getdoc9cf4. 265KB Jun 04 2011 12:04:45 AM
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Vol. 11 (2006), Paper no. 23, pages 563–584.
Journal URL
http://www.math.washington.edu/~ejpecp/
Parabolic SPDEs degenerating on the boundary of
non-smooth domain
Kyeong-Hun Kim
Department Of Mathematics
Korea university
1 anam-dong sungbuk-gu, Seoul
South Korea 136-701
email : kyeonghun@korea.ac.kr
Abstract
Degenerate stochastic partial differential equations of divergence and non-divergence forms
are considered in non-smooth domains. Existence and uniqueness results are given in
weighted Sobolev spaces, and Hölder estimates of the solutions are presented
Key words: SPDEs degenerating on the boundary, Weighted Sobolev spaces
AMS 2000 Subject Classification: Primary 60H15, 35R60.
Submitted to EJP on October 28 2005, final version accepted June 2 2006.
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1
Introduction
We are dealing with an Lp -theory of parabolic stochastic partial differential equations (SPDEs)
of the types
(1.1)
du = (aij uxi xj + bi uxi + cu + f ) dt + (σ ik uxi + ν k u + g k ) dwtk
k
k
k
ik
i
i
i
ij
¯
(1.2)
du = (Di (a uxj + b̄ u + f ) + b uxi + cu + f ) dt + (σ uxi + ν u + g ) dwt
considered for t > 0 and x ∈ G. Here wtk are independent one-dimensional Wiener processes
and G is a bounded domain in Rd .
In this article we assume that the equations have the ”degeneracy α” near ∂G : ∃δ0 , K > 0 such
that for any λ ∈ Rd ,
δ0 ρ2α (x)|λ|2 ≤ (aij (t, x) − αij (t, x))λi λj ≤ Kρ2α (x)|λ|2
(1.3)
P
where ρ(x) := dist(x, ∂G) and αij := 21 k σ ik σ jk . Note that if α = 0 then the equations are
uniformly nondegenerate. In this case, unique solvability of the equations in appropriate Banach
spaces has been widely studied in many articles. See, for instance, [3], [4], [5], [8], [10], [14], [15]
and [17].
Our motivation of considering SPDEs with such degeneracy comes from several articles related to
PDEs with different types of degeneracies. We refer to [16], [19] and [20] for degenerate elliptic
equations. For parabolic PDEs we refer to [1], [18] (and references therein), where interior
Schauder estimates for equations with the degeneracy α < 1/2 were established.
An Lp -theory of equation (1.1) with the degeneracy α = 1 can be found in [12]. In this article,
we extend the results in [12]. We prove the unique solvability of equations (1.1) and (1.2)
with arbitrary degeneracy α ∈ [1, ∞) in appropriate Sobolev spaces. Also we give some Hölder
estimates of the solutions.
One of main applications of the theory of SPDEs is a nonlinear filtering problem. Consider a
pair of diffusion processes (Xt , Yt ) ∈ Rd × Rd1 −d ,
dXt = ρα (Xt )b(t, Xt , Yt )dt + ρα (Xt )r(t, Xt , Yt )dWt ,
dYt = B(t, Xt , Yt )dt + R(t, Yt )dWt ,
X(0) = X0
Y (0) = Y0 ,
where Wt is d1 -dimensional Wiener process and b, r, B, R are Lipschitz continuous matrices.
The nonlinear filtering problem is computing the conditional density πt of Xt given by the
observations {Ys : s ≤ t}. It was shown in [8] that when α = 0, there exists a conditional density
πt and πt satisfies a SPDE of type (1.1). Based on our Lp -theory, one can easily construct the
corresponding results when α ≥ 1. The motivations of considering the case α > 0 were discussed
at length in [12]. We only mention that usually the process Xt evolves in a bounded region
due to, for instance, mechanical restrictions, and therefore the above model is suitable when the
process Xt stays in the bounded domain. Note that since ρα (α ≥ 1) is Lipschitz continuous
in Rd (ρ(x) := 0 if x 6∈ G), by the unique solvability of the above SDE, if X0 is in G then the
process Xt never cross the boundary of G.
Here are notations used in the article. As usual Rd stands for the Euclidean space of points
x = (x1 , ..., xd ) and Br (x) := {y ∈ Rd : |x−y| < r}. For i = 1, ..., d, multi-indices β = (β1 , ..., βd ),
βi ∈ {0, 1, 2, ...}, and functions u(x) we set
uxi = ∂u/∂xi = Di u,
Dβ u = D1β1 · ... · Ddβd u,
564
|β| = β1 + ... + βd .
We also use the notation Dm for a partial derivative of order m with respect to x.
The author is sincerely grateful to the referee for giving several useful comments.
2
Main results
Let (Ω, F, P ) be a complete probability space, and {Ft , t ≥ 0} be an increasing filtration of σfields Ft ⊂ F, each of which contains all (F, P )-null sets. By P we denote the predictable σ-field
generated by {Ft , t ≥ 0} and we assume that on Ω we are given independent one-dimensional
Wiener processes wt1 , wt2 , ..., each of which is a Wiener process relative to {Ft , t ≥ 0}.
Choose and fix a smooth function ψ such that ψ(x) ∼ ρ(x) (see (2.9)). We rewrite equations
(1.1) and (1.2) in the following forms.
du = (ψ 2α aij uxi xj + ψ α bi uxi + cu + f ) dt
+ (ψ α σ ik uxi + ν k u + g k ) dwtk ,
and
(2.4)
du = (Di (ψ 2α aij uxj + ψ α b̄i u + f¯i ) + ψ α bi uxi + cu + f ) dt
+ (ψ α σ ik uxi + ν k u + g k ) dwtk ,
(2.5)
Here, i and j go from 1 to d, and k runs through {1, 2, ...}. The coefficients aij , b̄i , bi , c, σ ik , ν k
and the free terms f¯i , f, g k are random functions depending on t and x. Throughout the article,
for functions defined on Ω × [0, T ] × G, the argument ω ∈ Ω will be omitted.
To describe the assumptions of f¯i , f and g we use Sobolev spaces introduced in [8], [9] and [13].
If θ ∈ R and n is a nonnegative integer, then
Hpn = Hpn (Rd ) = {u : u, Du, ..., Dn u ∈ Lp },
0
Lp,θ (G) := Hp,θ
(G) = Lp (G, ρθ−d dx),
n
Hp,θ
(G) := {u : u, ρux , ..., ρn Dn u ∈ Lp,θ (G)}.
(2.6)
In general, by Hpγ = Hpγ (Rd ) = (1−∆)−γ/2 Lp we denote the space of Bessel potential. We define
kukHpγ = k(1 − ∆)γ/2 ukLp .
γ
The space Hp,θ
(G) is defined as the set of all distributions u on G such that
kukpH γ (G)
p,θ
:=
∞
X
enθ kζ−n (en ·)u(en ·)kpH γ < ∞,
p
n=−∞
where {ζn : n ∈ Z} is a sequence of smooth functions such that
X
|Dm ζn (x)| ≤ N (m)emn ,
ζn ≥ const > 0,
n
ζn ∈ C0∞ (Gn ),
Gn := {x ∈ G : e−n−1 < ρ(x) < e−n+1 }.
565
(2.7)
(2.8)
If Gn is empty set, then we put ζn = 0. One can construct the function ζn , for instance, by
mollifying the indicator function of Gn . It is known that up to equivalent norms the space
γ
Hp,θ
(G) and its norm are independent of {ζn } (see Lemma 2.1(iv)).
We also use the above notations for ℓ2 -valued functions g = (g1 , g2 , ...). We define
kgkHpγ = kgkHpγ (ℓ2 ) = k|(1 − ∆)γ/2 g|ℓ2 kLp .
kgkH γ
p,θ (G)
=
∞
X
enθ kζ−n (en ·)g(en ·)kpH γ .
p
n=−∞
Fix a smooth function ψ in G such that
sup |ρ(x)m Dm+1 ψ(x)| < ∞,
x
ρ(x) ≤ N ψ(x) ≤ N ρ(x),
P
For instance one can take ψ(x) = n e−n ζn (x).
∀x ∈ G.
(2.9)
γ
In the following lemma we collect some properties of Hp,θ
(G) (see [9] and [13] for detail). For
ν ∈ (0, 1], we denote
|u|C(X) = sup |u(x)|,
[u]C ν (X) = sup
X
x6=y
|u(x) − u(y)|
.
|x − y|ν
Lemma 2.1. (i) Assume that γ − d/p = m + ν for some m = 0, 1, ... and ν ∈ (0, 1]. Let i, j be
γ
multi-indices such that |i| ≤ m, |j| = m. Then for any u ∈ Hp,θ
(G), we have
ψ |i|+θ/p Di u ∈ C(G),
ν
ψ m+ν+θ/p Dj u ∈ Cloc
(G),
|ψ |i|+θ/p Di u|C(G) + [ψ m+ν+θ/p Dj u]C ν (G) ≤ N kukH γ
p,θ (G)
.
γ
γ−1
γ
(ii) ψD, Dψ : Hp,θ
(G) → Hp,θ
(G) are bounded linear operators, and for any u ∈ Hp,θ
(G)
kukH γ
p,θ (G)
kukH γ
p,θ (G)
≤ N kψux kH γ−1 (G) + N kukH γ−1 (G) ≤ N kukH γ
p,θ
p,θ (G)
p,θ
≤ N k(ψu)x kH γ−1 (G) + N kukH γ−1 (G) ≤ N kukH γ
p,θ
,
p,θ (G)
p,θ
(2.10)
.
γ
γ
(iii) For any ν, γ ∈ R, ψ ν Hp,θ
(G) = Hp,θ−pν
(G) and
kukH γ
p,θ−pν (G)
≤ N kψ −ν ukH γ
p,θ (G)
≤ N kukH γ
p,θ−pν (G)
.
(iv) Let {ξn } be a sequence of C0∞ (G) functions such that
|Dm ξn | ≤ N enm ,
supp ξn ⊂ {x ∈ G : e−n−k0 < ρ(x) < e−n+k0 }
γ
for some k0 > 0. Then for any u ∈ Hp,θ
(G)
X
kξ−n (en x)u(en x)kpH γ
And, if in addition
P
p,θ (G)
n
≤ N kukpH γ
p,θ (G)
≥ δ > 0, then
X
kξ−n (en x)u(en x)kpH γ
kukpH γ (G) ≤ N
.
n ξn (x)
p,θ
p,θ (G)
n
566
.
(2.11)
| τ ]] = {(ω, t) : 0 <
Now we define stochastic Banach spaces. For any stopping time τ , denote (0,
t ≤ τ (ω)},
γ
| τ ]], P, Hpγ ),
| τ ]], P, H (G)),
Hγp (τ ) = Lp ( (0,
Hγp,θ (G, τ ) = Lp ( (0,
p,θ
L... (...) = H0... (...),
γ,α
Up,θ
(G) = ψ
Upγ = Lp (Ω, F0 , Hpγ−2/p ),
− p2 (1−α)+1
γ−2/p
Lp (Ω, F0 , Hp,θ
(G)).
γ+2,α
Definition 2.2. We write u ∈ Hγ+2,α
(G, τ ) if u ∈ ψHγ+2
(G) and for
p,θ
p,θ (G, τ ), u(0, ·) ∈ Up,θ
some f ∈ ψ −1+2α Hγp,θ (G, τ ), g ∈ ψ α Hγ+1
p,θ (G, τ, ℓ2 )
du = f dt + g k dwtk ,
(2.12)
in the sense of distribution. In other words, for any φ ∈ C0∞ (G), the equality
(u(t, ·), φ) = (u(0, ·), φ) +
Z
t
(f (s, ·), φ) ds +
0
∞ Z
X
k=1
t
0
(g k (s, ·), φ) dwsk
holds for all t ≤ τ with probability 1. In this situation we also write f = Du, g = Su. Let
γ+2,α
Hγ+2,α
(G, τ ) ∩ {u : u(0, ·) = 0}.
p,θ,0 (G, τ ) = Hp,θ
The norm in Hγ+2,α
(G, τ ) is introduced by
p,θ
kukHγ+2,α (G,τ ) = [|u|]Hγ+2,α (G,τ ) + ku(0, ·)kU γ+2,α (G) ,
p,θ
p,θ
p,θ
where
[|u|]Hγ+2,α (G,τ ) := kψ −1 ukHγ
p,θ (G,τ )
p,θ
ku(0, ·)kp γ+2,α
Up,θ
+ kψ 1−2α DukHγ
p,θ (G,τ )
2
(G)
= Ekψ p
(1−α)−1
u(0, ·)kp
+ kψ −α SukHγ+1 (G,τ ) ,
p,θ
γ+2−2/p
Hp,θ
(G)
.
Remark 2.3. Up to equivalent norms, the space Hγ+2,α
(G, τ ) is independent of the choice of ψ,
p,θ
−1
and for instance the norm kψ ukHγ+2 (G,τ ) can be replaced by kukHγ+2 (G,τ ) . Also note that if
p,θ
p,θ−p
u ∈ ψHγ+2
p,θ (G, τ ), then by Lemma 2.1
ψ 2α ∆u ∈ ψ −1+2α Hγp,θ (G, τ ),
ψ α Du ∈ ψ α Hγ+1
p,θ (G, τ ).
Thus considering equation (2.4), we find that the spaces for Du and Su are defined naturally.
To state our assumptions on the coefficients, we take some notations from [2]. Denote ρ(x, y) =
ρG (x, y) = ρ(x) ∧ ρ(y). For δ ∈ (0, 1), and k = 0, 1, 2, ..., define
(0)
(0)
[f ]k = [f ]k,G = sup ρk (x)|Dk f (x)|,
x∈G
(0)
(0)
[f ]k+δ = [f ]k+α,G = sup ρk+α (x, y)
x,y∈G
|β|=k
567
|Dβ f (x) − Dβ f (y)|
,
|x − y|α
(0)
k
X
(0)
|f |k = |f |k,G =
(0)
[f ]j,G ,
(0)
(0)
(0)
(0)
|f |k+α = |f |k+α,G = |f |k,G + [f ]k+α,G .
j=0
Dβ f
By
we mean either classical derivatives or Sobolev ones and in the latter case sup’s in the
above are understood as ess sup’s. We also use the same notations for ℓ2 -valued functions.
