On the Lq(Lp)-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains

0.47 MB·English

by  Petru A. Cioica

On the L (L )-regularity and Besov smoothness of stochastic
q p
parabolic equations on bounded Lipschitz domains
∗
Petru A. Cioica, Kyeong-Hun Kim, Kijung Lee, Felix Lindner
December 4, 2012
3
1
0
2

n Abstract
a
J We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet

7 boundary condition on bounded Lipschitz domains Rd with both theoretical and numeri-
cal purpose. We use N.V. Krylov’s framework of stoOch⊆astic parabolic weighted Sobolev spaces
R] Hγ,q( ,T). The summability parameters p and q in space and time may differ. Existence and
p,θ O
uniqueness of solutions in these spaces is established and the Ho¨lder regularity in time is anal-
P
. ysed. Moreover,we prove a generalembedding of weighted Lp( )-Sobolev spaces into the scale
h ofBesovspacesBα ( ),1/τ =α/d+1/p,α>0.ThisleadstoOaHo¨lder-Besovregularityresult
t τ,τ O
a for the solution process. The regularity in this Besov scale determines the order of convergence
m
that can be achieved by certain nonlinear approximationschemes.
[

1
Keywords: Stochastic partial differential equations, Lipschitz domain, L (L )-theory, weighted
v q p
0 Sobolevspaces, Besov spaces,quasi-Banach spaces,embeddingtheorems,H¨older regularity intime,
8 nonlinear approximation, wavelets, adaptive numerical methods,squarerootof Laplacian operator.
1
1 AMS 2010 Subject Classification: Primary: 60H15; secondary: 46E35, 35R60.
.
1
0
3 1 Introduction
1
v: Let Rd be a bounded Lipschitz domain, T (0, ) and let (wk) , k N, be independent
Xi one-Odim⊆ensional standard Wiener processes defi∈ned o∞n a probabilityt st∈p[a0,cTe](Ω,∈ ,P). We are inter-
F
ested in theregularity ofthesolutions toparabolicstochastic partialdifferential equations (SPDEs,
r
a
for short) with zero Dirichlet boundary condition of the form
du = (aiju +f)dt+(σiku +gk)dwk on Ω [0,T] ,
xixj xi t × ×O
u= 0 on Ω (0,T] ∂ , (1.1)

× × O

u(0, ) = u on Ω , 
0
· ×O
∗ThisworkhasbeensupportedbytheDeutscheForschungsgemeinschaft (DFG,grantsDA360/13-1, DA360/13-
2, DA 360/11-1, DA 360/11-2, SCHI 419/5-1, SCHI 419/5-2) and a doctoral scholarship of the Philipps-Universit¨at
Marburg. The research of the third author has been also supported by Basic Science Research Program through
the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology
(2011-0005597).
1
where the indices i and j run from 1 to d and the index k runs through N = 1,2,... . Here and in
{ }
the sequel we use the summation convention on the repeated indices i,j,k. The coefficients aij and
σik depend on (ω,t) Ω [0,T]. The force terms f and gk depend on (ω,t,x) Ω [0,T] . By
∈ × ∈ × ×O
the nature of the problem, in particular by the bad contribution of the infinitesimal differences of
the Wiener processes, the second spatial derivatives of the solution may blow up at the boundary
∂ even if the boundary is smooth, see, e.g., [28]. Hence, a natural way to deal with problems
O
of type (1.1) is to consider u as a stochastic process with values in weighted Sobolev spaces on
that allow the derivatives of functions from these spaces to blow up near the boundary. This
O
approach has been initiated and developed by N.V. Krylov and collaborators, first as an L -theory
2
for smooth domains (see [28]), then as an L -theory (p 2) for the half space ([33, 34]), for
p
O ≥
smooth domains ([23, 27]), and for general bounded domains allowing Hardy’s inequality such as
bounded Lipschitz domains ([26]). Existence and uniqueness of solutions have been established
within specific stochastic parabolic weighted Sobolev spaces, denoted by Hγ ( ,T) in [26]. These
p,θ O
spaces consist of elements u˜ of the form du˜ = f˜dt + g˜kdwk, where u˜, f˜ and g˜k, considered as
t
stochastic processes with values in certain weighted L ( )-Sobolev spaces, are L -integrable w.r.t.
p p
O
P dt. We refer to Section 3 for the exact definition.
⊗
In this article we treat regularity issues concerning the solution u of problem (1.1) which arise,
besides others, in the context of adaptive numerical approximation methods.
