Carleman Estimates for Parabolic Operators with Discontinuous and Anisotropic Diffusion Coefficients, an Elementary Approach

0.22 MB·English

by  Qi Lü

Carleman Estimates for Parabolic Operators
with Discontinuous and Anisotropic Diffusion
Coefficients, an Elementary Approach∗
3 Qi Lu¨† and Xu Zhang‡
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Abstract
J

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Byusingsomedeeptoolsfrommicrolocalanalysis,J.LeRousseauandL.Robbiano(In-

] vent. Math., 183(2011), 245–336) establishedseveralCarlemanestimates forparabolic
C
operators with isotropic diffusion coefficients which have jumps at interfaces. In this
O
paper, we revisit the same problem but for the general case of anisotropic diffusion
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h coefficients. Our main tools are a pointwise estimate for parabolic operators and a
t
a suitable chosen weight function.
m
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2010 Mathematics Subject Classification. Primary 35K05; Secondary 35K20, 35B37.
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6 Key Words. Carleman estimate, parabolic operator, diffusion coefficient, pointwise esti-
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mate, weight function.
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∗This work is partially supported by the NSFC under grants 11231007and 11101070.
†School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu
610054,China. e-mail: [email protected].
‡School of Mathematics, Sichuan University, Chengdu 610064,China. e-mail: zhang [email protected].
1
1 Introduction
Let Ω ∈ Rn (n ∈ N) be a bounded domain with the boundary Γ ∈ C2, and T > 0. Let
S be a C2 hypersurface in Ω such that S ∩Γ = ∅ and Ω\S is composed of two connected
domains Ω and Ω . Denote by ν(x)(= (ν (x),··· ,ν (x))) (resp. ν˜(x)) the unit outward
1 2 1 n
normal vector of Ω (resp. Ω ) at x ∈ Γ (resp. x ∈ Γ ) (Hence, ν(x) = −ν˜(x) for any
1 2 1 2
x ∈ S). For each δ > 0 and G ⊂ Rn, we define
O (G) = {x ∈ Rn | dist(x,G) < δ}.
δ
In what follows, we assume that
(aij) ∈ C2([0,T]×Ω ;Rn×n), (a˜ij) ∈ C2([0,T]×Ω ;Rn×n), (1.1)
1≤i,j≤n 1 1≤i,j≤n 2
are fixed, satisfying
aij(t,x) = aji(t,x), ∀ (t,x) ∈ [0,T]×Ω , i,j = 1,2,··· ,n,
1
(1.2)
a˜ij(t,x) = a˜ji(t,x), ∀ (t,x) ∈ [0,T]×Ω , i,j = 1,2,··· ,n,
2
and for some constant s > 0,
0
n
aij(t,x)ξiξj ≥ s |ξ|2, ∀ (t,x,ξ) ∈ [0,T]×Ω ×Rn,
0 1
i,j=1
X (1.3)
n
a˜ij(t,x)ξiξj ≥ s |ξ|2, ∀ (t,x,ξ) ∈ [0,T]×Ω ×Rn,
0 2
i,j=1
X
where ξ = (ξ1,··· ,ξn).
We consider the following transmission problem for a parabolic equation:
n
y + (aijy ) = f in (0,T)×Ω ,
1,t 1,xi xj 1 1

i,j=1
 X

 n

 y + (a˜ijy ) = f in (0,T)×Ω ,
 2,t 2,xi xj 2 2

 i,j=1
 X


 y1 = 0 on (0,T)×(Γ1 \S),


 y = 0 on (0,T)×(Γ \S), (1.4)
 2 2


 y = y +β on (0,T)×S,
1 2 1
 n n

 aijy ν = a˜ijy ν +β on (0,T)×S,
 1,xi j 2,xi j 2

 iX,j=1 iX,j=1

 y (T) = y0 in Ω ,
 1 1 1


 y (T) = y0 in Ω ,
 2 2 2




 2
where
f ∈ L2((0,T)×Ω ), f ∈ L2((0,T)×Ω ),
1 1 2 2
 β1 ∈ L2(0,T;H23(S))∩H1(0,T;H12(S)), β2 ∈ H1(0,T;L2(S)), (1.5)


 y0 ∈ L2(Ω ), y0 ∈ L2(Ω ).
1 1 2 2
The equation (1.4) is well-posed in the class:

(y ,y ) y ∈ C([0,T];L2(Ω ))∩C((0,T];H1(Ω )),y | = 0,i = 1,2 .
1 2 i i i i (0,T)×(Γi\S)
n (cid:12) o
In what follows(cid:12), we assume that ω is an open subset of Ω such that
(cid:12)
ω ∩Ω 6= ∅, i = 1,2. (1.6)
i
The main purpose of this paper is to derive the following type of a priori estimates, i.e.,
Carleman estimates for solutions to (1.4):
2 T
(W y2+W |∇y |2)dxdt
i1 i i2 i
i=1 Z0 ZΩi
X
2 T
(1.7)
≤ C W y2 + W f2 dxdt
i3 i i4 i
hXi=1 Z0 (cid:16)Zω∩Ωi ZΩi (cid:17)
+|W β |2 +|W β |2 ,
5 1 L2(0,T;H23(S))∩H1(0,T;H21(S)) 6 2 H1(0,T;L2(S))
i
for some suitably chosen (parameterized) weight functions W , W , W , W (i = 1,2), W
i1 i2 i3 i4 5
and W (which are positive almost everywhere). Here and henceforth, C denotes a generic
6
positive constant which may vary from one line to another.
Carleman estimate is a basic tool to solve many problems in partial differential equations,
say uniqueness in Cauchy problems ([5, 15]), inverse problems ([6, 8]), control problems
([4, 13, 14]), and so on. Especially, there are many works addressing Carleman estimates
for parabolic equations with smooth diffusion coefficients (e.g., [2, 4, 12]). Nevertheless,
very little are known for the Carleman estimate for parabolic equations with non-smooth
diffusion coefficients. In this respect, as far as we know, [1] is the first paper establishing
global Carleman estimates for the equation (1.4) when
(aij) = aI , (a˜ij) = a˜I (1.8)
1≤i,j≤n n 1≤i,j≤n n
for some time-independent functions a ∈ C2(Ω ) and a˜ ∈ C2(Ω ) (I stands for the n× n
1 2 n
identity matrix), and the following monotonicity condition holds:
a| ≥ a˜| . (1.9)
S S
By means of the microlocal techniques, the recent work [7] obtained several interesting local
and global Carleman estimates for the equation (1.4) when S ∈ C∞ and the assumption
(1.8) holds for some time-dependent functions a ∈ C∞([0,T]×Ω ) and a˜ ∈ C∞([0,T]×Ω )
1 2
without the monotonicity condition as (1.9).
3
In [7], an open problem was posed to derive Carleman estimates for the equation (1.4) in
thepresenceofjumpsofthediffusion(coefficient)matrixattheinterfaceS. Thepresentwork
aims to give an affirmative answer to this problem under considerably weak regularities on
the involved data, say aij, a˜ij and S. For this purpose, we need to derive a careful pointwise
estimate for parabolic operators, and to construct a suitable weight function that will be
used in the Carleman estimates. It turns out that our approach is rather elementary.
The rest of this paper is organized as follows. In Section 2, as a preliminary, we establish
a crucial pointwise estimate for a class of parabolic operators. As another key preliminary,
we present a construction of the desired weight function in Section 3. Then, in Section 4,
we show a global Carleman estimate for the equation (1.4) with time-independent diffusion
coefficients. Finally, in Section 5, we derive local and global Carleman estimates for (1.4)
with general diffusion coefficients.
2 A pointwise estimate for parabolic operators
LetΩ ⊂ Rn beaboundeddomainwiththeC2-boundary. Foranyfunctionψ ∈ C2([0,T]×Ω),
parameters λ > 0 and µ > 0, and positive number d > |ψ|L∞((0,T)×Ωb), put
b b
eµψ eµψ −eµd
ϕ = , α = , θ = eλα. (2.1)
t(T −t) t(T −t)
In what follows, for a positive integer r, we denote by O(µr) a function of order µr for large
µ (which is independent of λ and T); by O (λr) a function of order λr for fixed µ and for
µ
large λ, which is independent of T, either.
For any (bij) ∈ C1([0,T]×Ω;Rn×n) satisfying
1≤i,j≤n
bij(t,x) = bji(t,x), b∀ (t,x) ∈ [0,T]×Ω, i,j = 1,2,··· ,n, (2.2)
n
b
we have the following pointwise estimate for the parabolic operator ∂ + bij∂ .
t xixj
i,j=1
X
Lemma 2.1 Let u ∈ C2((0,T)×Ω) and v = θu. Then, for any ε > 0, it holds that
n
b 2
θ2 u + biju
t xixj
(cid:12) iX,j=1 (cid:12)
(cid:12) (cid:12)
ε(cid:12) (cid:12)1 n
≥ |v |2 +divV + M + cijv v +Bv2 (2.3)
8λϕ t 2 t xi xj
i,j=1
X
λ
−C ελµ2 + +λµ+λµ+µ2 ϕ|∇v|2.
ε
(cid:16) (cid:17)
Here
n n
M = λ2µ2ϕ2 bijψ ψ v2 −λα v2 − bijv v , (2.4)
xi xj t xi xj
i,j=1 i,j=1
X X
4
V = (V1,··· ,Vn),
n n n