Fix a function δ0 (τ ) ≥ 0 defined on [0, ∞) such that δ0 (τ ) > 0 unless τ ∈ {0, 1, 2, ...}. For τ ≥ 0
define
τ + = τ + δ0 (τ ),
and fix some constants
δ0 , K ∈ (0, ∞),
γ ∈ R.
Assumption 2.4. (i) For each x ∈ G, the coefficients aij (t, x), b̄i (t, x), bi (t, x) c(t, x), σ ik (t, x)
and ν k (t, x) are predictable functions of (ω, t).
(ii) For any x, t, ω and λ ∈ Rd ,
δ0 |λ|2 ≤ (aij (t, x) − αij (t, x))λi λj ≤ K|λ|2 ,
where αij =
1
2
P
k
(2.13)
σ ik σ jk .
Assumption 2.5. For any ε > 0, there exists δ = δ(ε) > 0 such that
sup(|aij (t, x) − aij (t, y)| + |σ i (t, x) − σ i (t, y)|ℓ2 ) ≤ ε
ω,t
whenever x, y ∈ G and |x − y| ≤ δ(ε)ρ(x, y).
Assumption 2.6. For any t > 0 and ω ∈ Ω,
(0)
(0)
(0)
|aij (t, ·)||γ|+ + |ψ 1−α bi (t, ·)||γ|+ + |ψ 2(1−α) c(t, ·)||γ|+
(0)
(0)
+|σ i (t, ·)||γ+1|+ + |ψ 1−α ν(t, ·)||γ+1|+ ≤ K.
Remark 2.7. Assumption 2.5 is much weaker than uniform continuity of aij and σ i . For instance,
let G = (0, 1) and a(t, x) = 2 + sin(ln x(1 − x)). Then one can easily check that a satisfies
Assumptions 2.5 and 2.6 for any γ ∈ R.
Here are our main results. From this point on we assume that
τ ≤ T,
α ∈ [1, ∞),
p ∈ [2, ∞).
Theorem 2.8. Let Assumptions 2.4, 2.5 and 2.6 be satisfied. Then
γ+2,α
(i) for any f ∈ ψ −1+2α Hγp,θ (G, τ ), g ∈ ψ α Hγ+1
(G), equation (2.4) with
p,θ (G, τ ) and u0 ∈ Up,θ
initial data u0 admits a unique solution u (in the sense of distribution) in the class Hγ+2,α
(G, τ ),
p,θ
(ii) for this solution
kukp γ+2,α
Hp,θ
(G,τ )
≤ N (kψ 1−2α f kpHγ
p,θ (G,τ )
+ kψ −α gkp γ+1
Hp,θ (G,τ )
where the constant N depends only on d, γ, p, θ, δ0 , K and T .
568
+ ku0 kp γ+2,α
Up,θ
(G)
),
(2.14)
Note that in the following theorem Assumption 2.6 is not assumed.
Theorem 2.9. Let Assumptions 2.4 and 2.5 be satisfied, and
|ψ 1−α b̄i | + |ψ 1−α bi | + |ψ 2(1−α) c| + |ψ 1−α ν| ≤ K,
∀ω, t, x.
(2.15)
Then
1,α
α
(i) for any f¯i ∈ ψ 2α Lp,θ (G, τ ), f ∈ ψ −1+2α H−1
p,θ (G, τ ), g ∈ ψ Lp,θ (G, τ ) and u0 ∈ Up,θ (G),
equation (2.5) with initial data u0 admits a unique solution u (in the sense of distribution) in
the class H1,α
p,θ (G, τ ),
(ii) for this solution
kukp 1,α
Hp,θ (G,τ )
≤ N (kψ −2α f¯kpLp,θ (G,τ ) + kψ 1−2α f kpH−1 (G,τ )
p,θ
+ kψ
−α
gkpLp,θ (G,τ )
+
ku0 kp 1,α ),
Up,θ (G)
(2.16)
where the constant N depends only on d, γ, p, θ, δ0 , K and T .
Now we state the regularity of the solutions in terms of Hölder continuity in time and space, both
inside the domain and near the boundary. The following results are immediate consequences of
Lemma 2.1, Remark 3.2 and Theorem 3.3.
Corollary 2.10. Let u ∈ Hγ+2,α
(G, τ ) be the solution of Theorem 2.8 or Theorem 2.9. Let
p,θ
2/p < µ < β < 1,
γ + 2 − β − d/p = m + ν,
for some m = 0, 1, ..., ν ∈ (0, 1]. Then for any multi-indices i, j such that |i| ≤ m, |j| = m
E
|ψ |i|−1+θ/p Di (u(t) − u(s))|pC(G)
sup
|t − s|pµ/2−1
0≤s 0.
(3.23)
If k ≤ m and en x ∈ suppζ−n (e·), then (since ρ(en x) ∼ en ),
(0)
|enk (Dk a)(en x)| ≤ N ρk (en x)|(Dk a)(en x)| ≤ N |a||γ|+ .
(3.24)
Obviously, (3.23) and (3.24) prove (3.22). Next let δ 6= 0. To show
|Dm a(en x)ζ−n (en x) − Dm a(en y)ζ−n (en )| ≤ N |x − y|δ , ∀x, y ∈ Rd ,
we may assume that |x − y| ≤ e−4 and en x ∈ suppζ−n (e·). In this case, en y ∈ B̄e−4+n (en x) ⊂ G
and ρ(en x) ∼ ρ(en x, en y) ∼ en . Thus, due to (3.23),
|Dm a(en x)ζ−n (en x) − Dm a(en y)ζ−n (en )|
≤N
X
ρk (en x, en y)|(Dk a)(en x) − (Dk a)(en y)||Dm−k (ζ−n (en x))|
k≤m
+N
X
enk |(Dk a)(en y)||Dm−k (ζ−n (en x)) − Dm−k (ζ−n (en y))|
k≤m
(0)
≤ N |a||γ|+ (e−nδ |en x − en y|δ + |x − y|) ≤ N |x − y|δ .
The lemma is proved.
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Remark 3.2. Let θ1 ≤ θ2 . By Lemmas 2.1 and 3.1
kukH γ
p,θ2 (G)
≤ N kψ (θ2 −θ1 )/p ukH γ
p,θ1 (G)
≤ N kukH γ
p,θ1 (G)
.
Consequently, if α1 ≤ α2 then
kukHγ,α1 (G,τ ) ≤ N kukHγ,α2 (G,τ ) .
p,θ
p,θ
The following results are due to Lototsky ([12]).
Theorem 3.3. (i) For any t ≤ T ,
kψ
−1
ukp γ+1
Hp,θ (G,t)
≤ N (d, γ, p, T )
Z
t
0
kukp γ+2,1
Hp,θ
(G,s)
ds.
(ii) Let
2/p < µ < β < 1.
Then
Ekukp µ/2−1/p
C
γ+2−β
([0,τ ],Hp,θ−p
(G))
≤ N (µ, β, d, γ, p, T )kukp γ+2,1
Hp,θ
(G)
.
We choose and fix smooth functions ξn such that |Dm ξn | ≤ N (m)enm , suppξn ⊂ (Gn−1 ∪ Gn ∪
Gn+1 ) and ξn = 1 on the support of ζn .
Lemma 3.4. Let Assumptions 2.4(ii) and 2.5 be satisfied. By I we denote d ×d identity matrix.
Define
−2nα 2α n
2
2
aij
ψ (e x)ξ−n
(en x)aij (e2n(1−α) t, en x) + (1 − ξ−n
(en x))I,
n (t, x) = e
σnik (t, x) = e−nα ψ α (en x)ξ−n (en x)σ ik (e2n(1−α) t, en x).
Then
(i) For any λ ∈ Rd ,
ik jk i j
4α
2
e−4α δ0 |λ|2 ≤ (aij
n − 1/2σn σn )λ λ ≤ e K|λ| .
(ii) For any ε > 0, there exists δ = δ(ǫ) > 0 such that
ij
i
i
sup sup(|aij
n (t, x) − an (t, y)| + |σn (t, x) − σn (t, y)|) < ε,
n
ω,t
whenever x, y ∈ Rd and |x − y| < δ.
(iii)
i
sup sup(|aij
n (t, ·)|B |γ|+ + |σn |B |γ+1|+ ) < ∞.
n
ω,t
571
(3.25)
Proof. (i) is obvious and (3.25) follows from the same arguments as in the proof of Lemma 3.1.
Thus we only give a proof of the second assertion. Let δ ≤ e−4 and |x − y| < δ. Without loss of
generality, we assume that ξ−n (en x) 6= 0. Observe that
en y ∈ B̄ := B̄en δ (en x) ⊂ G,
|en x − en y| ≤ δen ≤ N0 δρ(en x, en y),
and for any z ∈ Ben δ (en x) we have ρ(z) ∼ en . Thus,
|ξ−n (en x) − ξ−n (en y)| ≤ |x − y|en sup |D(ξ−n )(z)| ≤ N1 |x − y|,
z
|ψ 2α (en x) − ψ 2α (en y)| ≤ sup |Dψ 2α (z)||en x − en y| ≤ N e2nα |x − y|,
z∈B
and
|e−2nα ψ 2α (en x)a(en x)ξ−n (en x) − e−2nα ψ 2α (en y)a(en y)ξ−n (en y)|
≤ e−2nα ψ 2α (en x)ξ−n (en x)|a(en x) − a(en y)|
+|a(en y)|e−2nα ψ 2α (en x)|ξ−n (en x) − ξ−n (en y)|
+|a(en y)ξ−n (en y)|e−2nα |ψ 2α (en x) − ψ 2α (en y)|
≤ N2 (|a(en x) − a(en y)| + δ + δ).
Note that the constant Ni are independent of x, y and n. So, if ε > 0 is given, then it is
enough to take δ > 0 such that (N1 + 2N2 )δ < ε/2 and N2 |a(t, x) − a(t, y)| ≤ ε/3 whenever
|x − y| < N0 δρ(x, y).
We handle σni similarly. The lemma is proved.
The following lemma is taken from [13].
Lemma 3.5. Let {φk : k = 1, 2, ...} be a collection of C0∞ (G) functions such that for each m > 0
X
sup
ρm (x)|Dm φk (x)| ≤ M (m) < ∞.
x∈G k
Then there exists a constant N = N (d, γ, M ) such that for any f ∈ Hγp,θ (G),
X
kφk f kpH γ
p,θ (G)
k
If in addition
X
≤ N kf kpH γ
p,θ (G)
.
|φk (x)|p ≥ c > 0,
k
then
kf kpH γ
p,θ (G)
≤ N (d, γ, M, c)
X
k
572
kφk f kpH γ
p,θ (G)
.
4
Proof of Theorem 2.8
As usual, we may assume that τ ≡ T (see [8]). For a moment, we assume that bi = c = ν k = 0.
ik
Take aij
n and σn from Lemma 3.4. Denote
cn := e−2n(1−α) ,
wtk (n) := e−n(1−α) wek2n(1−α) t
Then for each n, wtk (n) are independent one dimensional Wiener processes. By Theorem 5.1 in
γ+2
[8], for any f ∈ Hγp (cn T ), g ∈ Hγ+1
the equation
p (cn T ) and u0 ∈ Up
k
k
ik
du = (aij
n uxi xj + f )dt + (σn uxi + g )dwt (n) u(0, ·) = u0 ,
(4.26)
has a unique solution u ∈ Hγ+2
p (cn T ) and u satisfies
kukHγ+2
≤ N (kf kHγp (cn T ) + kgkHpγ+1 (cn T ) + ku0 kUpγ+2 ),
(cn T )
p
(4.27)
where the constant N depends only d, p, γ, δ0 , K, cn T, |an |B |γ|+ , |σn |B |γ+1|+ and uniform continuity of an , σn .
By Sn (f, g, u0 ) we denote the the solution of (4.26). Define
S̄n (f, g, u0 )(t, x) = Sn (f, g, u0 )(cn t, e−n x).
From now on, without loss of generality, we assume that
X
2
(x) = 1, ∀x ∈ G.
ζ−n
n
Remember the fact that a function v satisfies
dv = f dt + g k dwtk ,
t≤T
if and only if vc (t, x) := v(c2 t, cx) (c > 0) satisfies
dvc = c2 f (c2 t, cx) dt + cg(c2 t, cx)d(c−1 wck2 t ),
t ≤ c−2 T.
It follows that if u ∈ Hγ+2,α
(G, T ) is a solution of equation (2.4), then vn (t, x) :=
p,θ
−1
n
(ζ−n u)(cn t, e x) satisfies
k
k
k
ik
dvn = (aij
n vnxi xj + An u + fn )dt + (σn vnxi + Bn u + gn )dwt (n),
where
n
−1
n
2n
An u(t, x) := −2aij
n e uxi (cn t, e x)ζ−nxj (e x)
n
2n
−1
n
−aij
n e u(cn t, e x)ζ−nxi xj (e x),
n
n
Bnk u(t, x) := −σnik en u(c−1
n t, e x)ζ−nxi (e x),
fn (t, x) := e2n(1−α) f (e2n(1−α) t, en x)ζ−n (en x),
gnk (t, x) := en(1−α) g(e2n(1−α) t, en x)ζ−n (en x),
u0n := u0 (en x)ζ−n (en x).
573
(4.28)
Consequently,
n
vn (t, x) := (ζ−n u)(c−1
n t, e x) = Sn (An u + fn , Bn u + gn , u0n )
and
u=
X
ζ−n (ζ−n u) =
n
X
ζ−n S̄n (An u + fn , Bn u + gn , u0n ).
(4.29)
n
To proceed further, we need the following lemma.