The starting point of our considerations was the question whether we can improve the Besov
regularity results in [4] in time direction. In [4] the spatial regularity of u is measured in the scale
of Besov spaces
1 α 1
Bα ( ), = + , α > 0, ( )
τ,τ O τ d p ∗
where p 2 is fixed. Note that for α > (p 1)d/p the sumability parameter τ becomes less than
≥ −
one, so that in this case Bα ( ) is not a Banach space but a quasi-Banach space. It is a known
τ,τ O
result from approximation theory that the smoothness of a target function f L ( ) within the
p
∈ O
scale ( ) determines the rate of convergence that can be achieved by adaptive and other nonlinear
∗
approximation methods if the approximation error is measured in L ( ); see [5, Chapter 4], [13]
p
O
or the introduction of [4]. Based on the L -theory in [26], it is shown in [4] that the solution u to
p
problem (1.1) satisfies
1 α 1
u L (Ω [0,T], ,P dt;Bα ( )), = + , (1.2)
∈ τ × P ⊗ τ,τ O τ d p
for certain α > 0 depending on the smoothness of u , f and gk, k N. In general, the spatial
0
∈
regularity of u in the Sobolev scale Ws( ), s 0, which determines the order of convergence for
p O ≥
uniformapproximationmethodsinL ( ),isstrictlylessthanthespatialregularity ofuinthescale
p
O
( ). It can be due to, e.g., the irregular behaviour of the noise at the boundary or the irregularities
∗
of the boundary itself; see [37, Chapter 4] for the latter case. This justifies the use of nonlinear
approximation methods such as adaptive wavelet methods for the numerical treatment of SPDEs,
cf. [3, 2]. The proof of (1.2) relies on characterizations of Besov spaces by wavelet expansions and
onweighted Sobolevnormestimates foru,resultingfromthesolvability of theproblem(1.1)within
the spaces Hγ ( ,T).
p,θ O
An obvious approach to improve (1.2) with respect to regularity in time is to try to combine
the existing H¨older estimates in time for the elements of the spaces Hγ ( ,T) (see [26, Theorem
p,θ O
2
2.9]) with the wavelet arguments in [4]. However, it turns out that a satisfactory result requires a
more subtle strategy in three different aspects.
Firstly, we need an extension of the L -theory in [26] to an L (L )-theory for SPDEs dealing
p q p
with stochastic parabolic weighted Sobolev spaces Hγ,q( ,T) with possibly different summability
p,θ O
parameters q and p in time and space respectively. These spaces consist of elements u˜ of the form
du˜ = f˜dt+g˜kdwk, where u˜, f˜and g˜k, considered as stochastic processes with values in suitable
t
weighted L ( )-Sobolev spaces, are L -integrable w.r.t. P dt. Such an extension is needed to
p q
O ⊗
obtain betterH¨older estimates intimeinasecondstep.Satisfactory existence anuniquenessresults
concerning solutions in the spaces Hγ,q( ,T) have been established in [25] for domains with C1-
p,θ O O
boundary. Unfortunately, the techniques used there do not work on general Lipschitz domains.
Also, the L (L )-results that have been obtained in [47] within the semigroup approach to SPDEs
q p
do not directly suit our purpose: On the one hand, for general Lipschitz domains the domains
O
of the fractional powers of the leading linear differential operator cannot be characterized in terms
of Sobolev or Besov spaces as in the case of a smooth domains ; see, e.g., the introduction of [4]
O
for details. On the other hand, even in the case of a smooth domain we need regularity in terms
O
weighted Sobolev spaces to obtain the optimal regularity in the scale ( ).
Secondly, once we have established the solvability of SPDEs within∗ the spaces Hγ,q( ,T), we
p,θ O
have to exploit the L (L )–regularity of the solution and derive improved results on the H¨older
q p
regularity in time for large q. For = Rd this has been done by Krylov [32]. It takes quite delicate
O +
argumentstoapplytheseresultstothecaseofboundedLipschitzdomainsviaaboundaryflattening
argument.
Thirdly, in order to obtain a reasonable H¨older-Besov regularity result, it is necessary to gener-
alize the wavelet arguments applied in [4] to a wider range of smoothness parameters. This requires
more sophisticated estimates.
In this article we tackle and solve the tasks described above. We organize the article as follows.
In Section 2 we recall the definition and basic properties of the (deterministic) weighted Sobolev
spaces Hγ (G) introduced in [38] (see also [43, Chapter 6]) on general domains G Rd with
p,θ ⊆
non-empty boundary. In Section 3 we give the definition of the spaces Hγ,q(G,T) and specify the
p,θ
concept of a solution for equations of type (1.1) in these spaces. Moreover, we show that if we have
a solution u Hγ,q(G,T) with low regularity γ 0, butf and thegk’s have high L (L )-regularity,
∈ p,θ ≥ q p
then we can lift up the regularity of the solution (Theorem 3.8). In this sense the spaces Hγ,q(G,T)
p,θ
are the right ones for our regularity analysis of SPDEs.
