Analyticity for some degenerate evolution equations defined on domains with corners

0.35 MB·English

by  Angela A. Albanese

ANALYTICITY FOR SOME DEGENERATE EVOLUTION
EQUATIONS DEFINED ON DOMAINS WITH CORNERS
ANGELA A.ALBANESE, ELISABETTA M. MANGINO
3
1 Abstract. Westudytheanalyticityof thesemigroupsgenerated bysomeclasses
0
ofdegeneratesecondorderdifferentialoperatorsinthespaceofcontinuousfunction
2
on a domain with corners. These semigroups arise from the theory of dynamics of

n populations.
a
J

3
2

] 1. Introduction
A
In this paper we deal with the class of degenerate second order elliptic differential
F
. operators
h
t d
a
m L = Γ(x) [γi(xi)xi∂x2i +bi(x)∂xi], x ∈ Qd = [0,M]d, (1.1)
[ Xi=1

1 where M > 0, Γ, bi and γi, for i = 1,...,d, are continuous functions on Qd and on
v [0,M] respectively and b = (b ,...,b ) is an inward pointing drift. The operator (1.1)
1 d
9
arises in the theory of Fleming–Viot processes, namely measure–valued processes that
4
canbeviewed as diffusionapproximations of empirical processesassociated withsome
4
5 classes of discrete time Markov chains in population genetics. We refer to [21, 22, 25]
1. for more details on the topic. Recent applications of Fleming-Viot processes to the
0 studyofthevolatility-stabilized markets canbefoundin[32]. Fromtheanalytic point
3 of view, the interest in the operator (1.1) relies on the fact that it is of degenerate
1
type and its domain presents edges and corners, hence, the classical techniques for the
:
v
study of (parabolic) elliptic operators on smooth domains cannot be applied.
i
X In the one-dimensional case, the study of such type of degenerate (parabolic) el-
r liptic problems on C([0,1]) started in the fifties with the papers by Feller [23, 24],
a
where it is pointed out that the behaviour on the boundary of the diffusion process
associated with the degenerate operator constitutes one of its main characteristics.
The subsequent work of Clément and Timmermans [15] clarified which conditions on
the coefficients of the operator (1.1) guarantee the generation of a C –semigroup in
0
C([0,1]). The problem of the regularity of the generated semigroup in C([0,1]) has
been considered by several authors, [6, 10, 9, 29, 2]. In particular, Metafune [29] es-
tablished theanalyticity of thesemigroup undersuitable conditions on thecoefficients
of the operator, obtaining, among other results, the analyticity of the semigroup gen-
erated by x(1 x)D2 on C([0,1]), which was a problem left open for a long time. We
−
refer to [11] for a survey on this topic.
Key words and phrases. Degenerate elliptic second orderoperator, domain with corners, analytic
C0–semigroup, space of continuousfunctions.
Mathematics Subject Classification 2010: Primary 35K65, 35B65, 47D07; Secondary 60J35.
1
2 ANGELAA.ALBANESE,ELISABETTAM.MANGINO
Thelatterresultwasextendedtothemultidimensionalcasein[4],wheretheauthors
proved the analyticity of the semigroup generated by the operator
d
1
Au(x) = x (δ x )∂2 u(x)
2 i ij − j xixj
i,j=1
X
on C(Sd), where Sd is the d-dimensional canonical simplex. On this topic we refer
to the papers [1, 2, 3, 4, 14, 12, 20, 33, 34, 35, 36] and the references quoted therein
(in particular, see the introduction of [4] for a brief survey of the main results on this
operator).
In [13] Cerrai and Clément established Schauder estimates for (1.1) under suitable
Hölder continuity hypothesis on its coefficients. Analogous estimates, but with differ-
ent tecniques, where established in [8] (see also [7]) for the same operator defined on
the orthant Rd and in [18, 19] for similar operators defined on domains with corners.
+
The aim of this paper is to present some results about generation, sectoriality and
gradient estimates for the resolvent of a suitable realization of (1.1) in C(Qd). To
this end, we start with the analysis in the particular case that the functions b are
i
costant and Γ = 1, first in the one-dimensional case and then, via a tensor product
argument, in the multi-dimensional setting. Much attention is paid to the costants
appearing in the analyticity and gradient estimates, showing their uniformity for b
i
belonging to an interval [0,B]. These results strongly rely on estimates proved in
[5]. As a consequence, we can treat the case of non-costant drift with a perturbation
argument under theassumption that thereexists δ > 0and C > 0 suchthat, forevery
i= 1,...,d and x, x′ Qd with x < δ and x′ = 0, we have
∈ i i
b (x) b (x′) C√x ,
i i i
| − | ≤
FinallywetreatthecasethatΓisnotacostantfunction,byapplyinga"freezingco-
efficients" proof. An important role in this argument will be played by the uniformity
of the costants in the resolvent estimates.
