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308 P. Gwiazda and A.

Swierczewska-Gwiazda

and consequently the sequences {M (

∗∇u)} and {M

∗

(

∗ χ)} are uniformly

integrable. Thus by Lemma 2.1,



∗∇u

−→ ∇u modularly in L

(Q),



∗ χ

∗

−→ χ modularly in L

∗

(Q).

By Proposition 2.3 we conclude that

lim

j→∞



(

∗χ) · (

∗∇u)dxdt=



χ ·∇udxdt. (4.6)

We are aiming to show that

u(s)

−

u





χ ·∇udxdt =0, (4.7)

which according to (4.3)–(4.6) holds for some 0 <s

<T, not necessarily equal to

zero. To pass to the limit with s

→ 0 we need to establish the weak continuity

of u in L

(Ω) w.r.t. time. For this purpose we consider the sequence {

} and

provide uniform estimates. By P

we mean the orthogonal projection of L

(Ω)

on the ﬁrst n eigenvectors of the Laplace operator. Let ϕ ∈ L

∞

(0,T; W

r,2

(Ω)),

ϕ

∞

(0,T ;W

r,2

)

≤ 1, where r>

+ 1 and observe that

,ϕ

= −



A(t, x, ∇u

) ·∇(P

ϕ)dx.

Since P

ϕ

r,2

≤ϕ

r,2

and W

r−1,2

(Ω) ⊂ L

∞

(Ω) we estimate as follows:



A(t, x, ∇u

) ·∇(P

ϕ)dxdt

≤



A(t, ·, ∇u

)

(Ω)

∇(P

ϕ)

∞

(Ω)

≤ c



A(t, ·, ∇u

)

(Ω)

P

ϕ

r,2

dt ≤ cA(·, ·, ∇u

)

(Q)

ϕ

∞

(0,T ;W

r,2

)

(4.8)

Hence we conclude that

is bounded in L

(0,T; W

−r,2

(Ω)). From the energy es-

timates and Lemma 2.2 we conclude existence of a monotone, continuous function

L : R

→ R

,withL(0) = 0 which is independent of n and



A(t, ·, ∇u

)

(Ω)

≤ L(|s

− s

for any s

∈ [0,T]. Consequently, estimate (4.8) provides that



,ϕ

≤ L(|s

−s

for all ϕ with supp ϕ ⊂ (s

) ⊂ [0,T]andϕ

∞

(0,T ;W

r,2

)

≤ 1. Since

u

) −u

)

−r,2

=sup

ψ

r,2

≤1



(t)

,ψ

(4.9)

Parabolic Equations in Anisotropic Orlicz Spaces 309

then

sup

n∈N

u

) − u

)

−r,2

≤ L(|s

−s

|) (4.10)

which provides that the family of functions u

:[0,T] → W

−r,2

(Ω) is equicontinu-

ous. Together with a uniform bound in L

∞

(0,T; L

(Ω)) it yields that the sequence

} is relatively compact in C([0,T]; W

−r,2

(Ω)) and u ∈ C([0,T]; W

−r,2

(Ω)).

Consequently we can choose a sequence {s

}

, s

→ 0

as i →∞such that

u(s

)

i→∞

−→ u(0) in W

−r,2

(Ω). (4.11)

The limit coincides with the weak limit of {u(s

)} in L

(Ω) and hence we conclude

that

lim inf

i→∞

u(s

)

(Ω)

≥u



(Ω)

. (4.12)

Consequently we obtain from (4.1) for any Lebesgue point s of u that

lim sup

n→∞



A(t, x, ∇u

) ·∇u

u



−lim inf

k→∞

u

(s)

(4.13)

≤

u



−

u(s)

(4.12)

≤ lim inf

i→∞



u(s

)

−

u(s)



= lim

i→∞



χ ·∇udxdt =



χ ·∇udxdt.

Having the above estimate we can complete the proof using the monotonicity of

A,namely



(A(t, x, ¯v) −A(t, x, ∇u

)) · (¯v −∇u

)dxdt ≥ 0 (4.14)

for all ¯v ∈ L

∞

(Q). Observe that for ¯v ∈ L

∞

(Q)italsoholdsthatA(t, x, ¯v) ∈

∞

(Q). Indeed, assume the opposite, i.e., A(t, x, ¯v) is unbounded. Then, since M

is nonnegative, by (A2), the following estimate holds:

|¯v|≥

∗

(x, A(t, x, ¯v))

|A(t, x, ¯v)|

Taking into account the superlinear growth of M we observe that the right-hand

side tends to inﬁnity, which contradicts that ¯v ∈ L

∞

(Q). Before passing to the

limit with n →∞, we rewrite (4.14):



