λ1|ξ|2 ≤
n
X
i,j=1
aij(x)ξiξj ≤ Λ2|ξ|2, 0 < Λ1, Λ2.
2010 Mathematics Subject Classification. 35R11; 35B60; 47A75.
Key words and phrases. Spectral fractional elliptic operators; Caffarelli-Silvestre- Stinga type extension; Non-local operators; Weighted eigenvalue problems; Unique continuation from a measurable set; Fractional Schrödinger equation.
1
Assume that a0, m(x) ∈ Lr(Ω) for some r > n/2. It is known that the eigenvalues of (1.1), depending on m, form a countable sequence:
· · · ≤ µ−2(m) ≤ µ−1(m) < 0 < µ1(m) ≤ µ2(m) ≤ · · · .
If m is non-negative (or non-positive), then this sequence is bounded below (or bounded above). In view of the variational characterization of eigenvalues, we can observe that each µk, k ∈ Z \ {0}, is nonincreasing in the weight function m, i.e., if m(x) ≤ ˆm(x) a.e., then µk( ˆm) ≤ µk(m). It was proved in [FG92] that µk(m) is strictly decreasing in m if and only if the corresponding eigenfunction enjoys the unique continuation property from a set of positive measure. We say that µk(m) is strictly monotonically decreasing in m if m(x) ≤ ˆm(x) a.e.
and {x : ˆm(x) − m(x) > 0} has positive measure, then µk( ˆm) < µk(m).
For the corresponding eigenfunction uk(x), we say that uk(x) has the measurable unique continuation property (MUCP) if u = 0 identically in Ω whenever uk(x) = 0 in E ⊂ Ω with the Lebesgue measure of E,
|E| > 0. A similar result was proved for the biharmonic operator ∆2 in [TCRD12].
The equivalence of strict monotonicity of eigenvalues and MUCP was further extended to nonlocal operators in [FI19] where the authors considered the eigenvalue problem
(1.3)
( LKu = µm(x)u in Ω, u = 0 in Rn\ Ω,
where LK is a nonlocal operator of the following general form (1.4) LKu(x) = p.v.
Z
Rn
(u(x) − u(y))K(x − y) dy, where the kernel K satisfies
(K1) ρ(x)K ∈ L1(Rn), where ρ(x) = min{|x|2, 1};
(K2) K(x) ≥ α|x|−(n+2s) for all x ∈ Rn\ {0} and s ∈ (0, 1);
(K3) K(−x) = K(x) for all x ∈ Rn\ {0}.
In particular, when K(x) = |x|−(n+2s), LK is known to be the fractional Laplacian and (1.3) is the eigenvalue problem for the regional fractional Laplacian. Since LK is nonlocal, to define LKu(x) for x ∈ Ω, u = 0 in Rn\ Ω serves as the zero Dirichlet condition.
The main theme of this work is to establish the equivalence of strict monotonicity of eigenvalues and MUCP for the spectral elliptic oper- ator. To define the operator, let us denote the second order elliptic
operator
Lu(x) = −
n
X
i,j=1
∂j(aij(x)∂iu(x)).
It is a standard result that there exists a sequence of positive Dirich- let eigenvalues and orthonormal eigenfunctions {λk, φk}∞k=0 with φk ∈ H01(Ω) for L in Ω. For s ∈ (0, 1), we define the spectral fractional elliptic operator
(1.5) Lsu(x) :=
∞
X
k=0
λskukφk(x) in Ω, where u(x) =P∞
k=0ukφk ∈ H01(Ω), i.e., uk = (u, φk). We also consider spectral fractional elliptic operator Lγ with the fractional power γ ∈ R+ \ N (see the precise definition of Lγ in Section 2). In this work, we will not discuss the classical case where γ ∈ N. Let µk(m) be the eigenvalue of
Lγψk(x) = µk(x)m(x)ψk(x) in Ω
with the corresponding eigenfunction ψk(x) belonging to a certain func- tion space (see Section 3). We then prove that µk is strictly monoton- ically decreasing in m if and only if ψk enjoys MUCP in Ω. We want to point out the spectral elliptic operator Lγ can not be written in the form of LK in (1.4) with a suitable kernel K. One can easily observe that if such kernel K exists for Lγ, then it cannot satisfy Property (K2) of K given above. In other words, our result here does not follow from that in [FI19].