γ+2
Lemma 4.1. Fix f ∈ ψ −1+2α Hγp,θ (G, T ), g ∈ ψ α Hγ+1
p,θ (G, T ) and u0 ∈ Up,θ (G). Then a sufficiently high power of the operator
X
R:u→
ζ−n S̄n (An u + fn , Bn u + gn , u0n )
(4.30)
n
is a contraction in Hγ+2,α
(G, T ) ∩ {u : u(0, ·) = u0 }, and the unique solution u ∈ Hγ+2,α
(G, T )
p,θ
p,θ
of (4.29) satisfies the estimate (2.14).
Proof. For simplicity, we use the notations Sn and S̄n instead of Sn (An u + fn , Bn u + gn , u0n )
and S̄n (An u + fn , Bn u + gn , u0n ), respectively.
Note that ζn ζm = 0 if |n − m| > 1. By Lemmas 3.5 and 3.1,
X
X
≤N
kζ−n S̄n kp γ+2,α
kRukp γ+2,α
≤N
kζ−n Rukp γ+2,α
Hp,θ
(G,T )
(G,T )
Hp,θ
n
Hp,θ
n
(G,T )
.
(4.31)
By definition,
kζ−n S̄n kp γ+2,α
Hp,θ
(G,T )
= kψ −1 ζ−n S̄n kp γ+2
Hp,θ (G,T )
+kψ −α S(ζ−n S̄n )kp γ+1
Hp,θ (G,T )
+ kψ 1−2α D(ζ−n S̄n )kpHγ
p,θ (G,T )
+ kζ−n u0 kp γ+2,α
Up,θ
(G)
.
Remember that ku(e±1 x)kH ν ∼ ku(x)kH ν and supn |ζ−n (en x)|B ν < ∞ for each ν > 0. Thus (cf.
Lemma 5.2 in [8]),
X
X
kζ−n S̄n kp γ+2
≤N
en(θ−p) kζ−n (en x)S̄n (t, en x)kp γ+2
n
Hp,θ−p (G,T )
≤N
X
Hp
n
en(θ−p+2−2α) kSn (t, x)kp γ+2
Hp
n
(cn T )
(T )
.
By writing the equation for ζ−n (en x)Sn , we find that v̄n := ζ−n S̄n satisfies
dv̄n = D(ζ−n S̄n ) dt + S(ζ−n S̄n ) dwtk
= [ψ 2α aij v̄nxi xj − 2ψ 2α aij (S̄n ζ−nxi )xj + ψ 2α aij S̄n ζ−nxi xj
2
−2ψ 2α aij uxi ζ−nxj ζ−n − ψ 2α aij uζ−nxi xj ζ−n + ζ−n
f ] dt
2
g] dwtk .
+[ψ α σ ik v̄nxi − ψ α σ ik S̄n ζ−nxi − ψ α σ ik uζ−nxi ζ−n + ζ−n
574
(4.32)
Thus, by Lemmas 2.1 and 3.1,
X
n
≤N
X
n
+N
X
n
+N
X
n
kψ 1−2α D(ζ−n S̄n )kpHγ
p,θ (G,T )
kψ(ζ−n S̄n )xx kpHγ
p,θ
p,θ
p,θ
X
Hp,θ (G,T )
Here,
X
X
kψ(S̄n ζ−nx )x kpHγ
p,θ (G,T )
X
n
X
n
Hp,θ−p (G,T )
+kψ −1 ukp γ+1
=
+N
(G,T )
+N
(G,T )
kζ−n S̄n kp γ+2
n
X
n
kψ −1 S̄n ψ 2 ζ−nxx kpHγ
kψ −1 uψ 2 ζ−nxx kpHγ
≤N
+N
(G,T )
+N
X
kux ψζ−nx kpHγ
kψ 1−2α f ζ−n kpHγ
p,θ (G,T )
kS̄n ζ−nx kp γ+1
Hp,θ (G,T )
n
+ N kψ 1−2α f kpHγ
p,θ (G,T )
Hp,θ (G,T )
em(θ−p) kS̄n (em x)em ζ−nx (em x)ζ−m (em x)kp γ+1
Hp
X
≤N
en(θ−p) kS̄n (t, x)(en x)kp γ+2
Hp
n
=N
X
en(θ−p+2−2α) kSn (t, x)kp γ+2
Hp
n
P
n kψ
p
−α S(ζ
−n S̄n )k γ+1
Hp,θ (G,T )
kRukp γ+2,α
Hp,θ
.
kζ−nx S̄n kp γ+1
n
n,m
We estimate
p,θ (G,T )
(G,T )
Hp,θ (G,T )
+kψ −α gkp γ+1
+
(T )
(cn T )
.
similarly, and conclude that
≤ N kψ −1 ukp γ+1
X
(T )
Hp,θ (G,T )
+ N kψ 1−2α f kpHγ
p,θ (G,T )
+ ku0 kp γ+2
Up,θ (G)
en(θ−p+2−2α) kSn (t, x)kp γ+2
Hp
n
(cn T )
.
(4.33)
Since G is bounded, we may assume that ζ−n = 0 for all n > 1. For each n ≤ 0, we have
cn T := e−2n(1−α) T ≤ T.
(4.34)
Thus by (4.27), there exists a constant N independent of n (due to Lemma 3.4 and (4.34)) such
that for each n ≤ 0,
kSn (t, x)kp γ+2
≤ N kAn u + fn kpHγ (c T )
Hp
+N kBn u +
(cn T )
p
gn kp γ+1
Hp (cn T )
575
+
n
N ku0n kp γ+2 .
Up
Also, by Lemma 2.1,
X
n≤0
=
X
n≤0
en(θ−p+2−2α) (kAn ukpHγ (c
p
nT )
+ kBn ukp γ+1
Hp
Hp
(T )
≤ N kψ −1 ukp γ+1
.
p
X
n≤0
+N
X
n≤0
+N
X
enθ kux (en x)en ζ−nx (en x)kpHγ (T )
p
en(θ−p) ku(en x)e2n ζ−nxx (en x)kpHγ (T )
p
en(θ−p) ku(en x)en ζ−nx (en x)kp γ+1
Hp
n≤0
≤ N kux kpHγ
+ N kukp γ+1
p,θ (G,T )
X
n≤0
Hp,θ−p (G,T )
en(θ−p+2−2α) (kfn kpHγ (c
≤N
p
X
n≤0
+N
nT )
+ kgn kp γ+1
Hp
(cn T )
)
en(θ+p(1−2α)) kf (t, en x)ζ−n (en x)kpHγ (T )
X
p
en(θ−pα) kg(t, en x)ζ−n (en x)kp γ+1
Hp
≤ N kψ 1−2α f kpHγ
p,θ (G,T )
X
(T )
Hp,θ (G,T )
n≤0
and
)
en(θ−p) k(An u)(cn t, x)|pHγ (T ) + k(Bn u)(cn t, x)kp γ+1
≤N
Similarly,
(cn T )
n(θ+p( p2 (1−α)−1))
e
+ N kψ −α gkp γ+1
(T )
Hp,θ (G,T )
,
ku0 (en x)ζ−n (en x)kp γ+2 ≤ N ku0 kp γ+2
Up
n
Up,θ (G)
.
Hence, coming back to (4.33), we get
kRukp γ+2,α
Hp,θ
(G,τ )
≤ N (kψ −1 ukp γ+1
Hp
+ kψ −α gkp γ+1
Hp
(T )
(T )
+ kψ 1−2α f kpHγ (T )
+ ku0 kp γ+2
Up,θ (G)
p
).
(4.35)
Note that H̄γ+2,α
(G, T ) := Hγ+2,α
(G, T ) ∩ {u : u(0, ·) = u0 } is a complete Banach space and
p,θ
p,θ
contains R0 (thus not empty), where
X
R0 :=
ζ−n S̄n (fn , gn , u0n ).
n
By (4.35) and Theorem 3.3, for any u, v ∈ H̄γ+2,α
(G, T ),
p,θ
Ru − Rv =
X
ζ−n S̄n (An (u − v), Bn (u − v), 0),
n
576
kRu − RvkHγ+2,α (G,T ) ≤ N kψ −1 (u − v)kp γ+1
Hp,θ (G,T )
p,θ
≤N
Z
T
ku − vkp γ+2,α
Hp,θ
0
(G,s)
ds.
(4.36)
(4.36) shows that there exists m0 > 0 such that Rm is a contraction in H̄γ+2,α
(G, T ), and all
p,θ
the assertions of the lemma follow from this and (4.35). For more technical details, we refer to
the proof of Theorem 6.4 in [8]. The lemma is proved.
Let u be the solution of (4.29). We will show that u satisfies equation (2.4). Obviously u(0, ·) =
u0 , and (by definition) for some f0 ∈ ψ −1+2α Hγp,θ (G, T ) and g0 ∈ ψ α Hγ+1
p,θ (G, T )
du = f0 dt + g0k dwtk .
Observe that u satisfies equation (2.4) with f¯ := f0 − ψ 2α aij uxi xj and ḡ k := g0k − ψ α σ ik uxi
instead of f and g k , respectively. By the above arguments (see (4.29))
X
u=
ζ−n S̄n (An u + f¯n , Bn u + ḡn , u0n ),
n
where f¯n , ḡn , u0n are defined from f¯, ḡ, u0 as in (4.28). Also,
X
0=
ζ−n S̄n (f˜n , g̃n , 0),
(4.37)
n
where f˜ = f − f¯, g̃ = g − ḡ and f˜n , g̃n are defined as before.
Define the operators Ān and B̄n such that
Ān u = 2aij ψ 2α uxi ζ−nxj − aij ψ 2α uζ−nxi xj ,
B̄n u = ψ α σ ik uζ−nxi .
From (4.37),
0=D
X
ζ−n S̄n (f˜n , g̃n , 0) = f˜ −
n
0=S
X
X
Ān S̄n (f˜n , g̃n , 0),
n
ζ−n S̄n (f˜n , g̃n , 0) = g̃ −
n
X
B̄n S̄n (f˜n , g̃n , 0).
n
Therefore, to show f˜ = g̃ k = 0, we only need to prove that a sufficiently high power of the
operator
X
X
R̄ : (f, g) → (
Ān S̄n (fn , gn , 0),
B̄n S̄n (fn , gn , 0))
n
is a contraction in
γ,α
Fp,θ
(G, T )
:=
n
ψ −1+2α Hγp,θ (G, T )
× ψ α Hγ+1
p,θ (G, T ).
By Lemma 3.5,
k
X
n
ψζ−nx S̄nx kpHγ
p,θ (G,T )
≤N
X
m,n
|m−n|≤1
577
2
kψζ−m
ζ−nx S̄nx kpHγ
p,θ (G,T )
=N
X
m,n
|m−n|≤1
≤N
X
kψ
2
kψζ−nx (S̄n ζ−m
)x − 2ψ −1 S̄n ζ−m ψ 2 ζ−mx ζ−nx kpHγ
p,θ (G,T )
−1
m,n
|m−n|≤1
ζ−m S̄n kp γ+1
Hp,θ (G,T )
Z
≤N
X
T
0
kζ−m S̄n kp γ+2,α
As in the proof of Lemma 4.1 (see (4.31) and (4.35)),
X
kζ−m S̄n kp γ+2,α
≤ N kψ 1−2α f kpHγ
Hp,θ
m,n
|m−n|≤1
We estimate other terms in
p,θ (G,s)
(G,s)
P
n Ān S̄n
and
kR̄(f, g)kpF γ (G,T )
p,θ
P
n B̄n S̄n
Z
≤N
T
Hp,θ
m,n
|m−n|≤1
+ N kψ −α gkp γ+1
ds.
Hp,θ (G,s)
.
similarly (actually much easily) and get
k(f, g)kpF γ
p,θ (G,s)
0
(G,s)
ds.
This shows that a sufficiently high power of R̄ is a contraction and f¯ = f, ḡ = g.
For general case (previously we assumed that bi = c = ν k = 0), having the method of continuity
in mind, we only show that (2.14) holds true given that a solution u ∈ Hγ+2,α
(G, T ) already
p,θ
exists. Let ū ∈ Hγ+2,α
(G, T ) be the solution of
p,θ
dū = ψ 2α ∆ū dt,
Then
kūkp γ+2,α
Hp,θ
ū(0, ·) = u0 .
≤ N ku0 kp γ+2
(G,T )
Up,θ (G)
.
Thus by considering u − ū, as usual, we may assume that u0 = 0.
By the previous results (when bi = c = ν k = 0),
kukp γ+2,α
Hp,θ
where
(G,T )
≤ N (kψ 1−2α f˜kpHγ
p,θ (G,T )
f˜ = ψ α bi uxi + cu + f,
By Lemma 3.1,
kψ 1−2α f˜kpHγ
p,θ (G,T )
≤ N kψ 1−2α f kpHγ
p,θ (G,T )
+ kψ −α g̃kp γ+1
Hp,θ (G,T )
g̃ i k = ν k u + g k .
+ kψ −α g̃kp γ+1
Hp,θ (G,T )
+ N kψ −α gkp γ+1
Hp,θ (G,T )
+ N kψ −1 ukp γ+1
Hp,θ (G,T )
Thus, by Theorem 3.3 for each t ≤ T ,
kukp γ+2,α
Hp,θ
+N kψ
−α
(G,T )
≤ N kψ 1−2α f kpHγ
p,θ (G,T )
gkp γ+1
Hp,θ (G,T )
),
+
Z
0
t
kukp γ+2,α
Hp,θ
(G,s)
ds.
This and Gronwall’s inequality lead to (2.14). The theorem is proved.
578
.
5
Proof of Theorems 2.9
Consider the operators
L0 u := ψ 2α ∆u = Di (ψ 2α uxi ) − 2αψ 2α−1 ψxi uxi ,
L1 u := Di (ψ 2α aij uxj + b̄i u) + bi uxi + cu,
Λk1 u : ψ α σ ik uxi + ν k u.
One can easily check that the coefficients of the operators Lλ := (1 − λ)L0 + λL1 and Λλ := λΛ1
satisfy Assumptions 2.4, 2.5 and (2.15). Also note that
kψ 1−2α f¯xi kH −1 (G) ≤ N kf¯i kLp,θ−2pα (G) ≤ N kψ −2α f¯i kLp,θ (G) .
p,θ
By Theorem 2.8, the equation
du = (L0 u + f¯xi i + f )dt + (Λk0 u + g k )dwtk
has a unique solution u ∈ H1,α
p,θ (G, τ ). Thus by the method of continuity, we only need to prove
that the estimate (2.16) holds true given that a solution u ∈ H1,α
p,θ (G, τ ) of equation (2.5) already
exists.