Vj = bijv v −λµϕ 2bijv bkℓψ v −bijψ bkℓv v
 xi t xi xk xℓ xi xk xℓ

 Xi=1 h (cid:16) kX,ℓ=1 kX,ℓ=1 (cid:17) (2.5)
 n n

 − 2λµ2ϕbijvxi bkℓψxkψxℓv −λ3µ3ϕ3bijψxiv2 bkℓψxkψxℓ
k,ℓ=1 k,ℓ=1
X X



 +λ2µϕαtbijψxiv2 ,

 n i

cij = λµϕ µbijbkℓψ ψ −bij(bkℓψ ) −bij bkℓψ +bij
xk xℓ xk xℓ xk xℓ t
kX,ℓ=1(cid:16) (cid:17) (2.6)
n
= λµ2ϕbij bkℓψ ψ +λϕO(µ),
k ℓ
k,ℓ=1
X
and
n n n
2
B ≥ λ3µ4ϕ3 bijψ ψ +λ3µ3ϕ3 bijψ ψ bkℓψ
xi xj xi xj xk
(cid:16)iX,j=1 (cid:17) (cid:16)iX,j=1 kX,ℓ=1 (cid:17)xℓ
n n
1 1
− 2λ2µ2ϕ(ϕ −α ) bijψ ψ − λ2µ2ϕ2 bijψ ψ + λα
t t xi xj 2 t xi xj 2 tt
i,j=1 i,j=1
X X
n n 2 n 2 (2.7)
−λ2µϕα (bijψ ) −2λ2µ2ϕ2 bijψ −2λ2µ4ϕ2 bijψ ψ
t xi xj xixj xi xj
iX,j=1 (cid:16)iX,j=1 (cid:17) (cid:16)iX,j=1 (cid:17)
−Cλµ4ϕ−Cλ2µ4ϕ
n
2
= λ3µ4ϕ3 bijψ ψ +λ3ϕ3O(µ3)+λ2ϕ2O(µ2)+O (λ2).
xi xj µ
(cid:16)iX,j=1 (cid:17)
Proof: We borrow some idea from [3, 8, 9, 10]. It is an easy matter to see that
θu = v −λα v, θu = v −λµϕψ v,
t t t xi xi xi (2.8)
( θuxixj = vxixj −2λµϕψxivxj +λ2µ2ϕ2ψxiψxjv −λµ2ϕψxiψxjv −λµϕψxixjv.
Write
n n
I = bijv +λ2µ2ϕ2 bijψ ψ v −λα v,
1 xixj xi xj t

i,j=1 i,j=1
 X X

 n n


 I2 = vt −2λµϕ bijψxivxj −2λµ2ϕ bijψxiψxjv,

 i,j=1 i,j=1
 X X
n n n
 I = θ u + biju −λµ2ϕ bijψ ψ v +λµϕ bijψ v.
 3 t xixj xi xj xixj