Even though Lγ is defined in a bounded domain Ω, it is a nonlocal operator. The proof of the uniqueness continuation property is highly nontrivial. However, it was shown in [ST10] that Ls for s ∈ (0, 1) can be expressed as the Dirichlet-to-Neumann map of an extension problem in the spirit of the fractional Laplacian (−∆)s established in [CS07].
The operator in the extension problem is a local, but degenerate, elliptic operator. Combining [ST10] and [Yan13], the spectral fractional elliptic operator Lγ can also be described as the Dirichlet-to-Neumann map of an extension problem. Having established the extension problem for Lγ, we can prove the MUCP using some results from [GR19] involving Carleman estimates.
The paper is organized as follows. In Section 2, we discuss the def- inition of the spectral fractional elliptic operator Lγ in detailed. We also describe the corresponding extension problem, especially the case of γ > 1. In Section 3, we state and prove main results of the paper.
We will discuss the unique continuation property for Lγ in Section 4.
2. The Fractional Operator Lγ
Let Ω be a bounded Lipschitz domain in Rn. We now give a formal definition of Lγ for γ ∈ R+\ N. Let bγc be the integer part of γ and s := γ − bγc, that is, we write
γ = bγc + s with bγc ∈ Z≥0 and s ∈ (0, 1).
To consider the fractional elliptic operator of higher power, we need to impose a higher regularity on the coefficients, namely,
(2.1) (aij) ∈ C2bγc,1(Ω).
Before giving the precise definition of Lγ, we first discuss some special Sobolev spaces [Gr16]. For s > 0, we define the space Hs(Ω) as the restriction of Hs(Rn) to Ω. Let us denote
Hes(Ω) =
u ∈ Hs(Ω), 0 < s < 1/2,
u ∈ Hs(Ω), u = 0 on ∂Ω, 1/2 < s < 5/2,
u ∈ Hs(Ω), u = Lu = · · · = Lku = 0 on ∂Ω, 2k + 1/2 < s < 2k + 5/2,
u ∈ Hs(Ω), u = Lu = · · · = Lk−1u = 0 on ∂Ω, Lku ∈ H1/2(Rn) with supp u ∈ Ω, s = 2k + 1/2.
Now we define Lγ for γ ∈ R+\ N as (2.2) Lγu(x) :=
∞
X
k=0
λγk(u, φk)φk(x) in Ω
for u ∈ dom(Lγ) = eH2γ(Ω). Thus, we have Lγ : dom(Lγ) → L2(Ω).
We can see that for u ∈ dom(Lγ) Lγu =
∞
X
k=0
λγk(u, φk)φk=
∞
X
k=0
λs+bγc−1k (u, λkφk)φk =
∞
X
k=0
λs+bγc−1k (Lu, φk)φk
=
∞
X
k=0
λs+bγc−2k (Lu, λkφk)φk=
∞
X
k=0
λs+bγc−2k (L2u, φk)φk= · · ·
=Ls(Lbγcu).
On the other hand, we also have that for u ∈ dom(Lγ) Lγu =
∞
X
k=0
λγk(u, φk)φk=
∞
X
k=0
λbγck (u, λskφk)φk =
∞
X
k=0
λbγck (Lsu, φk)φk= Lbγc(Lsu), which immediately implies
(2.3) Lγu = Ls(Lbγcu) = Lbγc(Lsu).
3. An Eigenvalue Problem: Strict Monotonicity and Unique Continuation
For γ ∈ R+\ N we consider the eigenvalue problem (3.1)
(Lγu = µm(x)u in Ω, u ∈ dom(Lγ),
where m ∈ L∞(Ω). We are interested in the connection between the strict monotonicity of the eigenvalues µ(m) and the weight function m.
The classical case γ = 1 was studied by de Figueredo and Gossez in [FG92].