Section 4 is devoted to the solvability of Eq. (1.1) in Hγ,q( ,T), Rd being a bounded
p,θ O O ⊆
Lipschitz domain. The focus lies on the case q > p 2 and we restrict our considerations to
≥
equations with additive noise, i.e. σik 0. In Subsection 4.2 we consider equations on domains
≡
with small Lipschitz constants and derive a result for general integrability parameters q p
≥ ≥
2 (Theorem 4.2). We use an L (L )-regularity result for deterministic parabolic equations from
q p
[16] and an estimate for stochastic integrals in UMD spaces from [46] to obtain a certain low
L (L )-regularity of the solution. Then the regularity is lifted up with the help of Theorem 3.8.
q p
In Subsection 4.2, we consider the stochastic heat equation on general bounded Lipschitz domains.
Here we use the results from [47] on maximal L -regularity of stochastic evolution equations (see
q
also [48] and [46]) to derive existence and uniqueness of a solution with low regularity. A main
ingredient will be to prove that the domain of the square root of the weak Dirichlet Laplacian on
L ( ) coincides with the closure of the test functions C∞( ) in the L ( )-Sobolev space of order
p O 0 O p O
one (Lemma 4.5). This stays true only for a certain range of p [2,p ) with p > 3. Thus, so does
0 0
∈
3
our result (Theorem 4.4). In a second step, we again lift up the regularity by using Theorem 3.8.
In both settings we derive suitable a-priori estimates.
In Section 5 we present our result on the H¨older regularity in time of the elements of Hγ,q( ,T)
p,θ O
(Theorem 5.1). It is an extension of the H¨older estimates in time for the elements of Hγ,q(T) =
p,θ
Hγ,q(Rd,T) in [31] to the case of bounded Lipschitz domains. The implications for the H¨older
p,θ +
regularity of the solutions of SPDEs are described in Theorem 5.3.
In Section 6 we pave the way for the analysis of the spatial regularity of the solutions of SPDEs
γ
in the scale ( ). We discuss the relationship between the weighted Sobolev spaces H ( ) and
∗ p,θ O
Besov spaces. Our main result in this section, Theorem 6.9, is a general embedding of the spaces
γ
H ( ), γ,ν > 0, into the Besov scale ( ). Its proof is an extension of the wavelet arguments in
p,d−νp O ∗
theproofof [4,Theorem 3.1], whereonly integer valued smoothnessparameters γ are considered.It
canalsobeseenasanextensionofandasupplementtotheBesovregularityresultsfordeterministic
elliptic equations in [10] and [7, 8, 9, 11]. To the best of our knowledge, no such general embedding
has been proven before. In the course of the discussion we also enlighten the fact that, for the
γ γ∧ν
relevant range of parameters γ and ν, the spaces H ( ) act like Besov spaces B ( ) with
p,d−νp O p,p O
zero trace on the boundary (Remark 6.7).
In Section 7 the results of the previous sections are combined in order to determine the H¨older-
Besov regularity of the elements of the stochastic parabolic spaces Hγ,q( ,T) and of the solutions
p,θ O
of SPDEs within these spaces. The related result in [4] is significantly improved in several aspects;
see Remark 7.3 for a detailed comparison. We obtain an estimate of the form
1 α 1
E u q C u q , = + ,
k kCκ([0,T];Bτα,τ(O)) ≤ k kHγp,,θq(O,T) τ d p
for certain α depending on the smoothness and weight parameters γ and θ and for certain κ
depending on q and α (Theorem 7.4). Using the a-priori estimates from Section 4, the right hand
side of the above inequality can be estimated by suitable norms of f and g if u is the solution to
the corresponding SPDE (Theorem 7.5).
Let us mention the related work [1] on the Besov regularity for the deterministic heat equation.
The authors study the regularity of temperatures in terms of anisotropic Besov spaces of type
α/2,α
B ((0,T) ), 1/τ = α/d+1/p. However, the range of admissible values for the parameter τ
τ,τ
×O
is a priori restricted to (1, ), so that α is always less than d(1 1/p). In our article the parameter
∞ −
τ in ( ) may be any positive number, including in particular the case where τ is less than 1 and
∗
where Bα ( ) is not a Banach space but a quasi-Banach space.