Again without loss of generality we assume that τ ≡ T and ζ−n = 0 for all n > 0. Also as in the
proof of Theorem 2.8, we assume that u0 = 0.
Step 1. We will show that there exists a constant ε0 = ε0 (d, p, δ0 , K) > 0 such that the theorem
holds true if T ≤ ε0 ≤ 1. As before, denote cn := e−2n(1−α) . By Lemma 2.1,
X
en(θ−p) ku(en x)ζ−n (en x)kpH1 (T )
kψ −1 ukpH1 (G,T ) ≤ N
p,θ
=N
X
p
n≤0
en(θ−p+2−2α) ku(e2n(1−α) t, en x)ζ−n (en x)kpH1 (cn T ) .
p
n≤0
(5.38)
Denote vn (t, x) = u(e2n(1−α) t, en x)ζ−n (en x). Then vn satisfies
i
¯i
dvn = (Di (aij
n vnxj + b̄n vn + fn ) + bn vnxi + cn vn + fn ) dt
+ (σnik vnxi + νnk vn + gnk ) dwtk (n),
where
ik
k
aij
n , σn , wt (n)
are defined as before and
n
b̄in (t, x) = en−2nα ψ α (en x)ξ−n (en x)b̄i (c−1
n t, e x),
n
bin (t, x) = en−2nα ψ α (en x)ξ−n (en x)bi (c−1
n t, e x),
n
n
cn (t, x) = e2n(1−α) c(c−1
n t, e x)ξ−n (e x),
n
n
νnk (t, x) = en(1−α) ν k (c−1
n t, e x)ξ−n (e x),
n
−1
n
n−2nα ¯i −1
n
f (cn t, en x)ζ−n (en x),
f¯ni (t, x) = −aij
n e ζ−nxj (e x)u(cn t, e x) + e
n
n(1−α) k −1
n
n
g (cn t, en x)ζ−n (en x),
gnk (t, x) = −σnik u(c−1
n t, e x)e ζ−nxi (e x) + e
579
(5.39)
−1
n
n
n
fn (t, x) = −en aij
n uxj (cn t, e x)e ζ−nxi (e x)
n
n
n
n−2nα ¯i −1
+b̄n u(c−1
f (cn t, en x)en ζ−nxi (en x)
n t, e x)e ζ−nxi (e x) + e
n
2n(1−α)
n
n
f (c−1
−bin en ζ−nxi (en x)u(c−1
n t, e x) + e
n t, e x)ζ−n (e x).
Note that ψ(en x) ∼ en on the support of ξ−n (en x). It follows from (2.15) that
sup sup (|b̄in | + bin | + |cn | + |νn |) < ∞.
n ω,t,x
By Theorem 2.12 in [4],
kvn kpH1 (cn T ) ≤ N (kf¯n kpLp (cn T ) + N kfn kpH−1 (c
p
p
nT )
+ kgn kpLp (cn T ) ).
(5.40)
−1
n
n
n
Actually, due to the term −en aij
n uxj (cn t, e x)e ζ−nxi (e x) of fn , (5.40) only yields
kψ −1 ukpH1
p,θ (G,T )
≤ N kψ −1 ukpH1
p,θ (G,T )
+ ....
Of course, this estimate is useless unless N < 1. The following argument below is to avoid
−1
n
n
n
estimating ken aij
.
n uxj (cn t, e x)e ζ−nxi (e x)kH−1
p (cn T )
Denote
n
−1
n
n
f˜n (t, x) = −en aij
n uxj (cn t, e x)e ζ−nxi (e x) ∈ Lp (cn T ).
By Theorem 5.1 in [8], the equation
du = (∆u + f˜n )dt,
u(0, ·) = 0
has a unique solution un ∈ H2p (cn T ), and (see Theorems 7.1 and 7.2 in [8])
kun kpH1 (cn T ) ≤ N (T )kf˜n kpLp (cn T ) ,
p
(5.41)
where N (T ) is independent of n (since cn T ≤ T ), and N (T ) ↓ 0 as T → 0.
v̂n := vn − un satisfies (5.39) with
¯n := f¯n + b̄n un + (aij − δ ij )unxi ,
fˆ
n
fˆn := fn − f˜n + bn unx + cn un ,
ĝn := gn + σ i unxi + νun
instead of f¯n , fn and gn , respectively. Thus by Theorem 2.12 in [4], there exists a constant N
depending only on d, p, δ0 and K (remember cn T ≤ T ≤ 1) such that
¯n kp
ˆ p
kv̂n kpH1 (cn T ) ≤ N (kfˆ
Lp (cn T ) + kfn kH−1 (c
p
p
nT )
+ kĝn kpLp (cn T ) ).
Consequently,
¯n kp
ˆ p
kvn kpH1 (cn T ) ≤ N (kfˆ
Lp (cn T ) + kfn kH−1 (c
p
p
nT )
+ kĝn kpLp (cn T ) + kun kH1p (cn T ) )
p
n
n
n
≤ N (T )ken ux (c−1
n t, e x)e ζ−nx (e x)kLp (cn T )
580
(5.42)
p
n
+N ken ζ−nx (en x)u(c−1
n t, e x)kLp (cn T )
p
n
n
n
n
+N ken(1−2α) f¯(c−1
n t, e x)[ζ−n (e x) + e ζ−nx (e x)]kLp (cn T )
p
n
n
+N ke2n(1−α) f (c−1
n t, e x)ζ−n (e x)kH−1 (c
p
nT )
p
n
n
+N ken(1−α) g(c−1
n t, e x)ζ−n (e x)kLp (cn T ) .
Coming back to (5.38), by Lemma 2.1, we get
kψ −1 ukpH1
p,θ (G,T )
≤ N kψ −1 ukpLp,θ (G,T ) + N kψ −2α f¯kpLp,θ (G,T )
+N kψ 1−2α f kpH−1 (G,T ) + N kψ −α gkpLp,θ (G,T ) + N N (T )kψ −1 ukpH1
p,θ (G,T )
p,θ
.
Now fix ε0 such that N N (T ) ≤ 1/2 for each T ≤ ε0 , then by Theorem 3.3 for each t ≤ T
kukp 1,α
Hp,θ (G,t)
≤N
Z
t
kukp 1,α
Hp,θ (G,s)
0
ds
+N (kψ −2α f¯kpLp,θ (G,T ) + kψ 1−2α f kpH−1 (G,T ) + kψ −α gkpLp,θ (G,T ) ).
p,θ
Gronwall’s inequality leads to (2.16).
Step 2. Consider the case T > ε0 . To proceed further, we need the following lemma.
Lemma 5.1. Let τ ≤ T be a stopping time. Let u ∈ Hγ+2,α
p,θ,0 (τ ), and
du(t) = f (t)dt + g k (t)dwtk .
Then there exists a unique ũ ∈ Hγ+2,α
p,θ,0 (T ) such that ũ(t) = u(t) for t ≤ τ (a.s) and, on (0, T ),
dũ = (ψ 2α ∆ũ(t) + f˜(t))dt + g k It≤τ dwtk ,
(5.43)
where f˜ = (f (t) − ψ 2α ∆u(t))It≤τ . Furthermore,
kũkHγ+2,α (G,T ) ≤ N kukHγ+2,α (G,τ ) ,
p,θ
p,θ
(5.44)
where N is independent of u and τ .
Proof. Note that f˜ ∈ ψ −1+2α Hγp,θ (G, T ) and gIt≤τ ∈ ψ α Hγ+1
p,θ (G, T ), so that, by Theorem 2.8,
equation (5.43) has a unique solution ũ ∈ Hγ+2,α
p,θ,0 (G, T ) and (5.44) holds. To show that ũ(t) =
u(t) for t ≤ τ , notice that, for t ≤ τ , the function v(t) = ũ(t) − u(t) satisfies the equation
dv = ψ 2α ∆v dt,
Theorem 2.8 shows that v(t) = 0 for t ≤ τ (a.e).
581
v(0, ·) = 0.
Now, to complete the proof, we repeat the arguments in [5]. Take an integer M ≥ 2 such that
T /M ≤ ε0 , and denote tm = T m/M . Assume that, for m = 1, 2, ..., M − 1, we have the estimate
(2.16) with tm in place of τ (and N depending only on d, p, δ0 , K and T ). We are going to use
the induction on m. Let um ∈ H1,α
p,θ,0 be the continuation of u on [tm , T ], which exists by Lemma
5.1 with γ = −1 and τ = tm . Denote vm := u − um , then (a.s) for any t ∈ [tm , T ], φ ∈ C0∞ (G)
(since dum = ψ 2α ∆um dt on [tm , T ] and vm (tm , ·) = 0)
Z t
i
(ψ 2α aij vmxj + ψ α b̄i vm + f¯m
, φxi )(s)ds
(vm (t), φ) = −
tm
+
Z
t
tm
(ψ α bi vmxi + cvm + fm , φ)(s)ds +
where
Z
t
tm
k
(ψ α σ ik vmxi + ν k vm + gm
, φ)(s)dwsk ,
i
= ψ 2α (aij − δ ij )umxj + ψ α b̄i um + f¯i ,
f¯m
fm = ψ α bi umxi + cum + f,
k
gm
= ψ α σ ik umxi + ν k um + g k .
Next instead of random processes on [0, T ] we consider processes given on [tm , T ] and, in a
γ
natural way, introduce spaces Hγ,α
p,θ (G, [tm , T ]), Lp,θ (G, [tm , t]), Hp,θ (G, [tm , T ]). Then we get a
counterpart of the result of step 1 and conclude that
Z tm+1
E
kψ −1 (u − um )(s)kpH 1 (G) ds
p,θ
tm
≤ NE
+N E
Z
Z
tm+1
tm
tm+1
tm
i
kψ −2α f¯m
(s)kpLp,θ (G) ds
kψ 1−2α fm (s)kpH −1 (G) + kψ −α gm (s)kpLp,θ (G) ds.
p,θ
Thus by the induction hypothesis,
Z tm+1
E
kψ −1 u(s)kpH 1
p,θ
0
+N E
Z
tm+1
tm
ds ≤ N E
(G)
Z
0
T
kψ −1 um (s)kpH 1
p,θ (G)
kψ −1 (u − um )(s)kpH 1
p,θ (G)
≤ N (kψ −2α f¯i kpLp,θ (G,tm+1 ) + kψ 1−2α f kpH−1 (G,t
p,θ
m+1 )
ds
ds
+ kψ α gkpLp,θ (G,tm+1 ) ).
We see that the induction goes through and thus the theorem is proved.
References
[1] D.G. Aronson and P. Besala, Parabolic equations with unbounded coefficients, J. Differential Equations, 3 (1967), no.1, 1-14. MR208160
[2] D. Gilbarg and N.S. Trudinger, “Elliptic partial differential equations of second order”, 2d
ed., Springer Verlag, Berlin, 1983. MR737190
582
[3] K. Kim, On stochastic partial differential equations with variable coefficients in C 1 domains, Stochastic process. Appl., 112 (2004), no.2, 261-283. MR2073414
[4] K. Kim, On Lp -theory of stochastic partial differential equations of divergence form in C 1
domains, Probab. Theory Relat. Fields 130 (2004), no.4, 473-492. MR2102888
[5] K. Kim and N.V. Krylov, On stochastic partial differential equations with variable coefficients in one dimension, Potential Anal., 21 (2004), no.3, 203-239.
[6] K. Kim and N.V. Krylov, On the Sobolev space theory of parabolic and elliptic equations
in C 1 domains, SIAM J. Math. Anal., 36 (2004), no.2, 618-642. MR2111792
[7] N.V. Krylov, Some properties of Traces for Stochastic and Deterministic Parabolic
Weighted Sobolev Spaces, Journal of Functional Analysis, 183 (2004), 1-41.
[8] N.V. Krylov, An analytic approach to SPDEs, pp. 185-242 in Stochastic Partial Differential
Equations: Six Perspectives, Mathematical Surveys and Monographs, 64 (1999), AMS,
Providence, RI. MR1661761
[9] N.V. Krylov, Weighted Sobolev spaces and Laplace equations and the heat equations in a
half space, Comm. in PDEs, 23 (1999), no.9-10, 1611-1653.
[10] N.V. Krylov and S.V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients
in a half space, SIAM J. on Math. Anal., 31 (1999), no.1, 19-33. MR1720129
[11] O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Utal’stseva, Linear and quasilinear
parabolic equations, Am. Math.Soc. (1968).
[12] S.V. Lototsky, Linear stochastic parabolic equations, degenerating on the boundary of a
domain, Electronic Journal of Probability, 6 (2001), no.24, 1-14. MR1873301
[13] S.V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for
degenerate elliptic equations, Methods and Applications of Analysis, 1 (2000), no.1, 195204. MR1796011
[14] S.V. Lototsky, Dirichlet problem for stochastic parabolic equations in smooth domains,
Stochastics and Stochastics Reports, 68 (1999), no.1-2, 145-175. MR1742721
[15] R. Mikulevicius and B. Rozovskii, A note on Krylov’s Lp -theory for systems of SPDEs,
Electron. J. Probab., 6 (2001), no. 12, 35 pp.
[16] O.A. Oleinik and E.V. Radkevič, Second order equations with nonnegative characteristic
form, AMS and Plenum press, Phode Island, New York, 1973
[17] E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes ,
Stochastics, 3(1979), 127-167. MR553909
[18] I.D. Pukal’skii, Estimates of solutions of parabolic equations degenerating on the boundary
of a domain, Mathematical Notes (Historical Archive), 22 (1977), no.4, 799-803.
[19] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Johann Amrosius Barth, Heidelberg, 1995
583
[20] M.I. Višik and V.V. Grušin, Boundary value problems for elliptic equations which are
degenerate on the boundary of the domain, Math. USSR Sb., 9 (1969), 423-454.