We first discuss the existence of discrete eigenvalues of (3.1). Since m is an indefinite weight, we will follow the approach used in [Fig82]
(or [FI19]). In view of (2.3), Lγ is a self-adjoint operator in L2(Ω) with domain dom(Lγ). The eigenvalue problem (3.1) can be expressed in the variational form:
(3.2) a[u, v] = µ Z
Ω
muv, for all v ∈ dom(Lγ), u ∈ dom(Lγ), where a[·, ·] : dom(Lγ) × dom(Lγ) → R is the bilinear form (inner product) defined by
a[u, v] :=
Z
Ω
(Lγu(x))v(x) dx.
Clearly, kukγ = a[u, u]1/2 induces a norm on dom(Lγ). However, dom(Lγ) is, in general, not complete in k · kγ. Thus, we need to con- sider a suitable extension of Lγ. Since Lγ is semibounded, Lγ can be extended in the Friedrichs sense to H, the completion of dom(Lγ) in k · kγ. Observe that
(3.3) a[u, v] =
Z
Ω
(Lγ/2u)(Lγ/2v) dx for u, v ∈ dom(Lγ). Thus, we have
H = eHγ(Ω).
To abuse the notation, we still denote its Friedrichs extension by Lγ. Note that the Friedrichs extension of Lγremains self-adjoint on eHγ(Ω).
For fixed u ∈ eHγ(Ω), the map v 7→
Z
Ω
muv is a bounded linear functional in eHγ(Ω). By the Riesz-Fréchet representation theorem,
there exists a unique element in eHγ(Ω), says T u, such that (3.4) a[T u, v] =
Z
muv for all v ∈ eHγ(Ω).
We can see that T is self-adjoint and bounded in eHγ(Ω). We can further prove that
Lemma 3.1. The operator T : eHγ(Ω) → eHγ(Ω) is compact.
Proof. Let {un} be a bounded sequence in eHγ(Ω). Then there exists a subsequence, still denoted {un}, such that
un → u weakly in eHγ(Ω).
The compact embedding of eHγ(Ω) ,→ L2(Ω) (see, for example, [DPV12]), implies
un → u strongly in L2(Ω).
Substituting u = un− u and v = T un− T u in (3.4), we obtain kT un− T uk2γ ≤ kmkL∞(Ω)kT un− T ukL2(Ω)kun− ukL2(Ω)→ 0.
Consequently, T has a set of countably many real eigenpairs {˜λk, uk} in which ˜λk can only accumulate at 0. Therefore, from (3.3), we have that
Z
Ω
mukvdx = a[T uk, v] = ˜λka[uk, v] = ˜λk Z
Ω
(Lγ/2uk)(Lγ/2v)dx, i.e.,
Z
Ω
(Lγ/2uk)(Lγ/2v)dx = µk Z
Ω
mukvdx
for all v ∈ eHγ(Ω), where µk = 1/˜λk. In other words, the eigenvalue problem (3.1) has a double sequence of eigenvalues
· · · ≤ µ−2 ≤ µ−1 < 0 < µ1 ≤ µ2 ≤ · · · ,
with the corresponding eigenfunctions {uk} in the weak sense. How- ever, since µkmuk ∈ L2(Ω), we have that Lγuk ∈ L2(Ω), which implies that uk ∈ eH2γ(Ω)(= dom(Lγ)). In other words, (µk, uk) solves (3.1) in the strong sense.
Repeating the arguments in [Fig82] (or in [FI19]), we can derive the following variational characterization of eigenvalues. For the sake of completeness, we will provide its proof in Appendix A.
Proposition 3.2. The sequence of eigenvalues
· · · ≤ µ−2 ≤ µ−1 < 0 < µ1 ≤ µ2 ≤ · · · , can be characterized by
1
µn(m) = max
Fn
inf
Z
Ω
mu2 : kukγ = 1, u ∈ Fn
, 1
µ−n(m) = min
Fn
sup
Z
Ω
mu2 : kukγ= 1, u ∈ Fn
, (3.5)
where Fn varies over all n-dimensional subspaces of eHγ(Ω). In partic- ular, we have
(3.6) 1
µk(m) = Z
Ω
mu2k with kukkγ = 1.
Following exactly the same argument as in [Fig82], we obtain the following result, which shows some properties of the eigenvalues.