As aby-product of theprevious results we obtain analogous results forthe operator
d
Γ(x) [γ (x )x (1 x )∂2 +b (x)∂ ], x [0,1]d.
i i i − i xi i xi ∈
i=1
X
This will be the starting point for a forthcoming paper on the analyticity of Fleming-
Viot type operators defined on the canonical simplex.
1.1. Notation. The function spaces considered in this paper consist of complex–
valued functions.
Let K Rd be a compact set. For n N we denote by Cn(K) the space of all
⊆ ∈
n–times continuously differentiable functions u on K such that lim Dαu(x) exists
x→x0
and is finite for all α n and x ∂K. In particular, C(K) denotes the space of
0
| | ≤ ∈
all continuous functions u on K. The norm on C(K) is the supremum norm and is
denotedby . Thenorm onCn(K)isdefinedby u = Dαu .
kk∞ kkn,∞ k kn,∞ |α|≤nk k∞
Moreover, we denote by C([0, ]) the Banach space of continuous functions on
∞ P
[0, [ converging at infinity, endowed with the supremum norm . Analogously,
∞
for∞every n N, Cn([0, ]) stands for the space of functions|u|·|| C([0, ]) with
∈ ∞ ∈ ∞
derivatives up to order n that have finite limits at . Finally Cn([0, [) denotes the
∞ c ∞
subspace of Cn([0, [) of functions with compact support and C ([0, [) denotes the
0
∞ ∞
space of continuous functions on [0, [ vanishing at .
∞ ∞
ANALYTICITYFORSOMEDEGENERATEEVOLUTIONEQUATIONS 3
For easy reading, in some cases we will adopt the notation ϕ(x)u to stilldenote
∞
k k
sup ϕ(x)u(x).
x∈K| |
For other undefined notation and results on the theory of semigroups we refer to
[17, 28, 31].
In the present paper we will use some results about injective tensor products of
Banach spaces. We refer to [26, 27, 37, 30] for definitions and basic results in this
topic and for related applications.
2. Auxiliary Results
2.1. The one–dimensional case. Let M,B R with M,B > 0 and let γ
∈ ∈
C([0,M]) be a strictly positive function. Set γ := min γ(x) > 0. Let b [0,B]
0 x∈[0,M]
∈
and consider the one–dimensional second order differential operator
Lγ,bu(x) = γ(x)xu′′(x)+bu′(x), x [0,M]. (2.1)
∈
According to [29, Proposition 3.1] (see also [11]), we define the domain of Lγ,b in the
following way: u D(Lγ,b) if, and only if, u C([0,M]) C2(]0,M]), u′(M) = 0 and
∈ ∈ ∩
lim Lγ,bu(x) =0 if b = 0, (2.2)
x→0+
u C1([0,δ]) and lim xu′′(x) = 0 if b > 0. (2.3)
∈ x→0+
Itis known from [29,10, 11, 15] that the operator Lγ,b with domain D(Lγ,b) generates
a bounded analytic C –semigroup (T(t)) of positive contractions and angle π/2 on
0 t≥0
C([0,M]).
We are here interested in proving estimates for the norm of the resolvent operators
of Lγ,b and of their gradient with constants which depend only on B. In order to do
this we need the following fact.
Remark 2.1. Let B, γ > 0. For every b [0,B] and γ R, γ γ consider the
0 0
∈ ∈ ≥
one–dimensional second order differential operator
Gγ,bu(x) =γu′′(x)+bu′(x), x [0, ), (2.4)
∈ ∞
with domain D := u C2([0, ]) : u′(0) = 0 and γ γ , b [0,B]. It is well
0
{ ∈ ∞ } ≥ ∈
known that the operator (Gγ,b,D) generates a bounded analytic semigroup of angle
π/2 in C([0, ]), see, e.g., [17, Theorem VI.4.3]. In particular, (Gγ,b,D) satisfies the
∞
following properties:
There exists c , c ,R > 0 depending only on B and on γ such that, for every λ C
1 2 0
∈
with Reλ > R and u C([0, ]),
∈ ∞
c
R(λ,Gγ,b) 1 (2.5)
k k ≤ λ
| |
c
(R(λ,Gγ,b)u)′ 2 u . (2.6)
∞ ∞
k k ≤ λ k k
| |
p
The proof is as follows.