τ,τ O
Notation and Conventions. Throughout this paper, always denotes a bounded Lipschitz
O
domain in Rd, d 1, as specified in Definition 2.5 below. General subsets of Rd are denoted by
≥
G. We write ∂G for their boundary (if it is not empty) and G◦ for the interior. N := 1,2,...
{ }
denotes the set of strictly positive integers whereas N := N 0 . Let (Ω, ,P) be a complete
0
∪ { } F
probability space and ,t 0 be an increasing filtration of σ-fields , each of which
t t
{F ≥ } F ⊂ F
contains all ( ,P)-null sets. By we denote the predictable σ-field generated by ,t 0
t
F P {F ≥ }
and we assume that (w1) ,(w2) ,... are independentone-dimensional Wiener processes
{ t t∈[0,T] t t∈[0,T] }
w.r.t. ,t 0 . For κ (0,1) and a quasi-Banach space (X, ) we denote by Cκ([0,T];X)
t X
{F ≥ } ∈ k·k
the H¨older space of continuous X-valued functions on [0,T] with finite norm defined
Cκ([0,T];X)
k·k
4
by
u(t) u(s)
X
[u] := sup k − k ,
Cκ([0,T];X) t s κ
s,t∈[0,T]
| − |
u := sup u(t) ,
C([0,T];X) X
k k k k
t∈[0,T]
u = u +[u] .
Cκ([0,T];X) C([0,T];X) Cκ([0,T];X)
k k k k
For1 < p < ,L (A,Σ,µ;X)denotesthespaceofµ-stronglymeasurableandp-Bochnerintegrable
p
∞
functions with values in X on a σ-finite measure space (A,Σ,µ), endowed with the usual L -Norm.
p
We writeL (G) instead of L (G, (G),λd;R)if G (Rd), where (G)and (Rd)aretheBorel-σ-
p p
B ∈B B B
fieldsonGandRd.RecalltheHilbertspaceℓ := ℓ (N) = a = (a1,a2,...) : a = ( ak 2)1/2 <
2 2 { | |ℓ2 k| |
with the inner product a,b = akbk, for a,b ℓ . The notation C∞(G) is used for the
∞} h iℓ2 i ∈ 2 0 P
space of infinitely differentiable test functions with compact support in a domain G Rd. For any
P ⊆
distribution f on G and any ϕ C∞(G), (f,ϕ) denotes the application of f to ϕ. Furthermore,
∈ 0
for any multi-index α = (α ,...,α ) Nd, we write Dαf = ∂|α|f for the corresponding
1 d ∈ 0 ∂xα1...∂xαd
1 d
(generalized) derivative w.r.t. x = (x ,...,x ) G, where α = α +...+α . By making slight
1 d 1 d
∈ | |
abuseofnotation,form N ,wewriteDmf forany(generalized)m-thorderderivativeoff andfor
0
∈
the vector of all m-th order derivatives of f. E.g. if we write Dmf X, where X is a function space
onG,wemeanDαf X forallα Nd with α = m.Wealsouset∈henotationf = ∂2f , f =
∈ ∈ 0 | | xixj ∂xi∂xj xi
∂f .Thenotation f (respectively f )isusedsynonymouslyforDf := D1f (respectively forD2f),
∂xi x xx
whereas f := u (respectively f := f ). Moreover, ∆f := f ,
k xkX ik xikX k xxkX i,jk xixjkX i xixi
whenever it makes sense. Given p [1, ) and m N , Wm(G) denotes the classical Sobolev
P ∈ ∞ ∈ P0 p P
space consisting of all f L (G) such that f := sup Dαf is finite. It is
∈ p | |Wpm(G) α∈Nd0,|α|=mk kLp(G)
normed via f := f + f . The closure of C∞( ) in W1( ) is denoted by
k kWpm(G) k kLp(G) | |Wpm(G) 0 O p O
W˚1( ) and is normed by f := ( f p )1/p. If we have two quasi-normed spaces
p O k kW˚p1(O) ik xikLp(O)
(X , ), i = 1,2, X ֒ X means that X is continuously linearly embedded in X . For a
i k ·kXi 1 → 2 P 1 2
compatible couple (X ,X ) of quasi-Banach spaces, [X ,X ] denotes the interpolation space of
1 2 1 2 η
exponent η (0,1) arising from the complex interpolation method. In general, N will denote a
∈
positive finite constant, which may differ from line to line. The notation N = N(a ,a ,...) is used
1 2
toemphasizethedependenceoftheconstantN onthesetofparameters a ,a ,... .Ingeneral,this
1 2
{ }
set will not contain all the parameters N depends on. A B means that A and B are equivalent.