Proposition 3.3. Let Ω±:= {x ∈ Ω : m(x) ≷ 0}, then (i) |Ω+| = 0 =⇒ there is no positive µn.
(ii) |Ω−| = 0 =⇒ there is no negative µ−n.
(iii) |Ω+| > 0 =⇒ there is a sequence of positive µn → +∞.
(iv) |Ω−| > 0 =⇒ there is a sequence of negative µ−n→ −∞.
Here, | · | denotes the Lebesgue measure of the set.
Proposition 3.4. Let m, ˆm ∈ L∞(Ω) such that m(x) ≤ ˆm(x) for x ∈ Ω. For a given n ∈ Z \ {0}, if the eigenvalues µn(m) and µn( ˆm) exist, then µn(m) ≥ µn( ˆm).
Proposition 3.5. µn(m) is a continuous function of m in the norm of L∞(Ω).
Proposition 3.4 and Proposition 3.5 are immediate consequences of (3.6).
Now we use ≤6≡ to denote that the inequality holds a.e. with strict inequality on a set of positive measure. The following results can be easily proved following the ideas in [FG92].
Proposition 3.6. Let m and ˆm be two weights with m ≤6≡ ˆm. For any j ∈ N, if the eigenfunction associated with µj(m) satisfies the MUCP, then the strict inequality µj(m) > µj( ˆm) holds.
Proposition 3.7. Let m be a given weight. Assume that µ(m) is an eigenvalue of (3.1) and the corresponding eigenfunction u(m) does not
satisfy the MUCP. Denote N = {x ∈ Ω : u(x) = 0}. Note that
|N | > 0. Then for any weight ˆm satisfying
{x ∈ Ω : | ˆm(x) − m(x)| > 0} ⊆ N ,
we obtain that u(m) is also an eigenfunction of some eigenvalue µ( ˆm) of (3.1) with weight ˆm and µ( ˆm) = µ(m).
Remark 3.8. Observe that µj(m) = −µ−j(−m), we can obtain an ana- logue result for negative eigenvalues.
Remark 3.9. Proposition 3.7 implies that for some j ∈ N, if the eigen- function corresponding to µj(m) does not satisfy the MUCP, then µj(m) = µ`( ˆm) for some ` ∈ N.
To make the paper self contained, we present the proofs of Proposi- tion 3.6 and 3.7 here.
Proof of Proposition 3.6. By Proposition 3.2, there exists Fj ⊂ dom(Lγ) with dim(Fj) = j such that
(3.7) 1
µj(m) = inf
u∈Fj,kukγ=1
Z
Ω
m|u|2. Pick any u ∈ Fj with kukγ = 1.
Case 1. If u achieves the infimum in (3.7), then u is an eigenfunction corresponding to µj(m), and by the MUCP assumption, we have that u > 0 a.e. and thus
1 µj(m) =
Z
Ω
m|u|2 <
Z
Ω
ˆ m|u|2.
Case 2. If u does not achieve the infimum in (3.7), then we have 1
µj(m) <
Z
Ω
m|u|2 ≤ Z
Ω
ˆ m|u|2. In view of both cases, we conclude that
1 µj(m) <
Z
Ω
ˆ
m|u|2, for all u ∈ Fj with kukγ = 1.
Since dim(Fj) = j < ∞, by a compactness argument, we then obtain 1
µj(m) < inf
u∈Fj,kukγ=1
Z
Ω
ˆ
m|u|2 ≤ max
Fj
inf
u∈Fj,kukγ=1
Z
Ω
ˆ
m|u|2 = 1 µj( ˆm),
which leads to the desired result.
Proof of Proposition 3.7. Let u ∈ dom(Lγ) be an eigenfunction associ- ated with eigenvalue µ(m) which vanishes on a set of positive measure N . In other words, we have and
Lγu = µ(m)mu = µ(m) ˆmu in Ω, that is, µ(m) is an eigenvalue of (3.1) with weight ˆm.
4. Remark on Unique Continuation Property
In this section, we would like to discuss the MUCP for the spectral fractional operator Lγ. Following the exactly same ideas in Appendix A2 of [GR19] and [ST10], Lγu with u ∈ dom(Lγ) can be determined by an extension problem. Firstly, we recall that the heat semigroup of L is defined by
(4.1) e−tLu :=
∞
X
k=0
e−tλk(u, φk)φk(x).