Set G := G1,0. Then, for every λ = λ eiθ ( ,0] with θ < π, we have
| | 6∈ −∞ | |
1 ∞
R(λ,G)u = e−µ|x−s|u(s)ds+ce−µx, u C([0, ]), (2.7)
2µ ∈ ∞
Z0
4 A.A.Albanese,E. M.Mangino
where µ2 = λ with Reµ > 0 andc = 1 ∞e−µsu(s)ds, see, e.g., [17, Theorem VI.4.3,
2µ 0
Theorem 4.2]. So, from (2.7) it follows that
R
3
R(λ,G) (2.8)
k k ≤ 2λ cos(θ/2)
| |
1
(R(λ,G)u)′ u , (2.9)
∞ ∞
k k ≤ λ cos(θ/2)k k
| |
for every λ = λ eiθ ( ,0] with θ < π apnd u C([0, ]).
| | 6∈ −∞ | | ∈ ∞
We now consider the operator Gγ,0 with γ γ and observe that
0
≥
R(λ,Gγ,0) = γ−1R(λ/γ,G), λ C ( ,0].
∈ \ −∞
This equality implies via (2.8) and (2.9) that
3 3
R(λ,Gγ,0) γ−1 = (2.10)
k k ≤ 2λ/γ cos(θ/2) 2λ cos(θ/2)
| | | |
1
(R(λ,Gγ,0)u)′ γ−1 u (2.11)
∞ ∞
k k ≤ λ /γcos(θ/2)k k
| |
1
u p,
∞
≤ γ λ cos(θ/2)k k
0
| |
for every λ = λ eiθ ( p,0] with θ < π and u C([0, ]).
| | 6∈ −∞ | | ∈ ∞
If we set Hbu = bu′ for b [0,B] and u C1([0, ]), then by (2.11) we obtain, for
∈ ∈ ∞
every λ = λ eiθ ( ,0] with θ <π and u C([0, ]), that
| | 6∈ −∞ | | ∈ ∞
B
HbR(λ,Gγ,0)u u . (2.12)
∞ ∞
k k ≤ γ λ cos(θ/2)k k
0
| |
By(2.12),foreveryλ = λ eiθ with θ < π/p2and λ > R =8B2/γ andb [0,B],the
0
| | | | | | ∈
operator HbR(λ,Gγ,0) has norm < 1/2 and sothe operator S (λ) := I HbR(λ,Gγ,0)
b
−
is invertible with inverse
∞
(S (λ))−1 = [HbR(λ,Gγ,0)]n (2.13)
b
n=1
X
so that (S (λ))−1 2. This estimate, combined with the identity λ Gγ,b =
b
k k ≤ −
λ Gγ,0 Hb = [I HbR(λ,Gγ,0)](λ Gγ,0) implies, for every λ = λ eiθ with
− − − − | |
θ < π/2 and λ > R = 8B2/γ and b [0,B], that
0
| | | | ∈
R(λ,Gγ,b) = R(λ,Gγ,0)(S (λ))−1. (2.14)
b
So, by (2.10), (2.11) and (2.14) we obtain, for every λ = λ eiθ with θ < π/2,
| | | |
λ > R = 8B2/γ , u C([0, ]) and b [0,B], that
0
| | ∈ ∞ ∈
3
R(λ,Gγ,b) (2.15)
k k ≤ λ cos(θ/2)
| |
2
(R(λ,Gγ,b)u)′ u . (2.16)
∞ ∞
k k ≤ γ λ cos(θ/2)k k
0
| |
(cid:3)
p
Proposition 2.2. Let B,M > 0 and let γ C([0,M]) be a strictly positive function
∈
with γ := min γ(x). Then, for every b [0,B], the following properties hold.
0 x∈[0,M]
∈
ANALYTICITYFORSOMEDEGENERATEEVOLUTIONEQUATIONS 5
(1) There exist d = d (B,γ),R = R(B,γ) >0 such that, for every u C([0,M])
0 0
and for every λ C with Reλ > R, we have ∈
∈
u
R(λ,Lγ,b)u d || ||∞, (2.17)
∞ 0
|| || ≤ λ
| |
u
√x(R(λ,Lγ,b)u)′ d || ||∞. (2.18)
∞ 0
|| || ≤ λ
| |
Moreover, limx→0+√x(R(λ,Lγ,b)u)′(x) = 0. Ipn particular, √x(R(λ,Lγ,b)u)′
extends continuously to [0,M].