 (cid:16) iX,j=1 (cid:17) iX,j=1 iX,j=1

 5


Then, we see I +I = I , which implies that
1 2 3
I2 ≥ I2 +2I I . (2.9)
3 2 1 2
By virtue of the Cauchy-Schwartz inequality, we have that
n n
2 2
I2 ≤ 2θ2 u + biju +4λ2µ4ϕ2 bijψ ψ v2
3 t xixj xi xj
(cid:16) iX,j=1 (cid:17) (cid:16)iX,j=1 (cid:17) (2.10)
n
2
+ 4λ2µ2ϕ2 bijψ v2.
xixj
(cid:16)iX,j=1 (cid:17)
Now,
n n
I I = bijv +λ2µ2ϕ2 bijψ ψ v −λα v
1 2 xixj xi xj t
(cid:16)iX,j=1 iX,j=1 (cid:17)
n n
× v −2λµϕ bijψ v −2λµ2ϕ bijψ ψ v
t xi xj xi xj
(cid:16) iX,j=1 iX,j=1 (cid:17)
n n n
= bijv v −2λµϕ bijv bkℓψ v
xixj t xixj xk xℓ
i,j=1 i,j=1 k,ℓ=1
X X X (2.11)
n n n
−2λµ2ϕ bijv bkℓψ ψ v +λ2µ2ϕ2 bijψ ψ vv
xixj xk xℓ xi xj t
i,j=1 k,ℓ=1 i,j=1
X X X
n n n
2
−2λ3µ3ϕ3 bijψ ψ v bkℓψ v −2λ3µ4ϕ3 bijψ ψ v2
xi xj xk xℓ xi xj
iX,j=1 kX,ℓ=1 (cid:16)iX,j=1 (cid:17)
n n
−λα vv +2λ2µϕα bijψ v v +2λ2µ2ϕα bijψ ψ v2.
t t t xi xj t xi xj
i,j=1 i,j=1
X X
Denote the terms in the right hand side of (2.11) by J ,i = 1,2,··· ,9, respectively. Then
i
J reads
1
n n n
J = bijv v − bijv v − bijv v
1 xi t xj xj xi t xi txj
i,j=1 i,j=1 i,j=1
X (cid:0) (cid:1) X X (2.12)
n n n n
1 1
= bijv v − bijv v − bijv v + bijv v .
xi t xj xj xi t 2 xi xj t 2 t xi xj
i,j=1 i,j=1 i,j=1 i,j=1
X (cid:0) (cid:1) X X (cid:0) (cid:1) X
6
By (2.2), we see that
n n
J = −2λµϕ bijv bkℓψ v
2 xixj xk xℓ
i,j=1 k,ℓ=1
X X
n n n
2
= − 2λµϕbijv bkℓψ v +2λµ2ϕ bijψ v
xi xk xℓ xi xj
iX,j=1(cid:16) kX,ℓ=1 (cid:17)xj (cid:16)iX,j=1 (cid:17)
n n n
+2λµϕ bijv bkℓψ v +bijv (bkℓψ ) v
xj xi xk xℓ xi xk xj xℓ
iX,j=1h kX,ℓ=1 kX,ℓ=1
n
+bijv bkℓψ v
xi xk xℓxj
kX,ℓ=1 i (2.13)
n n n
= − 2λµϕbijv bkℓψ v −λµϕbijψ bkℓv v
xi xk xℓ xi xk xℓ
iX,j=1(cid:16) kX,ℓ=1 kX,ℓ=1 (cid:17)xj
n n n
2
+2λµ2ϕ bijψ v + 2λµϕ bijv bkℓψ v
xi xj xj xi xk xℓ
(cid:16)iX,j=1 (cid:17) iX,j=1 kX,ℓ=1
n n n n
+ 2λµϕ bijv (bkℓψ ) v −λµ2ϕ bijv v bkℓψ ψ
xi xk xj xℓ xi xj xk xℓ
i,j=1 k,ℓ=1 i,j=1 k,ℓ=1
X X X X
n n n n
− λµϕ v v (bkℓψ ) − λµϕ bijv v bkℓψ .
xi xj xk xℓ xℓ xi xj xk
i,j=1 k,ℓ=1 i,j=1k,ℓ=1
X X X X
Also,
n n
J = −2λµ2ϕ bijv bkℓψ ψ v
3 xixj xk xℓ
i,j=1 k,ℓ=1
X X
n n n n
= − 2λµ2ϕbijv bkℓψ ψ v +2λµ2ϕ bijv v bkℓψ ψ (2.14)
xi xk xℓ xi xj xk xℓ
(cid:16)iX,j=1 kX,ℓ=1 (cid:17)xj iX,j=1 kX,ℓ=1
n n n n
+2λµ2ϕ bijv bkℓψ ψ v +2λµ2 bijv v ϕ bkℓψ ψ .
iX,j=1 xj xikX,ℓ=1 xk xℓ iX,j=1 xi (cid:16) kX,ℓ=1 xk xℓ(cid:17)xj
Further,
n
J = λ2µ2ϕ2 bijψ ψ vv
4 xi xj t
i,j=1
X
n n
1
= λ2µ2ϕ2 bijψ ψ v2 −λ2µ2ϕϕ bijψ ψ v2 (2.15)
2 xi xj t t xi xj
(cid:16) iX,j=1 (cid:17) iX,j=1
n n
1
− λ2µ2ϕ2 bijψ ψ v2 −λ2µ2ϕ2 bijψ ψ v2.
2 t xi xj xi xjt
i,j=1 i,j=1
X X
7
The J satisfies
5
n n
J = −2λ3µ3ϕ3 bijψ ψ v bkℓψ v
5 xi xj xk xℓ
i,j=1 k,ℓ=1
X X
n n n
2
= − λ3µ3ϕ3bijψ ψ bkℓψ v2 +3λ3µ4ϕ3 bijψ ψ v2 (2.16)
xi xj xk xi xj
iX,j=1(cid:16) kX,ℓ=1 (cid:17)xℓ (cid:16)iX,j=1 (cid:17)
n n
+λ3µ3ϕ3 bijψ ψ bkℓψ v2.
xi xj xk
(cid:16)iX,j=1 kX,ℓ=1 (cid:17)xℓ