Also, we define the operator
Lb := x−bn+1(∂xn+1xbn+1∂xn+1− xbn+1L) and the iterated operator
Ljb := (Lb)j for j ∈ N.
Proposition 4.1. Let γ ∈ R+ \ N and let u ∈ dom(Lγ). Then the Caffarelli-Silvestre-type extension of u(x), ˜u(x, xn+1), satisfies the sys- tem
(4.2)
Lbγc+11−2s u(x, x˜ n+1) = 0 in Ω × (0, ∞),
xn+1lim→0u(x, x˜ n+1) = u(x) for all x ∈ Ω,
xn+1lim→0Lk1−2su(x, x˜ n+1) = cn,γ,kLku(x) in Ω, for all k = 1, · · · , bγc, Lku(x, x˜ n+1) = 0 on ∂Ω × (0, ∞), for all k = 0, · · · , bγc,
xn+1lim→0x1−2sn+1∂n+1Lbγc1−2su(x, x˜ n+1) = cn,γLγu(x) in Ω,
xn+1lim→0x1−2sn+1∂n+1Lk1−2su(x, x˜ n+1) = 0 in Ω, for all k = 0, · · · , bγc − 1.
In fact, ˜u(x, xn+1) can be expressed explicitly by
˜
u(x, xn+1) := cγx2γn+1 Z ∞
0
e−tLu(x)e−
x2n+1 4t dt
t1+γ ∈ C2(bγc+1),1(Ω × (0, ∞)).
On the other hand, the extension solution ˜u(x, xn+1) can also be writ- ten as
˜
u(x, xn+1) := ˜cγ Z ∞
0
e−tLLγu(x)e−
x2n+1 4t dt
t1−γ.
Setting ˜u0(x, xn+1) := ˜u(x, xn+1), we can rewrite system (4.2) as the following one
(4.3)
L1−2su˜bγc= 0 in Ω × (0, ∞),
L1−2su˜j = ˜uj+1 in Ω × (0, ∞), for all j = 0, · · · , bγc − 1,
xn+1lim→0u˜j(x0, xn+1) = cn,γ,jLju(x) in Ω, for all j = 0, · · · , bγc,
˜
uj = 0 on ∂Ω × (0, ∞), for all j = 0, · · · , bγc.
xn+1lim→0x1−2sn+1∂n+1u˜bγc= cn,γLγu in Ω,
xn+1lim→0x1−2sn+1 ∂n+1u˜j = 0 in Ω, for all j = 0, · · · , bγc − 1.
All boundary conditions in (4.2) and (4.3) hold in L2 sense.
In [GR19], the authors study the fractional operator Lγ in Rn for γ ∈ R+\ N, where the fractional operator Lγ is defined in terms of the spectral decomposition. Precisely, we write
(Lf, g) = Z ∞
0
λdEf,g(λ), for all f ∈ dom(L), g ∈ L2(Rn), where dom(L) = {f ∈ L2(Rn) : R∞
0 λ2dEf,f(λ) < ∞} and dEf,g(λ) is the spectral measure corresponding to L. The fractional operator Lγ is now defined by
(Lγf, g) = Z ∞
0
λγdEf,g(λ), for all f ∈ dom(Lγ), g ∈ L2(Rn), where dom(Lγ) = {f ∈ L2(Rn) :R∞
0 λ2γdEf,f(λ) < ∞}. The extension problem related to Lγu for u ∈ dom(Lγ) is similar to (4.2) and (4.3) except that Ω is replaced by Rn and no boundary restrictions
Lku(x, x˜ n+1) = 0 on ∂Ω × (0, ∞), for all k = 0, · · · , bγc and
˜
uj = 0 on ∂Ω × (0, ∞), for all j = 0, · · · , bγc are required.