(2) If (T(t)) is the semigroup generated by (Lγ,b,D(Lγ,b)), then there exist K =
t≥0
K(B,γ),α = α(B,γ) >0 such that, for every u C([0,M]), we have
∈
tLγ,bT(t) Keαt, t 0 (2.19)
|| || ≤ ≥
Keαt
√x(T(t)u)′ u , 0< t <R−1, (2.20)
∞ ∞
|| || ≤ √t || ||
√x(T(t)u)′ Keαt u , t R−1, (2.21)
∞ ∞
|| || ≤ || || ≥
where R is the same constant which appears in part (1).
Moreover, lim √x(T(t)u)′(x) = 0 if t > 0. In particular, √x(T(t)u)′
x→0+
extends continuously to [0,M] if t > 0.
Proof. W.l.o.g. we may assume M = 1.
(1) For each n N and i 1,...,n 1 set Ii = i−1,i+1 and let ϕi n−1
∈ ∈ { − } n n n { n}i=1 ⊂
C∞(R) such that n−1(ϕi)2 1 on [0,1], supp(ϕi) i−1, i+1 for i= 2,...,n 2,
i=1 n ≡ n ⊂ (cid:2) n n (cid:3) −
supp(ϕ1) , 2 and supp(ϕn−1) n−2, . Observe that if v = n−1v ϕi
n ⊂ −∞Pn n ⊂ n ∞ (cid:2) (cid:3) i=1 i n
with v C([0,1]) for i = 1,...,n, then
i
∈ (cid:3) (cid:3) (cid:2) (cid:2) P
v 3 sup v . (2.22)
∞ i ∞
|| || ≤ || ||
i=1,...,n−1
For every i 1,...,n 2 we consider the operators
∈ { − }
i
Liu= γ xu′′(x)+bu′(x), u D(Li),
n n ∈ n
(cid:18) (cid:19)
with domain D(Li) defined as follows: if b = 0
n
D(Li) = u C([0, ]) C2(]0, [) lim Liu(x) = 0; lim Liu(x) = 0 ,
n { ∈ ∞ ∩ ∞ | x→0+ n x→+∞ n }
if b > 0
D(Li)= u C1([0, [) C2(]0, [) C([0, ]) lim xu′′(x) = 0, lim Liu(x) = 0 .
n { ∈ ∞ ∩ ∞ ∩ ∞ | x→0+ x→+∞ n }
For i= n 1 we consider the operator
−
n 1
Ln−1u= γ − u′′(x)+bu′(x), u D(Ln−1),
n n ∈ n
(cid:18) (cid:19)
with domain
D(Ln−1)= u C2([ ,1]) u′(1) = 0 .
n { ∈ −∞ | }
By[5, Corollary 4.2] andRemark 2.1, there existsd = d (B,γ ) > 0,R = R(B,γ ) >
1 1 0 0
0 such that, for every λ C with Reλ > R, we have
∈
d
R(λ,Li) 1, n N, i = 1,...,n 1. (2.23)
|| n || ≤ λ ∈ −
| |
6 A.A.Albanese,E. M.Mangino
Fix λ C, with Reλ > R. For each n N let S (λ): C([0,1]) C([0,1]) be the
n
∈ ∈ →
operator defined by
n−1
S (λ)f = ϕiR(λ,Li)(ϕi f), f C([0,1]).
n n n n ∈
i=1
X
By (2.22) and (2.23) we get, for every n N, that
∈
3d
S (λ)f 3 sup R(λ,Li)(ϕif) 1 f , f C([0,1]).
|| n ||∞ ≤ || n n ||≤ λ || ||∞ ∈
i=1,...,n−1
| |
Since R(λ,Li)(ϕif) D(Li) for every i = 1,...,n 1 and f C([0,1]), ϕn−1 0
n n ∈ n − ∈ n ≡
and ϕn−1 1 in an neighbourhood of 0 and in an neighbourhood of 1 respectively, we
n ≡
have ϕiR(λ,Li)(ϕi f) D(Lγ,b) and so we can consider (λ Lγ,b)(S (λ)f) for every
n n n ∈ − n
f C([0,1]). In particular, a straightforward calculation gives
∈
n−1
(λ Lγ,b)(S (λ)f) = f + ϕi(Li Lγ,b)R(λ,Li)(ϕif)
− n n n− n n
i=1
X
n−1 n−1
Lγ,b(ϕi)R(λ,Li)(ϕif) 2γ(x) (ϕi)′x(R(λ,Li)(ϕif))′
− n n n − n n n
i=1 i=1
X X
= f +Cn(λ)f +Cn(λ)f +Cn(λ)f, f C([0,1]), n N.