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Vol. 11 (2006), Paper no. 23, pages 563–584.
Journal URL
http://www.math.washington.edu/~ejpecp/
Parabolic SPDEs degenerating on the boundary of
non-smooth domain
Kyeong-Hun Kim
Department Of Mathematics
Korea university
1 anam-dong sungbuk-gu, Seoul
South Korea 136-701
email : kyeonghun@korea.ac.kr
Abstract
Degenerate stochastic partial differential equations of divergence and non-divergence forms
are considered in non-smooth domains. Existence and uniqueness results are given in
weighted Sobolev spaces, and Hölder estimates of the solutions are presented
Key words: SPDEs degenerating on the boundary, Weighted Sobolev spaces
AMS 2000 Subject Classification: Primary 60H15, 35R60.
Submitted to EJP on October 28 2005, final version accepted June 2 2006.
563
1
Introduction
We are dealing with an Lp -theory of parabolic stochastic partial differential equations (SPDEs)
of the types
(1.1)
du = (aij uxi xj + bi uxi + cu + f ) dt + (σ ik uxi + ν k u + g k ) dwtk
k
k
k
ik
i
i
i
ij
¯
(1.2)
du = (Di (a uxj + b̄ u + f ) + b uxi + cu + f ) dt + (σ uxi + ν u + g ) dwt
considered for t > 0 and x ∈ G. Here wtk are independent one-dimensional Wiener processes
and G is a bounded domain in Rd .
In this article we assume that the equations have the ”degeneracy α” near ∂G : ∃δ0 , K > 0 such
that for any λ ∈ Rd ,
δ0 ρ2α (x)|λ|2 ≤ (aij (t, x) − αij (t, x))λi λj ≤ Kρ2α (x)|λ|2
(1.3)
P
where ρ(x) := dist(x, ∂G) and αij := 21 k σ ik σ jk . Note that if α = 0 then the equations are
uniformly nondegenerate. In this case, unique solvability of the equations in appropriate Banach
spaces has been widely studied in many articles. See, for instance, [3], [4], [5], [8], [10], [14], [15]
and [17].
Our motivation of considering SPDEs with such degeneracy comes from several articles related to
PDEs with different types of degeneracies. We refer to [16], [19] and [20] for degenerate elliptic
equations. For parabolic PDEs we refer to [1], [18] (and references therein), where interior
Schauder estimates for equations with the degeneracy α < 1/2 were established.
An Lp -theory of equation (1.1) with the degeneracy α = 1 can be found in [12]. In this article,
we extend the results in [12]. We prove the unique solvability of equations (1.1) and (1.2)
with arbitrary degeneracy α ∈ [1, ∞) in appropriate Sobolev spaces. Also we give some Hölder
estimates of the solutions.
One of main applications of the theory of SPDEs is a nonlinear filtering problem. Consider a
pair of diffusion processes (Xt , Yt ) ∈ Rd × Rd1 −d ,
dXt = ρα (Xt )b(t, Xt , Yt )dt + ρα (Xt )r(t, Xt , Yt )dWt ,
dYt = B(t, Xt , Yt )dt + R(t, Yt )dWt ,
X(0) = X0
Y (0) = Y0 ,
where Wt is d1 -dimensional Wiener process and b, r, B, R are Lipschitz continuous matrices.
The nonlinear filtering problem is computing the conditional density πt of Xt given by the
observations {Ys : s ≤ t}. It was shown in [8] that when α = 0, there exists a conditional density
πt and πt satisfies a SPDE of type (1.1). Based on our Lp -theory, one can easily construct the
corresponding results when α ≥ 1. The motivations of considering the case α > 0 were discussed
at length in [12]. We only mention that usually the process Xt evolves in a bounded region
due to, for instance, mechanical restrictions, and therefore the above model is suitable when the
process Xt stays in the bounded domain. Note that since ρα (α ≥ 1) is Lipschitz continuous
in Rd (ρ(x) := 0 if x 6∈ G), by the unique solvability of the above SDE, if X0 is in G then the
process Xt never cross the boundary of G.
Here are notations used in the article. As usual Rd stands for the Euclidean space of points
x = (x1 , ..., xd ) and Br (x) := {y ∈ Rd : |x−y| < r}. For i = 1, ..., d, multi-indices β = (β1 , ..., βd ),
βi ∈ {0, 1, 2, ...}, and functions u(x) we set
uxi = ∂u/∂xi = Di u,
Dβ u = D1β1 · ... · Ddβd u,
564
|β| = β1 + ... + βd .
We also use the notation Dm for a partial derivative of order m with respect to x.
The author is sincerely grateful to the referee for giving several useful comments.
2
Main results
Let (Ω, F, P ) be a complete probability space, and {Ft , t ≥ 0} be an increasing filtration of σfields Ft ⊂ F, each of which contains all (F, P )-null sets. By P we denote the predictable σ-field
generated by {Ft , t ≥ 0} and we assume that on Ω we are given independent one-dimensional
Wiener processes wt1 , wt2 , ..., each of which is a Wiener process relative to {Ft , t ≥ 0}.
Choose and fix a smooth function ψ such that ψ(x) ∼ ρ(x) (see (2.9)). We rewrite equations
(1.1) and (1.2) in the following forms.
du = (ψ 2α aij uxi xj + ψ α bi uxi + cu + f ) dt
+ (ψ α σ ik uxi + ν k u + g k ) dwtk ,
and
(2.4)
du = (Di (ψ 2α aij uxj + ψ α b̄i u + f¯i ) + ψ α bi uxi + cu + f ) dt
+ (ψ α σ ik uxi + ν k u + g k ) dwtk ,
(2.5)
Here, i and j go from 1 to d, and k runs through {1, 2, ...}. The coefficients aij , b̄i , bi , c, σ ik , ν k
and the free terms f¯i , f, g k are random functions depending on t and x. Throughout the article,
for functions defined on Ω × [0, T ] × G, the argument ω ∈ Ω will be omitted.
To describe the assumptions of f¯i , f and g we use Sobolev spaces introduced in [8], [9] and [13].
If θ ∈ R and n is a nonnegative integer, then
Hpn = Hpn (Rd ) = {u : u, Du, ..., Dn u ∈ Lp },
0
Lp,θ (G) := Hp,θ
(G) = Lp (G, ρθ−d dx),
n
Hp,θ
(G) := {u : u, ρux , ..., ρn Dn u ∈ Lp,θ (G)}.
(2.6)
In general, by Hpγ = Hpγ (Rd ) = (1−∆)−γ/2 Lp we denote the space of Bessel potential. We define
kukHpγ = k(1 − ∆)γ/2 ukLp .
γ
The space Hp,θ
(G) is defined as the set of all distributions u on G such that
kukpH γ (G)
p,θ
:=
∞
X
enθ kζ−n (en ·)u(en ·)kpH γ < ∞,
p
n=−∞
where {ζn : n ∈ Z} is a sequence of smooth functions such that
X
|Dm ζn (x)| ≤ N (m)emn ,
ζn ≥ const > 0,
n
ζn ∈ C0∞ (Gn ),
Gn := {x ∈ G : e−n−1 < ρ(x) < e−n+1 }.
565
(2.7)
(2.8)
If Gn is empty set, then we put ζn = 0. One can construct the function ζn , for instance, by
mollifying the indicator function of Gn . It is known that up to equivalent norms the space
γ
Hp,θ
(G) and its norm are independent of {ζn } (see Lemma 2.1(iv)).
We also use the above notations for ℓ2 -valued functions g = (g1 , g2 , ...). We define
kgkHpγ = kgkHpγ (ℓ2 ) = k|(1 − ∆)γ/2 g|ℓ2 kLp .
kgkH γ
p,θ (G)
=
∞
X
enθ kζ−n (en ·)g(en ·)kpH γ .
p
n=−∞
Fix a smooth function ψ in G such that
sup |ρ(x)m Dm+1 ψ(x)| < ∞,
x
ρ(x) ≤ N ψ(x) ≤ N ρ(x),
P
For instance one can take ψ(x) = n e−n ζn (x).
∀x ∈ G.
(2.9)
γ
In the following lemma we collect some properties of Hp,θ
(G) (see [9] and [13] for detail). For
ν ∈ (0, 1], we denote
|u|C(X) = sup |u(x)|,
[u]C ν (X) = sup
X
x6=y
|u(x) − u(y)|
.
|x − y|ν
Lemma 2.1. (i) Assume that γ − d/p = m + ν for some m = 0, 1, ... and ν ∈ (0, 1]. Let i, j be
γ
multi-indices such that |i| ≤ m, |j| = m. Then for any u ∈ Hp,θ
(G), we have
ψ |i|+θ/p Di u ∈ C(G),
ν
ψ m+ν+θ/p Dj u ∈ Cloc
(G),
|ψ |i|+θ/p Di u|C(G) + [ψ m+ν+θ/p Dj u]C ν (G) ≤ N kukH γ
p,θ (G)
.
γ
γ−1
γ
(ii) ψD, Dψ : Hp,θ
(G) → Hp,θ
(G) are bounded linear operators, and for any u ∈ Hp,θ
(G)
kukH γ
p,θ (G)
kukH γ
p,θ (G)
≤ N kψux kH γ−1 (G) + N kukH γ−1 (G) ≤ N kukH γ
p,θ
p,θ (G)
p,θ
≤ N k(ψu)x kH γ−1 (G) + N kukH γ−1 (G) ≤ N kukH γ
p,θ
,
p,θ (G)
p,θ
(2.10)
.
γ
γ
(iii) For any ν, γ ∈ R, ψ ν Hp,θ
(G) = Hp,θ−pν
(G) and
kukH γ
p,θ−pν (G)
≤ N kψ −ν ukH γ
p,θ (G)
≤ N kukH γ
p,θ−pν (G)
.
(iv) Let {ξn } be a sequence of C0∞ (G) functions such that
|Dm ξn | ≤ N enm ,
supp ξn ⊂ {x ∈ G : e−n−k0 < ρ(x) < e−n+k0 }
γ
for some k0 > 0. Then for any u ∈ Hp,θ
(G)
X
kξ−n (en x)u(en x)kpH γ
And, if in addition
P
p,θ (G)
n
≤ N kukpH γ
p,θ (G)
≥ δ > 0, then
X
kξ−n (en x)u(en x)kpH γ
kukpH γ (G) ≤ N
.
n ξn (x)
p,θ
p,θ (G)
n
566
.
(2.11)
| τ ]] = {(ω, t) : 0 <
Now we define stochastic Banach spaces. For any stopping time τ , denote (0,
t ≤ τ (ω)},
γ
| τ ]], P, Hpγ ),
| τ ]], P, H (G)),
Hγp (τ ) = Lp ( (0,
Hγp,θ (G, τ ) = Lp ( (0,
p,θ
L... (...) = H0... (...),
γ,α
Up,θ
(G) = ψ
Upγ = Lp (Ω, F0 , Hpγ−2/p ),
− p2 (1−α)+1
γ−2/p
Lp (Ω, F0 , Hp,θ
(G)).
γ+2,α
Definition 2.2. We write u ∈ Hγ+2,α
(G, τ ) if u ∈ ψHγ+2
(G) and for
p,θ
p,θ (G, τ ), u(0, ·) ∈ Up,θ
some f ∈ ψ −1+2α Hγp,θ (G, τ ), g ∈ ψ α Hγ+1
p,θ (G, τ, ℓ2 )
du = f dt + g k dwtk ,
(2.12)
in the sense of distribution. In other words, for any φ ∈ C0∞ (G), the equality
(u(t, ·), φ) = (u(0, ·), φ) +
Z
t
(f (s, ·), φ) ds +
0
∞ Z
X
k=1
t
0
(g k (s, ·), φ) dwsk
holds for all t ≤ τ with probability 1. In this situation we also write f = Du, g = Su. Let
γ+2,α
Hγ+2,α
(G, τ ) ∩ {u : u(0, ·) = 0}.
p,θ,0 (G, τ ) = Hp,θ
The norm in Hγ+2,α
(G, τ ) is introduced by
p,θ
kukHγ+2,α (G,τ ) = [|u|]Hγ+2,α (G,τ ) + ku(0, ·)kU γ+2,α (G) ,
p,θ
p,θ
p,θ
where
[|u|]Hγ+2,α (G,τ ) := kψ −1 ukHγ
p,θ (G,τ )
p,θ
ku(0, ·)kp γ+2,α
Up,θ
+ kψ 1−2α DukHγ
p,θ (G,τ )
2
(G)
= Ekψ p
(1−α)−1
u(0, ·)kp
+ kψ −α SukHγ+1 (G,τ ) ,
p,θ
γ+2−2/p
Hp,θ
(G)
.
Remark 2.3. Up to equivalent norms, the space Hγ+2,α
(G, τ ) is independent of the choice of ψ,
p,θ
−1
and for instance the norm kψ ukHγ+2 (G,τ ) can be replaced by kukHγ+2 (G,τ ) . Also note that if
p,θ
p,θ−p
u ∈ ψHγ+2
p,θ (G, τ ), then by Lemma 2.1
ψ 2α ∆u ∈ ψ −1+2α Hγp,θ (G, τ ),
ψ α Du ∈ ψ α Hγ+1
p,θ (G, τ ).
Thus considering equation (2.4), we find that the spaces for Du and Su are defined naturally.
To state our assumptions on the coefficients, we take some notations from [2]. Denote ρ(x, y) =
ρG (x, y) = ρ(x) ∧ ρ(y). For δ ∈ (0, 1), and k = 0, 1, 2, ..., define
(0)
(0)
[f ]k = [f ]k,G = sup ρk (x)|Dk f (x)|,
x∈G
(0)
(0)
[f ]k+δ = [f ]k+α,G = sup ρk+α (x, y)
x,y∈G
|β|=k
567
|Dβ f (x) − Dβ f (y)|
,
|x − y|α
(0)
k
X
(0)
|f |k = |f |k,G =
(0)
[f ]j,G ,
(0)
(0)
(0)
(0)
|f |k+α = |f |k+α,G = |f |k,G + [f ]k+α,G .
j=0
Dβ f
By
we mean either classical derivatives or Sobolev ones and in the latter case sup’s in the
above are understood as ess sup’s. We also use the same notations for ℓ2 -valued functions.