In [GR19, Theorem 4], relying on the extension problem, the MUCP is established for the equation
(4.4) |Lγu| ≤
bγc
X
j=0
|qj(x)||∇ju| in Rn
under suitable assumptions on aij(x) and qj. Besides of the extension problem, another key ingredient in the proof of MUCP for (4.4) is the Carleman estimate for the extension problem. Since the extension problem is local, the same Carleman estimate can be used to prove the MUCP for
(4.5) Lγu = q(x)u in Ω
with q ∈ L∞(Ω). To state the MUCP result for (4.5), we first give the assumptions imposed on aij:
(C1) (aij) : Ω → Rn×n is symmetric, strictly positive definite and bounded;
(C2) (aij) ∈ C2bγc,1(Ω, Rn×nsym) with P2bγc+1
k=1 k∇kaijkL∞(Ω) δ for some sufficiently small parameter δ > 0;
(C3) aij(0) = δij.
Repeating the proof of Theorem 4 in [GR19], we can prove that Theorem 4.2 (MUCP for spectral fractional operator Lγ). Let u ∈ dom(Lγ) satisfy
Lγu(x) = q(x)u(x) in Ω,
where ajk satisfies the conditions (C1)–(C3) and q ∈ L∞(Ω). If there exists a measurable set E ⊂ Ω with |E| > 0 such that u = 0 in E, then u ≡ 0 in Ω.
For other UCP results for the fractional operators, we refer the reader to [FF14], [Rül15], [Yu17], etc. and references therein.
Acknoledgement
The authors are partially supported by MOST 108-2115-M-002-002- MY3.
Appendix A. The Min-Max Principle of Eigenvalues of Compact Operators
In this section, we shall prove the min-max principle in Proposition 3.2. The content of this section can be found in [Fig82] or [FG92]. Let H be a Hilbert space, and T : H → H be a compact symmetric linear operator. First of all, we recall a well-known facts about the compact linear operators.
Proposition A.1 (Existence of orthonormal eigenfunctions). If λn = sup{(T x, x) : kxk = 1, x ⊥ φ1, · · · , φn−1} > 0,
then there exists φn ∈ H with kφnk = 1 and φn ⊥ φ1, · · · , φn−1 such that
(T φn, φn) = λn and T φn = λnφn. Similarly, if
λ−n = sup{(T x, x) : kxk = 1, x ⊥ φ−1, · · · , φ−(n−1)} < 0,
then there exists φ−n∈ H with kφ−nk = 1 and φ−n⊥ φ−1, · · · , φ−(n−1)
such that
(T φ−n, φ−n) = λ−n and T φ−n = λ−nφ−n.
Using Proposition A.1, we can obtain the following min-max princi- ple.
Proposition A.2. For each positive integer n, λ±n can be character- ized as
λn= max
Fn
inf{(T x, x) : kxk = 1, x ∈ Fn}, (A.1)
λ−n= min
Fn
sup{(T x, x) : kxk = 1, x ∈ Fn}, (A.2)
where the maximum (minimum) is taken over all subspaces Fn of H with dim(Fn) = n.
Proof. Here we only prove (A.1). The proof of (A.2) is similar.
Given any subspace Fn of H with dim(Fn) = n, choose x ∈ Fn with kxk = 1 and x ⊥ φ1, · · · , φn−1. By Proposition A.1, we have (T x, x) ≤ λn. By arbitrariness of such x, we reach
inf{(T x, x) : kxk = 1, x ∈ Fn} ≤ λn for all subspace Fn of H with dim(Fn) = n.
This implies
(A.3) sup
Fn
inf{(T x, x) : kxk = 1, x ∈ Fn} ≤ λn.
By Proposition A.1, we can choose ˜Fn := span{φ1, · · · , φn}. For each x ∈ ˜F with kxk = 1, we can write
x =
n
X
i=1
xiφi with
n
X
i=1
x2i = 1.
For such x, we have
(T x, x) =
n
X
i=1
x2iλi ≥
n
X
i=1
x2iλn= λn.
This shows that the supremum in (A.3) is attained by ˜Fn. So we can write "max" rather than "sup".
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Department of Mathematics, National Taiwan University, Taipei 106, Taiwan.
Email address: d07221005@ntu.edu.tw
Institute of Applied Mathematical Sciences, NCTS, National Tai- wan University, Taipei 106, Taiwan.
Email address: jnwang@math.ntu.edu.tw