1 2 3 ∈ ∈
Applying again (2.22) and (2.23) we obtain
d
Cn(λ)f 3 f 1 sup Lγ,b(ϕi) , f C([0,1]), n N. (2.24)
|| 2 ||∞ ≤ || ||∞ λ || n ||∞ ∈ ∈
i=1,...,n−1
| |
On the other hand, by [5, Proposition 5.1(2)], Remark 2.1 and (2.22), there exists
d = d (B,γ ) > 0 such that
2 2 0
f
Cn(λ)f 3d γ sup (ϕi)′ || ||∞, f C([0,1]), n N. (2.25)
|| 3 ||∞ ≤ 2|| ||∞i=1,...,n−1|| n ||∞ λ ∈ ∈
| |
In order to estimate ||C1n(λ)||, we observe, for epvery n∈ N, that
ϕi(Li Lγ,b)R(λ,Li)(ϕif)=
n n− n n
ϕi γ i γ(x) x(R(λ,Li)(ϕif))′′(x) if i = 1,...,n 2,
= n n − n n −
ϕn−1 γ n−1 γ(x)x (R(λ,Ln−1)(ϕn−1f))′′(x) if i = n 1.
(cid:26) n(cid:2) (cid:0) (cid:1) n −(cid:3) n n −
Fixed ε > 0, we now(cid:2) c(cid:0)hoos(cid:1)e δ > 0 so(cid:3) that γ(x) γ(y) < ε if x y < δ and that
| − | | − |
γ(x) γ(y) +Γ 1 x < ε if x,y [1 δ,1], where Γ := max γ(x). If we
0 0 x∈[0,1]
| − | | − | ∈ −
take n N such that 2 < δ, then we have that γ(x) γ(i) < ε if x Ii and
∈ n | − n | ∈ n
i 1,...,n 2 and that γ n−1 γ(x)x < ε if x In−1. So, it follows from [5,
∈ { − } | n − | ∈ n
Proposition 5.1(2)], Remark 2.1 and (2.22) that
(cid:0) (cid:1)
1 d
Cn(λ)f 3ε 1+d +B 1 f , f C([0,1]). (2.26)
|| 1 ||∞ ≤ γ 1 λ || ||∞ ∈
0 (cid:18) | |(cid:19)
Therefore, combining (2.24), (2.25) and (2.26), we obtain
d
Cn(λ) + Cn(λ) + Cn(λ) 1 sup Lγ,b(ϕi) +
|| 1 || || 2 || || 3 || ≤ λ || n ||∞
i=1,...,n−1
| |
1 1 d
+3d γ sup (ϕi)′ +3ε 1+d +B 1 .
2|| ||∞i=1,...,n−1|| n ||∞ |λ| γ0 (cid:18) 1 |λ|(cid:19)
p
ANALYTICITYFORSOMEDEGENERATEEVOLUTIONEQUATIONS 7
Now, let ε > 0 be small enough that 3ε1+d1 < 1/4 and R′ > R such that
γ0
d 1 d 1
1 sup L(ϕi) +3d γ sup (ϕi)′ +B 1 <
λ i=1,...,n−1|| n ||∞ 2|| ||∞i=1,...,n−1|| n ||∞ λ λ 4
| | | | | |
for every λ C [0,+ ) with λ R′ (in particular, withpReλ > R′). So, R′ =
∈ \ ∞ | | ≥
R′(γ ,B). Since
0
Cn(λ)+Cn(λ)+Cn(λ) < 1/2,
k 1 2 3 k
the operator B(λ) = (λ Lγ,b)S (λ) is invertible in C([0,1]) with (B(λ))−1 2.
n
So, for every λ C with−Reλ > R′, we have R(λ,Lγ,b)= S (λ)B(λ)−k1 and k ≤
n
∈
6d
R(λ,Lγ,b) 1. (2.27)
k k ≤ λ
| |
Fixed λ C with Reλ > R′, it follows via [5, Proposition 5.1(2)], Remark 2.1 and
∈
(2.22) that, for every u C([0,1]), we have
∈
√x(R(λ,Lγ,b)u)′ = √x(S (λ)B(λ)−1u)′
n
|| || || ||
n−1 n−1
(ϕi)′R(λ,Li)(ϕi B(λ)−1u) + ϕi√x[R(λ,Li)(ϕi B(λ)−1u)]′
≤ || n n n ||∞ || n n n ||∞
i=1 i=1
X X
18nd sup (ϕi)′ 3nd sup ϕi
1 i=1,...,n−1|| n ||∞ + 2 i=1,...,n−1|| n||∞ u .