∼
2 Weighted Sobolev spaces
We start by recalling the definition and some basic properties of the (deterministic and stationary)
γ
weighted Sobolev spaces H (G) introduced in [38]. These spaces will serve as state spaces for the
p,θ
solution processes u= (u(t)) to SPDEs of type (1.1) and they will play a fundamental role in
t∈[0,T]
all the forthcoming sections.
For p (1, ) and γ R, let Hγ := Hγ(Rd) := (1 ∆)−γ/2L (Rd) be the spaces of Bessel
p p p
∈ ∞ ∈ −
potentials, endowed with the norm
kukHpγ := k(1−∆)γ/2ukLp(Rd) := kF−1[(1+|ξ|2)γ/2F(u)(ξ)]kLp(Rd),
5
where denotes the Fourier transform. It is well known that if γ is a nonnegative integer, then
F
Hγ = u L : Dαu L for all α Nd with α γ .
p ∈ p ∈ p ∈ 0 | | ≤
Let G Rd be an arb(cid:8)itrary domain with non-empty boundary ∂G. W(cid:9) e denote by ρ(x) :=
⊆
ρ (x) := dist(x,∂G) the distance of a point x G to the boundary ∂G. Furthermore, we fix a
G
∈
bounded infinitely differentiable function ψ defined on G such that for all x G,
∈
ρ(x) Nψ(x), ρ(x)m−1 Dmψ(x) N(m)< for all m N , (2.1)
0
≤ | | ≤ ∞ ∈
where N and N(m) do not depend on x G. For a detailed construction of such a function see,
∈
e.g., [43, Chapter 3, Section 3.2.3]. Let ζ C∞(R ) be a non-negative function satisfying
∈ 0 +
ζ(en+t) > c > 0 for all t R. (2.2)
∈
n∈Z
X
Note that any non-negative smooth function ζ C∞(R ) with ζ > 0 on [e−1,e] satisfies (2.2). For
∈ 0 +
x G and n Z, define
∈ ∈
ζ (x) := ζ(enψ(x)).
n
Then, there exists k > 0 such that, for all n Z, suppζ G := x G : e−n−k0 < ρ(x) <
0 n n
∈ ⊂ { ∈
e−n+k0 , i.e., ζ C∞(G ). Moreover, Dmζ (x) N(ζ,m)emn for all x G and m N , and
ζ}(x) δn>∈0 f0or alnl x G. For p| (1,n )|an≤d γ,θ R, we denote b∈y Hγ (G) th∈e sp0ace of
n∈Z n ≥ ∈ ∈ ∞ ∈ p,θ
all distributions u on G such that
P
u p := enθ ζ (en )u(en ) p < .
k kHpγ,θ(G) n∈Z k −n · · kHpγ ∞
X
It is well-known that
L (G) := H0 (G) = L (G,ρθ−ddx),
p,θ p,θ p
and that, if γ is a positive integer,
Hγ (G) = u L (G) : ρ|α|Dαu L (G) for all α Nd with α γ ,
p,θ ∈ p,θ ∈ p,θ ∈ 0 | | ≤
(cid:8) (cid:9)
u p ρ|α|Dαu pρθ−ddx;
k kHγ (G) ∼
p,θ |α|≤γZG
X (cid:12) (cid:12)
(cid:12) (cid:12) γ
see, e.g., [38, Proposition 2.2]. This is the reason why the space H (G) is called weighted Sobolev
p,θ
space of order γ, with summability parameter p and weight parameter θ.
For p (1, ) and γ R we write Hγ(ℓ ) for the collection of all sequences g = (g1,g2,...) of
p 2
distributio∈ns on∞Rd with ∈gk Hγ for each k N and
p
∈ ∈
∞
1/2
kgkHpγ(ℓ2) := kgkHpγ(Rd;ℓ2) := k|(1−∆)γ/2g|ℓ2kLp := |(1−∆)γ/2gk|2 Lp < ∞.
(cid:13)(cid:16)Xk=1 (cid:17) (cid:13)
(cid:13) (cid:13)
Analogously, for θ R, a sequence g = (g1,g2,...) of di(cid:13)stributions on G is in Hγ(cid:13)(G;ℓ ) if, and
∈ p,θ 2
only if, gk Hγ (G) for each k N and
∈ p,θ ∈
g p := enθ ζ (en )g(en ) p < .
k kHpγ(G;ℓ2) k −n · · kHpγ(ℓ2) ∞
n∈Z
X
γ
Now we present some useful properties of the space H (G) taken from [38], see also [29, 30].
p,θ
6
Lemma 2.1. (i) The space C∞(G) is dense in Hγ (G).