Fix a function δ0 (τ ) ≥ 0 defined on [0, ∞) such that δ0 (τ ) > 0 unless τ ∈ {0, 1, 2, ...}. For τ ≥ 0
define
τ + = τ + δ0 (τ ),
and fix some constants
δ0 , K ∈ (0, ∞),
γ ∈ R.
Assumption 2.4. (i) For each x ∈ G, the coefficients aij (t, x), b̄i (t, x), bi (t, x) c(t, x), σ ik (t, x)
and ν k (t, x) are predictable functions of (ω, t).
(ii) For any x, t, ω and λ ∈ Rd ,
δ0 |λ|2 ≤ (aij (t, x) − αij (t, x))λi λj ≤ K|λ|2 ,
where αij =
1
2
P
k
(2.13)
σ ik σ jk .
Assumption 2.5. For any ε > 0, there exists δ = δ(ε) > 0 such that
sup(|aij (t, x) − aij (t, y)| + |σ i (t, x) − σ i (t, y)|ℓ2 ) ≤ ε
ω,t
whenever x, y ∈ G and |x − y| ≤ δ(ε)ρ(x, y).
Assumption 2.6. For any t > 0 and ω ∈ Ω,
(0)
(0)
(0)
|aij (t, ·)||γ|+ + |ψ 1−α bi (t, ·)||γ|+ + |ψ 2(1−α) c(t, ·)||γ|+
(0)
(0)
+|σ i (t, ·)||γ+1|+ + |ψ 1−α ν(t, ·)||γ+1|+ ≤ K.
Remark 2.7. Assumption 2.5 is much weaker than uniform continuity of aij and σ i . For instance,
let G = (0, 1) and a(t, x) = 2 + sin(ln x(1 − x)). Then one can easily check that a satisfies
Assumptions 2.5 and 2.6 for any γ ∈ R.
Here are our main results. From this point on we assume that
τ ≤ T,
α ∈ [1, ∞),
p ∈ [2, ∞).
Theorem 2.8. Let Assumptions 2.4, 2.5 and 2.6 be satisfied. Then
γ+2,α
(i) for any f ∈ ψ −1+2α Hγp,θ (G, τ ), g ∈ ψ α Hγ+1
(G), equation (2.4) with
p,θ (G, τ ) and u0 ∈ Up,θ
initial data u0 admits a unique solution u (in the sense of distribution) in the class Hγ+2,α
(G, τ ),
p,θ
(ii) for this solution
kukp γ+2,α
Hp,θ
(G,τ )
≤ N (kψ 1−2α f kpHγ
p,θ (G,τ )
+ kψ −α gkp γ+1
Hp,θ (G,τ )
where the constant N depends only on d, γ, p, θ, δ0 , K and T .
568
+ ku0 kp γ+2,α
Up,θ
(G)
),
(2.14)
Note that in the following theorem Assumption 2.6 is not assumed.
Theorem 2.9. Let Assumptions 2.4 and 2.5 be satisfied, and
|ψ 1−α b̄i | + |ψ 1−α bi | + |ψ 2(1−α) c| + |ψ 1−α ν| ≤ K,
∀ω, t, x.
(2.15)
Then
1,α
α
(i) for any f¯i ∈ ψ 2α Lp,θ (G, τ ), f ∈ ψ −1+2α H−1
p,θ (G, τ ), g ∈ ψ Lp,θ (G, τ ) and u0 ∈ Up,θ (G),
equation (2.5) with initial data u0 admits a unique solution u (in the sense of distribution) in
the class H1,α
p,θ (G, τ ),
(ii) for this solution
kukp 1,α
Hp,θ (G,τ )
≤ N (kψ −2α f¯kpLp,θ (G,τ ) + kψ 1−2α f kpH−1 (G,τ )
p,θ
+ kψ
−α
gkpLp,θ (G,τ )
+
ku0 kp 1,α ),
Up,θ (G)
(2.16)
where the constant N depends only on d, γ, p, θ, δ0 , K and T .
Now we state the regularity of the solutions in terms of Hölder continuity in time and space, both
inside the domain and near the boundary. The following results are immediate consequences of
Lemma 2.1, Remark 3.2 and Theorem 3.3.
Corollary 2.10. Let u ∈ Hγ+2,α
(G, τ ) be the solution of Theorem 2.8 or Theorem 2.9. Let
p,θ
2/p < µ < β < 1,
γ + 2 − β − d/p = m + ν,
for some m = 0, 1, ..., ν ∈ (0, 1]. Then for any multi-indices i, j such that |i| ≤ m, |j| = m
E
|ψ |i|−1+θ/p Di (u(t) − u(s))|pC(G)
sup
|t − s|pµ/2−1
0≤s 0.
(3.23)
If k ≤ m and en x ∈ suppζ−n (e·), then (since ρ(en x) ∼ en ),
(0)
|enk (Dk a)(en x)| ≤ N ρk (en x)|(Dk a)(en x)| ≤ N |a||γ|+ .
(3.24)
Obviously, (3.23) and (3.24) prove (3.22). Next let δ 6= 0. To show
|Dm a(en x)ζ−n (en x) − Dm a(en y)ζ−n (en )| ≤ N |x − y|δ , ∀x, y ∈ Rd ,
we may assume that |x − y| ≤ e−4 and en x ∈ suppζ−n (e·). In this case, en y ∈ B̄e−4+n (en x) ⊂ G
and ρ(en x) ∼ ρ(en x, en y) ∼ en . Thus, due to (3.23),
|Dm a(en x)ζ−n (en x) − Dm a(en y)ζ−n (en )|
≤N
X
ρk (en x, en y)|(Dk a)(en x) − (Dk a)(en y)||Dm−k (ζ−n (en x))|
k≤m
+N
X
enk |(Dk a)(en y)||Dm−k (ζ−n (en x)) − Dm−k (ζ−n (en y))|
k≤m
(0)
≤ N |a||γ|+ (e−nδ |en x − en y|δ + |x − y|) ≤ N |x − y|δ .
The lemma is proved.
570
Remark 3.2. Let θ1 ≤ θ2 . By Lemmas 2.1 and 3.1
kukH γ
p,θ2 (G)
≤ N kψ (θ2 −θ1 )/p ukH γ
p,θ1 (G)
≤ N kukH γ
p,θ1 (G)
.
Consequently, if α1 ≤ α2 then
kukHγ,α1 (G,τ ) ≤ N kukHγ,α2 (G,τ ) .
p,θ
p,θ
The following results are due to Lototsky ([12]).
Theorem 3.3. (i) For any t ≤ T ,
kψ
−1
ukp γ+1
Hp,θ (G,t)
≤ N (d, γ, p, T )
Z
t
0
kukp γ+2,1
Hp,θ
(G,s)
ds.
(ii) Let
2/p < µ < β < 1.
Then
Ekukp µ/2−1/p
C
γ+2−β
([0,τ ],Hp,θ−p
(G))
≤ N (µ, β, d, γ, p, T )kukp γ+2,1
Hp,θ
(G)
.
We choose and fix smooth functions ξn such that |Dm ξn | ≤ N (m)enm , suppξn ⊂ (Gn−1 ∪ Gn ∪
Gn+1 ) and ξn = 1 on the support of ζn .
Lemma 3.4. Let Assumptions 2.4(ii) and 2.5 be satisfied. By I we denote d ×d identity matrix.
Define
−2nα 2α n
2
2
aij
ψ (e x)ξ−n
(en x)aij (e2n(1−α) t, en x) + (1 − ξ−n
(en x))I,
n (t, x) = e
σnik (t, x) = e−nα ψ α (en x)ξ−n (en x)σ ik (e2n(1−α) t, en x).
Then
(i) For any λ ∈ Rd ,
ik jk i j
4α
2
e−4α δ0 |λ|2 ≤ (aij
n − 1/2σn σn )λ λ ≤ e K|λ| .
(ii) For any ε > 0, there exists δ = δ(ǫ) > 0 such that
ij
i
i
sup sup(|aij
n (t, x) − an (t, y)| + |σn (t, x) − σn (t, y)|) < ε,
n
ω,t
whenever x, y ∈ Rd and |x − y| < δ.
(iii)
i
sup sup(|aij
n (t, ·)|B |γ|+ + |σn |B |γ+1|+ ) < ∞.
n
ω,t
571
(3.25)
Proof. (i) is obvious and (3.25) follows from the same arguments as in the proof of Lemma 3.1.
Thus we only give a proof of the second assertion. Let δ ≤ e−4 and |x − y| < δ. Without loss of
generality, we assume that ξ−n (en x) 6= 0. Observe that
en y ∈ B̄ := B̄en δ (en x) ⊂ G,
|en x − en y| ≤ δen ≤ N0 δρ(en x, en y),
and for any z ∈ Ben δ (en x) we have ρ(z) ∼ en . Thus,
|ξ−n (en x) − ξ−n (en y)| ≤ |x − y|en sup |D(ξ−n )(z)| ≤ N1 |x − y|,
z
|ψ 2α (en x) − ψ 2α (en y)| ≤ sup |Dψ 2α (z)||en x − en y| ≤ N e2nα |x − y|,
z∈B
and
|e−2nα ψ 2α (en x)a(en x)ξ−n (en x) − e−2nα ψ 2α (en y)a(en y)ξ−n (en y)|
≤ e−2nα ψ 2α (en x)ξ−n (en x)|a(en x) − a(en y)|
+|a(en y)|e−2nα ψ 2α (en x)|ξ−n (en x) − ξ−n (en y)|
+|a(en y)ξ−n (en y)|e−2nα |ψ 2α (en x) − ψ 2α (en y)|
≤ N2 (|a(en x) − a(en y)| + δ + δ).
Note that the constant Ni are independent of x, y and n. So, if ε > 0 is given, then it is
enough to take δ > 0 such that (N1 + 2N2 )δ < ε/2 and N2 |a(t, x) − a(t, y)| ≤ ε/3 whenever
|x − y| < N0 δρ(x, y).
We handle σni similarly. The lemma is proved.
The following lemma is taken from [13].
Lemma 3.5. Let {φk : k = 1, 2, ...} be a collection of C0∞ (G) functions such that for each m > 0
X
sup
ρm (x)|Dm φk (x)| ≤ M (m) < ∞.
x∈G k
Then there exists a constant N = N (d, γ, M ) such that for any f ∈ Hγp,θ (G),
X
kφk f kpH γ
p,θ (G)
k
If in addition
X
≤ N kf kpH γ
p,θ (G)
.
|φk (x)|p ≥ c > 0,
k
then
kf kpH γ
p,θ (G)
≤ N (d, γ, M, c)
X
k
572
kφk f kpH γ
p,θ (G)
.
4
Proof of Theorem 2.8
As usual, we may assume that τ ≡ T (see [8]). For a moment, we assume that bi = c = ν k = 0.
ik
Take aij
n and σn from Lemma 3.4. Denote
cn := e−2n(1−α) ,
wtk (n) := e−n(1−α) wek2n(1−α) t
Then for each n, wtk (n) are independent one dimensional Wiener processes. By Theorem 5.1 in
γ+2
[8], for any f ∈ Hγp (cn T ), g ∈ Hγ+1
the equation
p (cn T ) and u0 ∈ Up
k
k
ik
du = (aij
n uxi xj + f )dt + (σn uxi + g )dwt (n) u(0, ·) = u0 ,
(4.26)
has a unique solution u ∈ Hγ+2
p (cn T ) and u satisfies
kukHγ+2
≤ N (kf kHγp (cn T ) + kgkHpγ+1 (cn T ) + ku0 kUpγ+2 ),
(cn T )
p
(4.27)
where the constant N depends only d, p, γ, δ0 , K, cn T, |an |B |γ|+ , |σn |B |γ+1|+ and uniform continuity of an , σn .
By Sn (f, g, u0 ) we denote the the solution of (4.26). Define
S̄n (f, g, u0 )(t, x) = Sn (f, g, u0 )(cn t, e−n x).
From now on, without loss of generality, we assume that
X
2
(x) = 1, ∀x ∈ G.
ζ−n
n
Remember the fact that a function v satisfies
dv = f dt + g k dwtk ,
t≤T
if and only if vc (t, x) := v(c2 t, cx) (c > 0) satisfies
dvc = c2 f (c2 t, cx) dt + cg(c2 t, cx)d(c−1 wck2 t ),
t ≤ c−2 T.
It follows that if u ∈ Hγ+2,α
(G, T ) is a solution of equation (2.4), then vn (t, x) :=
p,θ
−1
n
(ζ−n u)(cn t, e x) satisfies
k
k
k
ik
dvn = (aij
n vnxi xj + An u + fn )dt + (σn vnxi + Bn u + gn )dwt (n),
where
n
−1
n
2n
An u(t, x) := −2aij
n e uxi (cn t, e x)ζ−nxj (e x)
n
2n
−1
n
−aij
n e u(cn t, e x)ζ−nxi xj (e x),
n
n
Bnk u(t, x) := −σnik en u(c−1
n t, e x)ζ−nxi (e x),
fn (t, x) := e2n(1−α) f (e2n(1−α) t, en x)ζ−n (en x),
gnk (t, x) := en(1−α) g(e2n(1−α) t, en x)ζ−n (en x),
u0n := u0 (en x)ζ−n (en x).
573
(4.28)
Consequently,
n
vn (t, x) := (ζ−n u)(c−1
n t, e x) = Sn (An u + fn , Bn u + gn , u0n )
and
u=
X
ζ−n (ζ−n u) =
n
X
ζ−n S̄n (An u + fn , Bn u + gn , u0n ).
(4.29)
n
To proceed further, we need the following lemma.