∞
≤ λ λ !|| ||
| | | |
If we choose d = max 18nd sup (ϕi )′ +p3nd sup ϕi ,6d ,
0 { 1 i=1,...,n|| n ||∞ 2 i=1,...,n−1|| n||∞ 1}
then the thesis now follows. We also have
n−1
lim √x(R(λ,Lγ,b)u)′(x) = lim √x (ϕi)′R(λ,Li)(ϕiB(λ)−1u)+
n n n
x→0+ x→0
i=1
X
n−1
+ (ϕi)′R(λ,Li)(ϕiB(λ)−1u)′ = 0,
n n n
!
i=1
X
by [5, Propositions 5.1(2) and 5.2] and Remark 2.1.
(2) Since the resolvent operators of Lγ,b satisfy the part (1) of this proposition, we
can apply [28, Proposition 2.1.11] to conclude that, for every λ C with λ = R and
∈ 6
arg(λ R) < π arctand , we have
0
| − | −
d
R(λ,Lγ,b) 0 ,
k k ≤ λ R
| − |
e
where d = 2d (1/(4d2)+1)−1/2. Then there exist K = K(B,γ)> 0 and α= α(B,γ)
0 0 0
such that
e t(Lγ,b α)T(t) Keαt, t 0
|| − || ≤ ≥
(see, f.i., [28, Proposition 2.1.1] and also the estimates in the relative proof). Since
(T(t)) contractive, it follows that
t≥0
tLT(t) (K +1)eαt, t 0.
|| || ≤ ≥
Finally, if u D(Lγ,b), then part (1) of this proposition ensures that, for every λ R,
∈ ∈
λ > R, we have
d
√xu′ 0 λu Lγ,bu .
∞ ∞
|| || ≤ √λ|| − ||
8 A.A.Albanese,E. M.Mangino
Asthesemigroup(T(t)) isanalyticandhence,T(t)f D(Lγ,b)foreveryf C[0,1]
t≥0
∈ ∈
and t > 0, it follows that
d d Keαt
√x(T(t)f)′ 0 λT(t)f Lγ,bT(t)f d √λ+ 0 f ,
∞ ∞ 0 ∞
|| || ≤ √λ|| − || ≤ t√λ || ||
(cid:18) (cid:19)
foreveryf C[0,1]andt > 0. So,ifwechooseλ = t−1 foreveryt < R−1 andλ = R+
∈
1forevery t R−1, then we gettheassertion. Moreover, lim √x(T(t)f(x))′ = 0,
x→0+
≥
for t > 0. Indeed, this property holds for every u D(Lγ,b) by part (1) of this
proposition. ∈ (cid:3)
Remark 2.3. Since the operator (Lγ,b,D(Lγ,b)) generates a bounded analytic C –
0
semigroup (T(t)) of positive contractions and angle π/2 on C([0,M]), for every
t≥0
θ (π/2,π) there exists M = M (θ)> 0 such that R(λ,Lγ,b) M /λ for all λ
0 0 0
∈ k k ≤ | | ∈
C 0 with arg(λ) < θ. Moreover, there exists M > 0 such that tLγ,bT(t) M
1 1
\{ } | | k k ≤
for every t 0. But, the constants M and M can depend on the functions b and γ.
0 1
≥
Corollary 2.4. Let B,M > 0 and let γ C([0,M]) be a strictly positive function.
∈
Then there exist ε > 0, C > 0 and D > 0 depending only on B and γ such that, for
every 0 < ε< ε, b [0,M] and u D(Lγ,b), we have
∈ ∈
C
√xu′ u +Dε Lγ,bu .
∞ ∞ ∞
k k ≤ εk k k k
Proof. Fixu D(Lγ,b)andλ CwithReλ > R,whereRisthecostantwhichappears
∈ ∈
in Proposition 2.2(1). Then there exists v C([0,M]) such that R(λ,Lγ,b)v = u and
∈
hence, by Propositon 2.2, we have that
d
√xu′ = √x(R(λ,Lγ,b)v)′ 0 λu Lγ,bu
∞ ∞ ∞
k k k k ≤ λ k − k
| |
1 p
d λ u + Lγ,bu , (2.28)
0 ∞ ∞
≤ | |k k λ k k !
p | |
p
where d depends only on B and γ. Now, the assertion follows from (2.28) and from
0
Proposition 2.2(1) by choosing ε= R−1 and, for 0< ε < ε, λ = 1/ε. (cid:3)
| |
p
Set C2([0,M]) = u C2([0,M]) : u′(M) = 0 . Then
⋄ { ∈ }
Proposition 2.5. Let b 0 and let γ C([0,M]) be a strictly positive function.