0 p,θ
(ii) Assume that γ d/p = m+ν for some m N , ν (0,1] and that i,j Nd are multi-indices
− ∈ 0γ ∈ ∈ 0
such that i m and j = m. Then for any u H (G), we have
| | ≤ | | ∈ p,θ
ψ|i|+θ/pDiu C(G), ψm+ν+θ/pDju Cν(G),
∈ ∈
|ψ|i|+θ/pDiu|C(G)+[ψm+ν+θ/pDju]Cν(G) ≤NkukHpγ,θ(G).
γ γ−1
(iii) u H (G) if, and only if, u,ψu H (G) and
∈ p,θ x ∈ p,θ
kukHpγ,θ(G) ≤ NkψuxkHpγ,−θ1(G) +NkukHpγ,−θ1(G) ≤ NkukHpγ,θ(G).
γ γ−1
Also, u H (G) if, and only if, u,(ψu) H (G) and
∈ p,θ x ∈ p,θ
kukHpγ,θ(G) ≤ Nk(ψu)xkHpγ,−θ1(G)+NkukHpγ,−θ1(G) ≤NkukHpγ,θ(G).
(iv) For any ν,γ R, ψνHγ (G) = Hγ (G) and
∈ p,θ p,θ−pν
kukHpγ,θ−pν(G) ≤ Nkψ−νukHpγ,θ(G) ≤ NkukHpγ,θ−pν(G).
(v) If γ (γ ,γ ) then, for any ε> 0, there exists a constant N = N(γ ,γ ,θ,p,ε), such that
0 1 0 1
∈
kukHpγ,θ(G) ≤ εkukHpγ,1θ(G) +N(γ0,γ1,θ,p,ε)kukHpγ,0θ(G).
Also, if θ (θ ,θ ) then, for any ε > 0, there exists a constant N = N(θ ,θ ,γ,p,ε), such that
0 1 0 1
∈
kukHpγ,θ(G) ≤ εkukHpγ,θ0(G)+N(θ0,θ1,γ,p,ε)kukHpγ,θ1(G).
(vi) There exists a constant c > 0 depending on p, θ, γ and the function ψ such that, for all
0
c c , the operator ψ2∆ c is a homeomorphism from Hγ+1(G) to Hγ−1(G).
≥ 0 − p,θ p,θ
γ
Remark 2.2. Assertions (vi) and (iv) in Lemma 2.1 imply the following: If u H (G) and
∈ p,θ−p
γ γ+2
∆u H (G), then u H (G) and there exists a constant N, which does not depend on u,
∈ p,θ+p ∈ p,θ−p
such that
kukHpγ,+θ−2p(G) ≤ Nk∆ukHpγ,θ+p(G)+NkukHpγ,θ−p(G).
γ
A proof of the following equivalent characterization of the weighted Sobolev spaces H (G) can
p,θ
be found in [38, Proposition 2.2].
Lemma 2.3. Let ξ : n Z C∞(G) be such that for all n Z and m N ,
{ n ∈ } ⊆ 0 ∈ ∈ 0
Dmξ N(m)cnm and suppξ x G :c−n−k0 < ρ(x) < c−n+k0 (2.3)
n n
| |≤ ⊆ { ∈ }
for some c> 1 and k > 0, where the constant N(m) does not depend on n Z and x G. Then,
0
γ ∈ ∈
for any u H (G),
∈ p,θ
cnθ ξ (cn )u(cn ) p N u p .
n∈Z k −n · · kHpγ ≤ k kHpγ,θ(G)
X
If in addition
ξ (x) δ > 0 for all x G (2.4)
n
≥ ∈
n∈Z
X
then the converse inequality also holds.
7
Remark 2.4. (i) It is easy to check that both
ξ(1) := e−n(ζ ) : n Z and ξ(2) := e−2n(ζ ) : n Z
n n xi ∈ n n xixj ∈
(cid:8) (cid:9) (cid:8) (cid:9)
satisfy (2.3) with c := e. Therefore,
enθ en(ζ ) (en )u(en ) p + e2n(ζ ) (en )u(en ) p N u p .
nX∈Z (cid:16)k −n xi · · kHpγ k −n xixj · · kHpγ(cid:17) ≤ k kHpγ,θ(G)
(ii) Given k 1, fix a function ζ˜ C∞(R ) with
1 ≥ ∈ 0 +
1
ζ˜(t) = 1 for all t 2−k1, N(0)2k1 ,
∈ N
h i
where N and N(0) are as in (2.1). Then, the sequence ξ :n Z C∞(G) defined by
{ n ∈ }⊆ 0
ξ := ζ˜(2nψ()), n Z,
n
· ∈
fulfils the conditions (2.3) and (2.4) from Lemma 2.3 with c = 2 and a suitable k > 0. Furthermore,
0
ξ (x) = 1 for all x ρ−1 2−n 2−k1,2k1 .