It is easy to see that
n
2
J = −2λ3µ4ϕ3 bijψ ψ v2, (2.17)
6 xi xj
(cid:16)iX,j=1 (cid:17)
1 1
J = −λα vv = − (λα v2) + λα v2, (2.18)
7 t t t t tt
2 2
and
n
J = 2λ2µ2ϕα bijψ ψ v2. (2.19)
9 t xi xj
i,j=1
X
Finally,
n
J = 2λ2µϕα bijψ v v
8 t xi xj
i,j=1
X
n n n
= λ2µϕα bijψ v2 −(λ2µϕ) α bijψ v2 −λ2µϕα bijψ v2
t xi xj t xi txj xi
(cid:16) iX,j=1 (cid:17)xj iX,j=1 iX,j=1
n
−λ2µϕα (bijψ ) v2
t xi xj
i,j=1
X
n n n
= λ2µϕα bijψ v2 −λ2µ2ϕα bijψ ψ v2 −λ2µ2ϕϕ bijψ ψ v2
t xi t xi xj t xi xj
(cid:16) iX,j=1 (cid:17)xj iX,j=1 iX,j=1
n
−λ2µϕα (bijψ ) v2.
t xi xj
i,j=1
X
(2.20)
Noting that for any a,b ∈ R, |a+b|2 ≥ 1a2 −b2, we find that
2
n n
1 1 2 2
I2 ≥ |v |2 −4λ2µ2ϕ2 bijψ v −4λ2µ4ϕ2 bijψ ψ v . (2.21)
2 2 4 t xi xj xi xj
(cid:12)iX,j=1 (cid:12) (cid:12)iX,j=1 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
8
Consequently, we have for some ε > 0 (which is fixed but very small), that
n
ε ε
I2 ≥ |v |2 −Cελµ2ϕ |bij|2 |∇ψ|2|∇v|2
2λϕ 2 4λϕ t C([0,T]×Ωb)
i,j=1
X
n
− Cλµ4ϕ |bij|2 |∇ψ|4|v|2 (2.22)
b
C([0,T]×Ω)
i,j=1
X
ε
≥ |v |2 −Cελµ2ϕ|∇v|2−Cλµ4ϕ|v|2.
4λϕ t
Further,
n n
ε C
bijv v ≤ |v |2 + λϕ |∇bij|2 |∇v|2
xj xi t 8λϕ t ε C([0,T]×Ωb)
(cid:12)iX,j=1 (cid:12) iX,j=1 (2.23)
(cid:12) (cid:12)
(cid:12) (cid:12) ε C
≤ |v |2 + λϕ|∇v|2.
8λϕ t ε
Also,
n n
2λµ2ϕ bijbkℓψ ψ +bij(bkℓψ ψ ) +µbijbkℓψ ψ vv
xj xk xℓ xk xℓ xj xk xℓ xi
(cid:12) iX,j=1kX,ℓ=1(cid:16) (cid:17) (cid:12)
(cid:12) (cid:12)
(cid:12) n n (cid:12) (2.24)
≤ C µ2ϕ |bij| |∇v|2+λ2µ4ϕ |∇bij| |v|2
b b
C([0,T]×Ω) C([0,T]×Ω)
(cid:16) iX,j=1 iX,j=1 (cid:17)
≤ Cµ2ϕ|∇v|2+Cλ2µ4ϕ|v|2,
and
n n n n
2λµϕ bijv bkℓψ v +2λµϕ bijv (bkℓψ ) v
xj xi xk xℓ xi xk xj xℓ
(cid:12) iX,j=1 kX,ℓ=1 iX,j=1 kX,ℓ=1 (cid:12) (2.25)
(cid:12) (cid:12)
(cid:12) n n (cid:12)
≤ Cλµϕ |∇bij| |∇v|2 +Cλµϕ |bij| |∇v|2 ≤ Cλµϕ|∇v|2.
b b
C([0,T]×Ω) C([0,T]×Ω)
i,j=1 i,j=1
X X
Combining (2.12)– (2.19), (2.9), (2.11), (2.21)–(2.25), we obtain the desired inequality
(2.3). This completes the proof of Lemma 2.1.
3 Construction of the weight function
In this section, we present a construction of the weight function that will be used to establish
the desired Carleman estimates. We have the following result.
Lemma 3.1 Assume that (1.1)–(1.3) and (1.6) hold. Let ω ⊂ ω∩Ω (i = 1,2) be nonempty
i i
open subsets such that ω ⊂ ω∩Ω . Then, there exist a function φ˜∈ C2(Ω ), a positive integer
i i 2
L and functions φℓ ∈ C2([t ,t ]×Ω ) with t = ℓT, ℓ = 0,1,··· ,L, such that
ℓ ℓ+1 1 ℓ L
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(1) φℓ > 0 in [t ,t ]×Ω and φ˜> 0 in Ω ;
ℓ ℓ+1 1 2
(2) φℓ = 0 on [t ,t ]×Γ and φ˜= 0 on Γ ;
ℓ ℓ+1 1 2
(3) |∇φℓ| > 0 in [t ,t ]×Ω \ω and |∇φ˜| > 0 in Ω \ω ;
ℓ ℓ+1 1 1 2 2
n n
(4) aijφℓ φℓ = a˜ijφ˜ φ˜ on [t ,t ]×S.
xi xj xi xj ℓ ℓ+1
i,j=1 i,j=1
X X
Proof: By [4, Lemma 1.1, p.4], there is a function φ˜∈ C2(Ω ) such that
2
φ˜> 0 in Ω ,
2
 φ˜= 0 on Γ , (3.1)
2