≥ ∈
Then the space C2([0,M]) is a core for the operator Lγ,b with domain D(Lγ,b) defined
⋄
according to (2.2) if b = 0 or to (2.3) if b > 0.
Proof. The assertion follows with the same argument of Proposition 3.1 in [5], with
some minor chages. (cid:3)
Remark 2.6. The inclusion (D(Lγ,b), )֒ C([0,M]) is compact (here,
k kLγ,b → k kLγ,b
denotes the graph norm), see [29, Theorem 4.1] or [10, Lemma 3.2]. So (Lγ,b,D(Lγ,b))
hascompactresolvent,[17,Proposition4.25]. Since(Lγ,b,D(Lγ,b))generatesabounded
analytic C –semigroup (T(t)) on C([0,M]) (and hence, a norm continuous C –
0 t≥0 0
semigroup) and has compact resolvent, (T(t)) is also compact, [17, Theorem 4.29].
t≥0
ANALYTICITYFORSOMEDEGENERATEEVOLUTIONEQUATIONS 9
3. The d-dimensional case with constant drift term
Set Qd = [0,M]d and, for each i = 1,...,d, define ∂(Qd) := x Qd x = 0
i i
{ ∈ | }
and ∂(Qd)i := x Qd x = M . Let B > 0 and fix b = (b ,b ,...,b ) [0,B]d
i 1 2 d
{ ∈ | } ∈
and γ = (γ ,γ ,...,γ ) C([0,M])d, with each γ strictly positive.
1 2 d i
∈
Foreachi 1,...,d setLγi,bi =γ (x )x ∂2 +b ∂ ,withdomainD(Lγi,bi)defined
∈ { } i i i xi i xi
according to (2.2) if b = 0 or to (2.3) if b > 0.
i i
We know that each operator (Lγi,bi,D(Lγi,bi)) generates a bounded analytic com-
pact C -semigroup (T (t)) of positive contractions and of angle π/2 in C([0,M]).
0 i t≥0
So, the injective tensor product (T(t)) = (ˆd T (t)) is also a bounded ana-
t≥0 ⊗ǫ,i=1 i t≥0
lytic compact C -semigroup of positive contractions and of angle π/2 in C([0,M]d) =
0
ˆ C([0,M]), see [30, Proposition, p.23, and p.24], [13, Appendix A] (see also [4,
d,ǫ
⊗
§2.2]). In particular, the infinitesimal generator ( γ,b,D( γ,b)) of (T(t)) is the
t≥0
L L
closure of the operator
d
Lγ,b = Lγi,bi I , (3.1)
⊗ ⊗j6=i xj
i=1
X (cid:0) (cid:1)
with domain d D(Lγi,bi), where I denote the identity map acting in C([0,M])
⊗i=1 xj
with respect to the variable x . Clearly, for every u d D(Lγi,bi), we have
j ∈ ⊗i=1
d
γ,bu(x) = γ (x )x ∂2 u+b ∂ u.
L i i i xi i xi
i=1
X
Moreover, if we define C2(Qd) = d u C2(Qd) : x ∂(Qd)i ∂ u(x) = 0 (such
⋄ ∩i=1{ ∈ ∀ ∈ xi }
a space is a Banach space when endowed with the supremum norm ), then the
2,∞
k k
following holds.
Proposition 3.1. Let b =(b ,b ,...,b ) [0, [d and γ = (γ ,γ ,...,γ ), with each
1 2 d 1 2 d
∈ ∞
γ a strictly positive continuous function on [0,M]. Then the space C2(Qd) is a core
i ⋄
for the operator ( γ,b,D( γ,b)).
L L
Proof. By Proposition 2.5 the space C2([0,M]) is a core for the one–dimensional
⋄
operator (Lγi,bi,D(Lγi,bi)) for every i = 1,...,d. So, d C2([0,M]) is a core for the
⊗i=1 ⋄
operator ( γ,b,D( γ,b)). On the other hand, it is known that d C2([0,M]) is dense
L L ⊗i=1 ⋄
in C2(Qd) with respect to the C2-norm which is clearly stronger than the graph-norm
⋄
of γ,b. So, it follows that C2(Qd) is a subspace of the domain of the closure of γ,b
andLis dense therein with resp⋄ect to the graph norm. L(cid:3)
We now prove that the operator ( γ,b,D( γ,b)) also shares similar gradient esti-
L L
mates with the analogous one–dimensional operator.