n
∈
In this paper, will always denote a bounded(cid:0)Lips(cid:2)chitz do(cid:3)m(cid:1) ain in Rd. More precisely:
O
Definition 2.5. We call a bounded domain Rd a Lipschitz domain if, and only if, for any
O ⊂
x = (x1,x′) ∂ , there exists a Lipschitz continuous function µ : Rd−1 R such that, upon
0 0 0 ∈ O 0 →
relabeling and reorienting the coordinate axes if necessary, we have
(i) B (x ) = x = (x1,x′) B (x ): x1 > µ (x′) , and
O∩ r0 0 { ∈ r0 0 0 }
(ii) µ (x′) µ (y′) K x′ y′ , for any x′,y′ Rd−1,
0 0 0
| − | ≤ | − | ∈
where r ,K are independent of x .
0 0 0
Remark 2.6. Recall that for a bounded Lipschitz domain Rd,
O ⊆
W˚1( ) = H1 ( )
p O p,d−p O
with equivalent norms. This follows from [35, Theorem 9.7] and Poincar´e’s inequality.
3 Stochastic parabolic weighted Sobolev spaces and SPDEs
In this section, we first introduce the stochastic parabolic spaces Hγ,q(G,T) for arbitrary domains
p,θ
G Rd with non-empty boundary in analogy to the spaces Hγ,q(T) = Hγ,q(Rd,T) from [31, 32].
⊆ p,θ p,θ +
Then we show that they are suitable to serve as solution spaces for equations of type (1.1) in the
following sense: If we have a solution u Hγ,q(G,T) with low regularity γ 0, but f and the gk’s
∈ p,θ ≥
have high L (L )-regularity, then we can lift up the regularity of the solution (Theorem 3.8).
q p
8
Definition 3.1. Let G be a domain in Rd with non-empty boundary. For p,q (1, ), γ,θ R
∈ ∞ ∈
and T (0, ) we define
∈ ∞
Hγ,q(G,T) := L (Ω [0,T], ,P dt;Hγ (G)),
p,θ q × P ⊗ p,θ
Hγ,q(G,T;ℓ ):= L (Ω [0,T], ,P dt;Hγ (G;ℓ )),
p,θ 2 q × P ⊗ p,θ 2
Uγ,q(G) := L (Ω, ,P;ψ1−2/qHγ−2/q(G)).
p,θ q F0 p,θ
If p = q we also write Hγ (G,T), Hγ (G,T;ℓ ) and Uγ (G) instead of Hγ,p(G,T), Hγ,p(G,T;ℓ )
p,θ p,θ 2 p,θ p,θ p,θ 2
γ,p
and U (G) respectively.
p,θ
From now on let
p [2, ), q [2, ), γ R, θ R.
∈ ∞ ∈ ∞ ∈ ∈
Definition 3.2. Let G be a domain in Rd with non-empty boundary. We write u Hγ,q(G,T)
∈ p,θ
if, and only if, u Hγ,q (G,T), u(0, ) Uγ,q(G), and there exist some f Hγ−2,q(G,T) and
∈ p,θ−p · ∈ p,θ ∈ p,θ+p
g Hγ−1,q(G,T;ℓ ) such that
∈ p,θ 2
du = fdt+gkdwk
t
in the sense of distributions. That is, for any ϕ C∞(G), with probability one, the equality
∈ 0
t ∞ t
(u(t, ),ϕ) = (u(0, ),ϕ)+ (f(s, ),ϕ)ds+ (gk(s, ),ϕ)dwk
· · · · s
Z0 k=1Z0
X
holds for all t [0,T], where the series is assumed to converge uniformly on [0,T] in probability.
In this situatio∈n we write Du:= f and Su:= g. The norm in Hγ,q(G,T) is defined as
p,θ
kukHγp,,θq(G,T) := kukHγp,,θq−p(G,T)+kDukHpγ,−θ+2,pq(G,T)+kSukHpγ,−θ1,q(G,T;ℓ2)+ku(0,·)kUpγ,,θq(G).