γ+2
Lemma 4.1. Fix f ∈ ψ −1+2α Hγp,θ (G, T ), g ∈ ψ α Hγ+1
p,θ (G, T ) and u0 ∈ Up,θ (G). Then a sufficiently high power of the operator
X
R:u→
ζ−n S̄n (An u + fn , Bn u + gn , u0n )
(4.30)
n
is a contraction in Hγ+2,α
(G, T ) ∩ {u : u(0, ·) = u0 }, and the unique solution u ∈ Hγ+2,α
(G, T )
p,θ
p,θ
of (4.29) satisfies the estimate (2.14).
Proof. For simplicity, we use the notations Sn and S̄n instead of Sn (An u + fn , Bn u + gn , u0n )
and S̄n (An u + fn , Bn u + gn , u0n ), respectively.
Note that ζn ζm = 0 if |n − m| > 1. By Lemmas 3.5 and 3.1,
X
X
≤N
kζ−n S̄n kp γ+2,α
kRukp γ+2,α
≤N
kζ−n Rukp γ+2,α
Hp,θ
(G,T )
(G,T )
Hp,θ
n
Hp,θ
n
(G,T )
.
(4.31)
By definition,
kζ−n S̄n kp γ+2,α
Hp,θ
(G,T )
= kψ −1 ζ−n S̄n kp γ+2
Hp,θ (G,T )
+kψ −α S(ζ−n S̄n )kp γ+1
Hp,θ (G,T )
+ kψ 1−2α D(ζ−n S̄n )kpHγ
p,θ (G,T )
+ kζ−n u0 kp γ+2,α
Up,θ
(G)
.
Remember that ku(e±1 x)kH ν ∼ ku(x)kH ν and supn |ζ−n (en x)|B ν < ∞ for each ν > 0. Thus (cf.
Lemma 5.2 in [8]),
X
X
kζ−n S̄n kp γ+2
≤N
en(θ−p) kζ−n (en x)S̄n (t, en x)kp γ+2
n
Hp,θ−p (G,T )
≤N
X
Hp
n
en(θ−p+2−2α) kSn (t, x)kp γ+2
Hp
n
(cn T )
(T )
.
By writing the equation for ζ−n (en x)Sn , we find that v̄n := ζ−n S̄n satisfies
dv̄n = D(ζ−n S̄n ) dt + S(ζ−n S̄n ) dwtk
= [ψ 2α aij v̄nxi xj − 2ψ 2α aij (S̄n ζ−nxi )xj + ψ 2α aij S̄n ζ−nxi xj
2
−2ψ 2α aij uxi ζ−nxj ζ−n − ψ 2α aij uζ−nxi xj ζ−n + ζ−n
f ] dt
2
g] dwtk .
+[ψ α σ ik v̄nxi − ψ α σ ik S̄n ζ−nxi − ψ α σ ik uζ−nxi ζ−n + ζ−n
574
(4.32)
Thus, by Lemmas 2.1 and 3.1,
X
n
≤N
X
n
+N
X
n
+N
X
n
kψ 1−2α D(ζ−n S̄n )kpHγ
p,θ (G,T )
kψ(ζ−n S̄n )xx kpHγ
p,θ
p,θ
p,θ
X
Hp,θ (G,T )
Here,
X
X
kψ(S̄n ζ−nx )x kpHγ
p,θ (G,T )
X
n
X
n
Hp,θ−p (G,T )
+kψ −1 ukp γ+1
=
+N
(G,T )
+N
(G,T )
kζ−n S̄n kp γ+2
n
X
n
kψ −1 S̄n ψ 2 ζ−nxx kpHγ
kψ −1 uψ 2 ζ−nxx kpHγ
≤N
+N
(G,T )
+N
X
kux ψζ−nx kpHγ
kψ 1−2α f ζ−n kpHγ
p,θ (G,T )
kS̄n ζ−nx kp γ+1
Hp,θ (G,T )
n
+ N kψ 1−2α f kpHγ
p,θ (G,T )
Hp,θ (G,T )
em(θ−p) kS̄n (em x)em ζ−nx (em x)ζ−m (em x)kp γ+1
Hp
X
≤N
en(θ−p) kS̄n (t, x)(en x)kp γ+2
Hp
n
=N
X
en(θ−p+2−2α) kSn (t, x)kp γ+2
Hp
n
P
n kψ
p
−α S(ζ
−n S̄n )k γ+1
Hp,θ (G,T )
kRukp γ+2,α
Hp,θ
.
kζ−nx S̄n kp γ+1
n
n,m
We estimate
p,θ (G,T )
(G,T )
Hp,θ (G,T )
+kψ −α gkp γ+1
+
(T )
(cn T )
.
similarly, and conclude that
≤ N kψ −1 ukp γ+1
X
(T )
Hp,θ (G,T )
+ N kψ 1−2α f kpHγ
p,θ (G,T )
+ ku0 kp γ+2
Up,θ (G)
en(θ−p+2−2α) kSn (t, x)kp γ+2
Hp
n
(cn T )
.
(4.33)
Since G is bounded, we may assume that ζ−n = 0 for all n > 1. For each n ≤ 0, we have
cn T := e−2n(1−α) T ≤ T.
(4.34)
Thus by (4.27), there exists a constant N independent of n (due to Lemma 3.4 and (4.34)) such
that for each n ≤ 0,
kSn (t, x)kp γ+2
≤ N kAn u + fn kpHγ (c T )
Hp
+N kBn u +
(cn T )
p
gn kp γ+1
Hp (cn T )
575
+
n
N ku0n kp γ+2 .
Up
Also, by Lemma 2.1,
X
n≤0
=
X
n≤0
en(θ−p+2−2α) (kAn ukpHγ (c
p
nT )
+ kBn ukp γ+1
Hp
Hp
(T )
≤ N kψ −1 ukp γ+1
.
p
X
n≤0
+N
X
n≤0
+N
X
enθ kux (en x)en ζ−nx (en x)kpHγ (T )
p
en(θ−p) ku(en x)e2n ζ−nxx (en x)kpHγ (T )
p
en(θ−p) ku(en x)en ζ−nx (en x)kp γ+1
Hp
n≤0
≤ N kux kpHγ
+ N kukp γ+1
p,θ (G,T )
X
n≤0
Hp,θ−p (G,T )
en(θ−p+2−2α) (kfn kpHγ (c
≤N
p
X
n≤0
+N
nT )
+ kgn kp γ+1
Hp
(cn T )
)
en(θ+p(1−2α)) kf (t, en x)ζ−n (en x)kpHγ (T )
X
p
en(θ−pα) kg(t, en x)ζ−n (en x)kp γ+1
Hp
≤ N kψ 1−2α f kpHγ
p,θ (G,T )
X
(T )
Hp,θ (G,T )
n≤0
and
)
en(θ−p) k(An u)(cn t, x)|pHγ (T ) + k(Bn u)(cn t, x)kp γ+1
≤N
Similarly,
(cn T )
n(θ+p( p2 (1−α)−1))
e
+ N kψ −α gkp γ+1
(T )
Hp,θ (G,T )
,
ku0 (en x)ζ−n (en x)kp γ+2 ≤ N ku0 kp γ+2
Up
n
Up,θ (G)
.
Hence, coming back to (4.33), we get
kRukp γ+2,α
Hp,θ
(G,τ )
≤ N (kψ −1 ukp γ+1
Hp
+ kψ −α gkp γ+1
Hp
(T )
(T )
+ kψ 1−2α f kpHγ (T )
+ ku0 kp γ+2
Up,θ (G)
p
).
(4.35)
Note that H̄γ+2,α
(G, T ) := Hγ+2,α
(G, T ) ∩ {u : u(0, ·) = u0 } is a complete Banach space and
p,θ
p,θ
contains R0 (thus not empty), where
X
R0 :=
ζ−n S̄n (fn , gn , u0n ).
n
By (4.35) and Theorem 3.3, for any u, v ∈ H̄γ+2,α
(G, T ),
p,θ
Ru − Rv =
X
ζ−n S̄n (An (u − v), Bn (u − v), 0),
n
576
kRu − RvkHγ+2,α (G,T ) ≤ N kψ −1 (u − v)kp γ+1
Hp,θ (G,T )
p,θ
≤N
Z
T
ku − vkp γ+2,α
Hp,θ
0
(G,s)
ds.
(4.36)
(4.36) shows that there exists m0 > 0 such that Rm is a contraction in H̄γ+2,α
(G, T ), and all
p,θ
the assertions of the lemma follow from this and (4.35). For more technical details, we refer to
the proof of Theorem 6.4 in [8]. The lemma is proved.
Let u be the solution of (4.29). We will show that u satisfies equation (2.4). Obviously u(0, ·) =
u0 , and (by definition) for some f0 ∈ ψ −1+2α Hγp,θ (G, T ) and g0 ∈ ψ α Hγ+1
p,θ (G, T )
du = f0 dt + g0k dwtk .
Observe that u satisfies equation (2.4) with f¯ := f0 − ψ 2α aij uxi xj and ḡ k := g0k − ψ α σ ik uxi
instead of f and g k , respectively. By the above arguments (see (4.29))
X
u=
ζ−n S̄n (An u + f¯n , Bn u + ḡn , u0n ),
n
where f¯n , ḡn , u0n are defined from f¯, ḡ, u0 as in (4.28). Also,
X
0=
ζ−n S̄n (f˜n , g̃n , 0),
(4.37)
n
where f˜ = f − f¯, g̃ = g − ḡ and f˜n , g̃n are defined as before.
Define the operators Ān and B̄n such that
Ān u = 2aij ψ 2α uxi ζ−nxj − aij ψ 2α uζ−nxi xj ,
B̄n u = ψ α σ ik uζ−nxi .
From (4.37),
0=D
X
ζ−n S̄n (f˜n , g̃n , 0) = f˜ −
n
0=S
X
X
Ān S̄n (f˜n , g̃n , 0),
n
ζ−n S̄n (f˜n , g̃n , 0) = g̃ −
n
X
B̄n S̄n (f˜n , g̃n , 0).
n
Therefore, to show f˜ = g̃ k = 0, we only need to prove that a sufficiently high power of the
operator
X
X
R̄ : (f, g) → (
Ān S̄n (fn , gn , 0),
B̄n S̄n (fn , gn , 0))
n
is a contraction in
γ,α
Fp,θ
(G, T )
:=
n
ψ −1+2α Hγp,θ (G, T )
× ψ α Hγ+1
p,θ (G, T ).
By Lemma 3.5,
k
X
n
ψζ−nx S̄nx kpHγ
p,θ (G,T )
≤N
X
m,n
|m−n|≤1
577
2
kψζ−m
ζ−nx S̄nx kpHγ
p,θ (G,T )
=N
X
m,n
|m−n|≤1
≤N
X
kψ
2
kψζ−nx (S̄n ζ−m
)x − 2ψ −1 S̄n ζ−m ψ 2 ζ−mx ζ−nx kpHγ
p,θ (G,T )
−1
m,n
|m−n|≤1
ζ−m S̄n kp γ+1
Hp,θ (G,T )
Z
≤N
X
T
0
kζ−m S̄n kp γ+2,α
As in the proof of Lemma 4.1 (see (4.31) and (4.35)),
X
kζ−m S̄n kp γ+2,α
≤ N kψ 1−2α f kpHγ
Hp,θ
m,n
|m−n|≤1
We estimate other terms in
p,θ (G,s)
(G,s)
P
n Ān S̄n
and
kR̄(f, g)kpF γ (G,T )
p,θ
P
n B̄n S̄n
Z
≤N
T
Hp,θ
m,n
|m−n|≤1
+ N kψ −α gkp γ+1
ds.
Hp,θ (G,s)
.
similarly (actually much easily) and get
k(f, g)kpF γ
p,θ (G,s)
0
(G,s)
ds.
This shows that a sufficiently high power of R̄ is a contraction and f¯ = f, ḡ = g.
For general case (previously we assumed that bi = c = ν k = 0), having the method of continuity
in mind, we only show that (2.14) holds true given that a solution u ∈ Hγ+2,α
(G, T ) already
p,θ
exists. Let ū ∈ Hγ+2,α
(G, T ) be the solution of
p,θ
dū = ψ 2α ∆ū dt,
Then
kūkp γ+2,α
Hp,θ
ū(0, ·) = u0 .
≤ N ku0 kp γ+2
(G,T )
Up,θ (G)
.
Thus by considering u − ū, as usual, we may assume that u0 = 0.
By the previous results (when bi = c = ν k = 0),
kukp γ+2,α
Hp,θ
where
(G,T )
≤ N (kψ 1−2α f˜kpHγ
p,θ (G,T )
f˜ = ψ α bi uxi + cu + f,
By Lemma 3.1,
kψ 1−2α f˜kpHγ
p,θ (G,T )
≤ N kψ 1−2α f kpHγ
p,θ (G,T )
+ kψ −α g̃kp γ+1
Hp,θ (G,T )
g̃ i k = ν k u + g k .
+ kψ −α g̃kp γ+1
Hp,θ (G,T )
+ N kψ −α gkp γ+1
Hp,θ (G,T )
+ N kψ −1 ukp γ+1
Hp,θ (G,T )
Thus, by Theorem 3.3 for each t ≤ T ,
kukp γ+2,α
Hp,θ
+N kψ
−α
(G,T )
≤ N kψ 1−2α f kpHγ
p,θ (G,T )
gkp γ+1
Hp,θ (G,T )
),
+
Z
0
t
kukp γ+2,α
Hp,θ
(G,s)
ds.
This and Gronwall’s inequality lead to (2.14). The theorem is proved.
578
.
5
Proof of Theorems 2.9
Consider the operators
L0 u := ψ 2α ∆u = Di (ψ 2α uxi ) − 2αψ 2α−1 ψxi uxi ,
L1 u := Di (ψ 2α aij uxj + b̄i u) + bi uxi + cu,
Λk1 u : ψ α σ ik uxi + ν k u.
One can easily check that the coefficients of the operators Lλ := (1 − λ)L0 + λL1 and Λλ := λΛ1
satisfy Assumptions 2.4, 2.5 and (2.15). Also note that
kψ 1−2α f¯xi kH −1 (G) ≤ N kf¯i kLp,θ−2pα (G) ≤ N kψ −2α f¯i kLp,θ (G) .
p,θ
By Theorem 2.8, the equation
du = (L0 u + f¯xi i + f )dt + (Λk0 u + g k )dwtk
has a unique solution u ∈ H1,α
p,θ (G, τ ). Thus by the method of continuity, we only need to prove
that the estimate (2.16) holds true given that a solution u ∈ H1,α
p,θ (G, τ ) of equation (2.5) already
exists.