Proposition 3.2. Let B > 0 and γ = (γ ,γ ,...,γ ) (C([0,M]))d, with each γ
1 2 d i
∈
strictly positive. Then, for every b [0,B]d, the following properties hold.
∈
(1) There exist K,α,t > 0 depending on B and on γ such that, for every u
∈
C(Qd) and i = 1,...,d, we have
t γ,bT(t) Keαt, t 0. (3.2)
|| L || ≤ ≥
Keαt
√x ∂ (T(t)u) u , 0< t < t. (3.3)
|| i xi ||∞ ≤ √t || ||∞
√x ∂ (T(t)u) Keαt u , t t. (3.4)
|| i xi ||∞ ≤ || ||∞ ≥
10 A.A.Albanese,E. M.Mangino
Moreover, for every i 1,...,d and u C(Qd), √x ∂ (T(t)u) C(Qd)
∈ { } ∈ i xi ∈
and
lim sup √x ∂ (T(t)u) = 0. (3.5)
xi→0+xj∈[0,M],j∈{1,...,d}\{i} i xi
(2) There exist d ,d ,R > 0 depending on B and on γ such that, for every λ C
1 2
∈
with Reλ > R , u C(Qd) and i = 1,...,d, we have
∈
u
R(λ, γ,b)u d || ||∞, (3.6)
∞ 1
|| L || ≤ λ
| |
u
√x ∂ (R(λ, γ,b)u) d || ||∞. (3.7)
|| i xi L ||∞ ≤ 2 λ
| |
Moreover, for every i 1,...,d and u C(pQd), √x ∂ (R(λ, γ,b)u)
∈ { } ∈ i xi L ∈
C(Qd) and
lim sup √x ∂ (R(λ, γ,b)u)(x) = 0. (3.8)
xi→0+xj∈[0,M],j∈{1,...,d}\{i} i xi L
(3) There exist C,D,ε > 0 depending on B and on γ such that, for every 0< ε <
ε, i= 1,...,d and u D( γ,b), we have
∈ L
C
√x ∂ u u +Dε γ,bu .
k i xi k∞ ≤ εk k∞ kL k∞
Proof. (1) By Proposition 2.2(2) there exists t,K,α > 0 depending on B and γ such
i
that the operators √x ∂ T (t), for i= 1,...,d, are bounded on C([0,M]) with norm
i xi i
less or equal to Keαt/√t if 0 < t < t and to Keαt if t t. Then the operators
≥
√x ∂ T(t) = T (t) ... (√x ∂ T (t)) ... T (t), i = 1,...,d,
i xi 1 ⊗ε ⊗ε i xi i ⊗ε ⊗ε d
are also bounded on C(Qd) with norm less or equal to Keαt/√t if 0 < t < t or to
b b b b
Keαt if t t, [26] (see also [13, Appendix A] or [4, §2.2]). So, inequalities (3.3)
≥
and (3.4) are satisfied. In particular, for every u C(Qd) and i = 1,...,d we have
∈
√x ∂ T(t)u C(Qd).
i xi ∈
Moreover, again by Proposition 2.2(2) we have that tLγi,biT (t) Keαt for every
i
k k ≤
t 0. Then, via (3.1) the linear operators
≥
d
tLγ,bT(t)= tLγi,biT (t) ( T (t))
i j6=i j
⊗ ⊗
i=1
X
are bounded on d C([0,M]) with norm less or equal to dKeαt for every t 0. So,
⊗i=1 ≥
(3.2) is satisfied on d C([0,M]). By the density of d C([0,M]) in C(Qd) and
⊗i=1 ⊗i=1
the continuity of the linear operators tLγ,bT(t) in C(Qd) (recall that (T(t)) is an
t≥0
analytic C –semigroup in C(Qd)) it follows that (3.2) is satisfied.
0
Finally, if u d C([0,M]), then (3.5) is clearly satisfied by Proposition 2.2(2).
∈ ⊗i=1
The density of d C([0,M]) in C(Qd) and the continuity of the linear operators
⊗i=1
√x ∂ T(t) in C(Qd) imply that (3.5) is valid for every u C(Qd).
i xi ∈
(2) By [28, Proposition 2.1.1] and (3.2) there exists d = d (B,γ) > 0 such that,
1 1
for every λ C, with Reλ > α and u C(Qd), we have
∈ ∈
u
R(λ, γ,b)u d || ||∞ .
∞ 1
|| L || ≤ λ α
| − |