 |∇φ˜| > 0 in Ω \ω .
2 2

Let us extend a˜ij (1 ≤ i,j ≤ n) tobe a C2 function on [0,T]×(O (S)∪Ω ) and denote by a˜ij
ε 2
the extended function, where ε is a sufficiently small positive number such that (a˜ij)
1≤i,j≤n
is still a uniformly positive definite matrix on [0,T]×(O (S)∪Ω ). Further, we extend the
ε 2
above φ˜to be a C2 function on O (S)∪Ω and denote by φ˜itself the extension. Since ∂φ˜ > 0
ε 2 ∂ν
on S (Recall that ν(x) = −ν˜(x) for x ∈ S), there is an ε ∈ (0,ε] so that
1
n
a˜ijφ˜ φ˜ > 0, for all (t,x) ∈ [0,T]×O (S).
xi xj ε1
i,j=1
X
Let us choose a (time-independent) ξˆ∈ C2(Ω ) such that ξˆ> 0 in Ω and ξˆ= 0 on Γ . It
1 1 1
follows from ∂ξˆ < 0 on S that there is an ε ∈ (0,ε ] such that
∂ν 2 1
n
aijξˆ ξˆ > 0, for all (t,x) ∈ [0,T]×O (S).
xi xj ε2
i,j=1
X
Then, we see that
n a˜ijφ˜ φ˜ 1
ς =△ i,j=1 xi xj 2 > 0 in [0,T]×(Ω ∩O (S))
n aijξˆ ξˆ 1 ε2
(cid:16)Pi,j=1 xi xj (cid:17)
and ς ∈ C2([0,T]×(Ω ∩PO (S))). We extend ς to be a positive C2 function on [0,T]×Ω ,
1 ε2 1
and still denote by ς the extension.
Put ξ = ςξˆ. Then, ξ ∈ C2([0,T]× Ω ), ξ > 0 in [0,T]× Ω and ξ = 0 on [0,T]× Γ .
1 1 1
Further,
n n n
aijξ ξ = ς2 aijξˆ ξˆ = a˜ijφ˜ φ˜ , for each (t,x) ∈ [0,T]×S. (3.2)
xi xj xi xj xi xj
i,j=1 i,j=1 i,j=1
X X X
In what follows, we will construct the desired function φℓ based on ξ. The method is very
similar to that of [4, Lemma 1.1, p.4 and pp. 20–22]. However, we give the detail here for
the sake of completeness.
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