Again without loss of generality we assume that τ ≡ T and ζ−n = 0 for all n > 0. Also as in the
proof of Theorem 2.8, we assume that u0 = 0.
Step 1. We will show that there exists a constant ε0 = ε0 (d, p, δ0 , K) > 0 such that the theorem
holds true if T ≤ ε0 ≤ 1. As before, denote cn := e−2n(1−α) . By Lemma 2.1,
X
en(θ−p) ku(en x)ζ−n (en x)kpH1 (T )
kψ −1 ukpH1 (G,T ) ≤ N
p,θ
=N
X
p
n≤0
en(θ−p+2−2α) ku(e2n(1−α) t, en x)ζ−n (en x)kpH1 (cn T ) .
p
n≤0
(5.38)
Denote vn (t, x) = u(e2n(1−α) t, en x)ζ−n (en x). Then vn satisfies
i
¯i
dvn = (Di (aij
n vnxj + b̄n vn + fn ) + bn vnxi + cn vn + fn ) dt
+ (σnik vnxi + νnk vn + gnk ) dwtk (n),
where
ik
k
aij
n , σn , wt (n)
are defined as before and
n
b̄in (t, x) = en−2nα ψ α (en x)ξ−n (en x)b̄i (c−1
n t, e x),
n
bin (t, x) = en−2nα ψ α (en x)ξ−n (en x)bi (c−1
n t, e x),
n
n
cn (t, x) = e2n(1−α) c(c−1
n t, e x)ξ−n (e x),
n
n
νnk (t, x) = en(1−α) ν k (c−1
n t, e x)ξ−n (e x),
n
−1
n
n−2nα ¯i −1
n
f (cn t, en x)ζ−n (en x),
f¯ni (t, x) = −aij
n e ζ−nxj (e x)u(cn t, e x) + e
n
n(1−α) k −1
n
n
g (cn t, en x)ζ−n (en x),
gnk (t, x) = −σnik u(c−1
n t, e x)e ζ−nxi (e x) + e
579
(5.39)
−1
n
n
n
fn (t, x) = −en aij
n uxj (cn t, e x)e ζ−nxi (e x)
n
n
n
n−2nα ¯i −1
+b̄n u(c−1
f (cn t, en x)en ζ−nxi (en x)
n t, e x)e ζ−nxi (e x) + e
n
2n(1−α)
n
n
f (c−1
−bin en ζ−nxi (en x)u(c−1
n t, e x) + e
n t, e x)ζ−n (e x).
Note that ψ(en x) ∼ en on the support of ξ−n (en x). It follows from (2.15) that
sup sup (|b̄in | + bin | + |cn | + |νn |) < ∞.
n ω,t,x
By Theorem 2.12 in [4],
kvn kpH1 (cn T ) ≤ N (kf¯n kpLp (cn T ) + N kfn kpH−1 (c
p
p
nT )
+ kgn kpLp (cn T ) ).
(5.40)
−1
n
n
n
Actually, due to the term −en aij
n uxj (cn t, e x)e ζ−nxi (e x) of fn , (5.40) only yields
kψ −1 ukpH1
p,θ (G,T )
≤ N kψ −1 ukpH1
p,θ (G,T )
+ ....
Of course, this estimate is useless unless N < 1. The following argument below is to avoid
−1
n
n
n
estimating ken aij
.
n uxj (cn t, e x)e ζ−nxi (e x)kH−1
p (cn T )
Denote
n
−1
n
n
f˜n (t, x) = −en aij
n uxj (cn t, e x)e ζ−nxi (e x) ∈ Lp (cn T ).
By Theorem 5.1 in [8], the equation
du = (∆u + f˜n )dt,
u(0, ·) = 0
has a unique solution un ∈ H2p (cn T ), and (see Theorems 7.1 and 7.2 in [8])
kun kpH1 (cn T ) ≤ N (T )kf˜n kpLp (cn T ) ,
p
(5.41)
where N (T ) is independent of n (since cn T ≤ T ), and N (T ) ↓ 0 as T → 0.
v̂n := vn − un satisfies (5.39) with
¯n := f¯n + b̄n un + (aij − δ ij )unxi ,
fˆ
n
fˆn := fn − f˜n + bn unx + cn un ,
ĝn := gn + σ i unxi + νun
instead of f¯n , fn and gn , respectively. Thus by Theorem 2.12 in [4], there exists a constant N
depending only on d, p, δ0 and K (remember cn T ≤ T ≤ 1) such that
¯n kp
ˆ p
kv̂n kpH1 (cn T ) ≤ N (kfˆ
Lp (cn T ) + kfn kH−1 (c
p
p
nT )
+ kĝn kpLp (cn T ) ).
Consequently,
¯n kp
ˆ p
kvn kpH1 (cn T ) ≤ N (kfˆ
Lp (cn T ) + kfn kH−1 (c
p
p
nT )
+ kĝn kpLp (cn T ) + kun kH1p (cn T ) )
p
n
n
n
≤ N (T )ken ux (c−1
n t, e x)e ζ−nx (e x)kLp (cn T )
580
(5.42)
p
n
+N ken ζ−nx (en x)u(c−1
n t, e x)kLp (cn T )
p
n
n
n
n
+N ken(1−2α) f¯(c−1
n t, e x)[ζ−n (e x) + e ζ−nx (e x)]kLp (cn T )
p
n
n
+N ke2n(1−α) f (c−1
n t, e x)ζ−n (e x)kH−1 (c
p
nT )
p
n
n
+N ken(1−α) g(c−1
n t, e x)ζ−n (e x)kLp (cn T ) .
Coming back to (5.38), by Lemma 2.1, we get
kψ −1 ukpH1
p,θ (G,T )
≤ N kψ −1 ukpLp,θ (G,T ) + N kψ −2α f¯kpLp,θ (G,T )
+N kψ 1−2α f kpH−1 (G,T ) + N kψ −α gkpLp,θ (G,T ) + N N (T )kψ −1 ukpH1
p,θ (G,T )
p,θ
.
Now fix ε0 such that N N (T ) ≤ 1/2 for each T ≤ ε0 , then by Theorem 3.3 for each t ≤ T
kukp 1,α
Hp,θ (G,t)
≤N
Z
t
kukp 1,α
Hp,θ (G,s)
0
ds
+N (kψ −2α f¯kpLp,θ (G,T ) + kψ 1−2α f kpH−1 (G,T ) + kψ −α gkpLp,θ (G,T ) ).
p,θ
Gronwall’s inequality leads to (2.16).
Step 2. Consider the case T > ε0 . To proceed further, we need the following lemma.
Lemma 5.1. Let τ ≤ T be a stopping time. Let u ∈ Hγ+2,α
p,θ,0 (τ ), and
du(t) = f (t)dt + g k (t)dwtk .
Then there exists a unique ũ ∈ Hγ+2,α
p,θ,0 (T ) such that ũ(t) = u(t) for t ≤ τ (a.s) and, on (0, T ),
dũ = (ψ 2α ∆ũ(t) + f˜(t))dt + g k It≤τ dwtk ,
(5.43)
where f˜ = (f (t) − ψ 2α ∆u(t))It≤τ . Furthermore,
kũkHγ+2,α (G,T ) ≤ N kukHγ+2,α (G,τ ) ,
p,θ
p,θ
(5.44)
where N is independent of u and τ .
Proof. Note that f˜ ∈ ψ −1+2α Hγp,θ (G, T ) and gIt≤τ ∈ ψ α Hγ+1
p,θ (G, T ), so that, by Theorem 2.8,
equation (5.43) has a unique solution ũ ∈ Hγ+2,α
p,θ,0 (G, T ) and (5.44) holds. To show that ũ(t) =
u(t) for t ≤ τ , notice that, for t ≤ τ , the function v(t) = ũ(t) − u(t) satisfies the equation
dv = ψ 2α ∆v dt,
Theorem 2.8 shows that v(t) = 0 for t ≤ τ (a.e).
581
v(0, ·) = 0.
Now, to complete the proof, we repeat the arguments in [5]. Take an integer M ≥ 2 such that
T /M ≤ ε0 , and denote tm = T m/M . Assume that, for m = 1, 2, ..., M − 1, we have the estimate
(2.16) with tm in place of τ (and N depending only on d, p, δ0 , K and T ). We are going to use
the induction on m. Let um ∈ H1,α
p,θ,0 be the continuation of u on [tm , T ], which exists by Lemma
5.1 with γ = −1 and τ = tm . Denote vm := u − um , then (a.s) for any t ∈ [tm , T ], φ ∈ C0∞ (G)
(since dum = ψ 2α ∆um dt on [tm , T ] and vm (tm , ·) = 0)
Z t
i
(ψ 2α aij vmxj + ψ α b̄i vm + f¯m
, φxi )(s)ds
(vm (t), φ) = −
tm
+
Z
t
tm
(ψ α bi vmxi + cvm + fm , φ)(s)ds +
where
Z
t
tm
k
(ψ α σ ik vmxi + ν k vm + gm
, φ)(s)dwsk ,
i
= ψ 2α (aij − δ ij )umxj + ψ α b̄i um + f¯i ,
f¯m
fm = ψ α bi umxi + cum + f,
k
gm
= ψ α σ ik umxi + ν k um + g k .
Next instead of random processes on [0, T ] we consider processes given on [tm , T ] and, in a
γ
natural way, introduce spaces Hγ,α
p,θ (G, [tm , T ]), Lp,θ (G, [tm , t]), Hp,θ (G, [tm , T ]). Then we get a
counterpart of the result of step 1 and conclude that
Z tm+1
E
kψ −1 (u − um )(s)kpH 1 (G) ds
p,θ
tm
≤ NE
+N E
Z
Z
tm+1
tm
tm+1
tm
i
kψ −2α f¯m
(s)kpLp,θ (G) ds
kψ 1−2α fm (s)kpH −1 (G) + kψ −α gm (s)kpLp,θ (G) ds.
p,θ
Thus by the induction hypothesis,
Z tm+1
E
kψ −1 u(s)kpH 1
p,θ
0
+N E
Z
tm+1
tm
ds ≤ N E
(G)
Z
0
T
kψ −1 um (s)kpH 1
p,θ (G)
kψ −1 (u − um )(s)kpH 1
p,θ (G)
≤ N (kψ −2α f¯i kpLp,θ (G,tm+1 ) + kψ 1−2α f kpH−1 (G,t
p,θ
m+1 )
ds
ds
+ kψ α gkpLp,θ (G,tm+1 ) ).
We see that the induction goes through and thus the theorem is proved.
References
[1] D.G. Aronson and P. Besala, Parabolic equations with unbounded coefficients, J. Differential Equations, 3 (1967), no.1, 1-14. MR208160
[2] D. Gilbarg and N.S. Trudinger, “Elliptic partial differential equations of second order”, 2d
ed., Springer Verlag, Berlin, 1983. MR737190
582
[3] K. Kim, On stochastic partial differential equations with variable coefficients in C 1 domains, Stochastic process. Appl., 112 (2004), no.2, 261-283. MR2073414
[4] K. Kim, On Lp -theory of stochastic partial differential equations of divergence form in C 1
domains, Probab. Theory Relat. Fields 130 (2004), no.4, 473-492. MR2102888
[5] K. Kim and N.V. Krylov, On stochastic partial differential equations with variable coefficients in one dimension, Potential Anal., 21 (2004), no.3, 203-239.
[6] K. Kim and N.V. Krylov, On the Sobolev space theory of parabolic and elliptic equations
in C 1 domains, SIAM J. Math. Anal., 36 (2004), no.2, 618-642. MR2111792
[7] N.V. Krylov, Some properties of Traces for Stochastic and Deterministic Parabolic
Weighted Sobolev Spaces, Journal of Functional Analysis, 183 (2004), 1-41.
[8] N.V. Krylov, An analytic approach to SPDEs, pp. 185-242 in Stochastic Partial Differential
Equations: Six Perspectives, Mathematical Surveys and Monographs, 64 (1999), AMS,
Providence, RI. MR1661761
[9] N.V. Krylov, Weighted Sobolev spaces and Laplace equations and the heat equations in a
half space, Comm. in PDEs, 23 (1999), no.9-10, 1611-1653.
[10] N.V. Krylov and S.V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients
in a half space, SIAM J. on Math. Anal., 31 (1999), no.1, 19-33. MR1720129
[11] O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Utal’stseva, Linear and quasilinear
parabolic equations, Am. Math.Soc. (1968).
[12] S.V. Lototsky, Linear stochastic parabolic equations, degenerating on the boundary of a
domain, Electronic Journal of Probability, 6 (2001), no.24, 1-14. MR1873301
[13] S.V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for
degenerate elliptic equations, Methods and Applications of Analysis, 1 (2000), no.1, 195204. MR1796011
[14] S.V. Lototsky, Dirichlet problem for stochastic parabolic equations in smooth domains,
Stochastics and Stochastics Reports, 68 (1999), no.1-2, 145-175. MR1742721
[15] R. Mikulevicius and B. Rozovskii, A note on Krylov’s Lp -theory for systems of SPDEs,
Electron. J. Probab., 6 (2001), no. 12, 35 pp.
[16] O.A. Oleinik and E.V. Radkevič, Second order equations with nonnegative characteristic
form, AMS and Plenum press, Phode Island, New York, 1973
[17] E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes ,
Stochastics, 3(1979), 127-167. MR553909
[18] I.D. Pukal’skii, Estimates of solutions of parabolic equations degenerating on the boundary
of a domain, Mathematical Notes (Historical Archive), 22 (1977), no.4, 799-803.
[19] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Johann Amrosius Barth, Heidelberg, 1995
583
[20] M.I. Višik and V.V. Grušin, Boundary value problems for elliptic equations which are
degenerate on the boundary of the domain, Math. USSR Sb., 9 (1969), 423-454.
584