If p = q we also write Hγ (G,T) instead of Hγ,p(G,T).
p,θ p,θ
Remark 3.3. Replacing G by Rd and omitting the weight parameter θ and the weight function ψ in
the definitions above, one obtains the spaces Hγ,q(T) = Hγ,q(Rd,T), Hγ,q(T;ℓ ) = Hγ,q(Rd,T;ℓ ),
p p p 2 p 2
Uγ,q = Uγ,q(Rd), and H γ,q(T) as introduced in [32, Definition 3.5]. The latter are denoted by
p p p
γ,q γ
(T) in [31]; if q =p they coincide with the spaces (T) introduced in [29, Definition 3.1].
p p
H H
We consider initial value problems of the form
du = (aiju +f)dt+(σiku +gk)dwk, u(0, ) = u , (3.1)
xixj xi t · 0
on an arbitrary domain G Rd with non-empty boundary. We use the following solution concept.
⊆
Definition 3.4. We say that a stochastic process u Hγ,q(G,T) is a solution of Eq. (3.1) if, and
∈ p,θ
only if,
u(0, ) = u , Du= aiju +f, and Su= σiku +gk ,
· 0 xixj xi k∈N
in the sense of Definition 3.2. (cid:0) (cid:1)
9
Remark 3.5. Here and in the sequel we use the summation convention on the repeated indices
i,j,k. The question, in which sense, for a bounded Lipschitz domain Rd, the elements of
Hγ,q( ,T) fulfil a zero Dirichlet boundary condition as in Eq. (1.1), will beOan⊆swered in Remark 6.7.
p,θ O
We make the following assumptions on the coefficients in Eq. (3.1). Throughout this paper,
whenever we will talk about this equation, we will assume that they are fulfilled.
Assumption 3.6. (i) The coefficients aij = aij(ω,t) and σik = σik(ω,t) are predictable. They do
not depend on x G. Furthermore, aij = aji for i,j 1,...,d .
∈ ∈ { }
(ii) There exist constants δ ,K > 0 such that for any (ω,t) Ω [0,T] and λ Rd,
0
∈ × ∈
δ λ 2 a¯ij(ω,t)λiλj K λ 2,
0
| | ≤ ≤ | |
where a¯ij(ω,t) := aij(ω,t) 1(σi·(ω,t),σj·(ω,t)) , with σi·(ω,t) = σik(ω,t) ℓ .
− 2 ℓ2 k∈N ∈ 2
We will use the following result taken from [31, Lemma 2.3]. (cid:0) (cid:1)
Lemma 3.7. Let p 2, m N, and, for i = 1,2,...,m,
≥ ∈
λ (0, ), γ R, u(i) γi+2(T), u(i)(0, ) = 0.
i ∈ ∞ i ∈ ∈ Hp ·
Denote Λi := (λi ∆)γi/2. Then
−
T m m T m
E Λ ∆u(i) p dt N E Λ f(i) p + Λ g(i) p Λ ∆u(j) p dt
k i kLp ≤ k i kLp k i x kLp(ℓ2) k j kLp
hZ0 Yi=1 i Xi=1 hZ0 (cid:16) (cid:17)jY=1 i
j6=i
T m
+N E Λ g(i) p Λ g(j) p Λ ∆u(k) p dt ,
k i x kLp(ℓ2)k j x kLp(ℓ2) k k kLp
1≤iX<j≤m hZ0 kY=1 i
k6=i,j
where f(i) := Du(i) arsu(i) , g(i)k := Sku(i) σrku(i) and L (ℓ ) := H0(ℓ ). The constant N
− xrxs − xr p 2 p 2
depends only on m, d, p, δ , and K.
0
Now we are able to prove that if we have a solution u Hγ+1,q(G,T) to Eq. (3.1) and if the
∈ p,θ
regularity of the forcing terms f and g is high then we can lift the regularity of the solution. Note
that in the next theorem there is no restriction, neither on the shape of the domain G Rd nor on
⊆
the parameters θ,γ R.
∈
Theorem 3.8. Let G Rd be an arbitrary domain with non-empty boundary. Let γ R, p 2
and q = pm for some m⊆ N. Let f Hγ,q (G,T), g Hγ+1,q(G,T;ℓ ) and let u H∈γ+1,q(G≥,T)
∈ ∈ p,θ+p ∈ p,θ 2 ∈ p,θ
be a solution to Eq. (3.1) with u = 0. Then u Hγ+2,q(G,T), and
0 ∈ p,θ
q q q q
u N u + f + g ,
k kHγp,+θ−2,pq(G,T) ≤ k kHγp,+θ−1,pq(G,T) k kHγp,,θq+p(G,T) k kHγp,+θ1,q(G,T;ℓ2)
(cid:16) (cid:17)
where the constant N (0, ) does not depend on u, f and g.
∈ ∞
10