A(t, x, ∇u

)·∇u

dxdt ≥



A(t, x, ∇u

)·¯vdxdt+



A(t, x, ¯v)·(∇u

−¯v)dxdt

(4.15)

hence



χ ·∇udxdt ≥



χ · ¯vdxdt +



A(t, x, ¯v) · (∇u − ¯v)dxdt (4.16)

310 P. Gwiazda and A.

Swierczewska-Gwiazda

and consequently



(A(t, x, ¯v) − χ) ·(¯v −∇u)dxdt ≥ 0. (4.17)

We ﬁx k>0 and deﬁne Q

= {(t, x) ∈ Q

: |∇u(t, x)|≤k a.e. in Q

Let now 0 <j<ibe arbitrary and h>0. We make the following choice of ¯v,

¯v =(∇u)1

+ h¯z1

with an arbitrary ¯z ∈ L

∞

(Q). Using (4.17) we obtain

−



(A(t, x, 0)−χ) ·∇udxdt+ h



(A(t, x, ∇u + h¯z)−χ) · ¯zdxdt ≥ 0. (4.18)

Since (A2) implies that A(t, x, 0) = 0, then obviously

−



(A(t, x, 0) −χ) ·∇udxdt =



χ ·∇u 1

dxdt

and since



|χ ·∇u|dxdt < ∞

we obtain, while passing to the limit with i →∞,

χ ·∇u 1

→ 0a.e. in Q.

Hence by the Lebesgue dominated convergence theorem

lim

i→∞



χ ·∇udxdt =0.

Letting i →∞in (4.18) and dividing by h yields



(A(t, x, ∇u + h¯z) − χ) · ¯zdxdt ≥ 0.

Observe that ∇u+h¯z →∇u a.e. in Q

as h → 0

and A(t, x, ∇u+h¯z) is uniformly

bounded in L

∞

), |Q

| < ∞, hence by Vitali’s theorem we conclude that

A(t, x, ∇u + h¯z) → A(t, x, ∇u)inL

)

and



(A(t, x, ∇u + h¯z) − χ) · ¯zdxdt →



(A(t, x, ∇u) − χ) · ¯zdxdt

as h → 0

.Consequently,



(A(t, x, ∇u) − χ) · ¯zdxdt ≥ 0

for all ¯z ∈ L

∞

(Q). The choice ¯z = −sgn (A(t, x, ∇u) − χ) yields



|A(t, x, ∇u) − χ|dxdt ≤ 0.

Parabolic Equations in Anisotropic Orlicz Spaces 311

Hence

A(t, x, ∇u)=χ a.e. in Q

. (4.19)

The above identity holds for arbitrary j, hence (4.19) holds a.e. in Q

. Since it

holds for almost all s such that 0 <s<T then

χ = A(t, x, ∇u)a.e. in Q,

which completes the proof of Theorem 1.1 

References

[1] Andrea Cianchi. A fully anisotropic Sobolev inequality. Paciﬁc J. Math., 196(2):283–

295, 2000.

[2] Thomas Donaldson. Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic

initial value problems. J. Diﬀerential Equations, 16:201–256, 1974.

[3] A. Elmahi and D. Meskine. Parabolic equations in Orlicz spaces. J. London Math.

Soc. (2), 72(2):410–428, 2005.

[4] Jean-Pierre Gossez. Some approximation properties in Orlicz-Sobolev spaces. Studia

Math., 74(1):17–24, 1982.

[5] Piotr Gwiazda and Agnieszka

Swierczewska-Gwiazda. On non-Newtonian ﬂuids with

a property of rapid thickening under diﬀerent stimulus. Math. Models Methods Appl.

Sci., 18(7):1073–1092, 2008.

[6] Julian Musielak. Orlicz spaces and modular spaces, volume 1034 of Lecture Notes in

Mathematics. Springer-Verlag, Berlin, 1983.

[7] Vesa Mustonen and Matti Tienari. On monotone-like mappings in Orlicz-Sobolev

spaces. Math. Bohem., 124(2-3):255–271, 1999.

[8] A. Novotn´y and Ivan Straˇskraba. Introduction to the mathematical theory of com-

pressible ﬂow. Oxford Lecture Series in Mathematics and its Applications 27. Oxford:

Oxford University Press., 2004.

Piotr Gwiazda and Agnieszka

Swierczewska-Gwiazda

Institute of Applied Mathematics and Mechanics

University of Warsaw

Banacha 2

02-097 Warsaw, Poland

e-mail: pgwiazda@mimuw.edu.pl

aswiercz@mimuw.edu.pl

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 313–341

 2011 Springer Basel AG

Maximal Parabolic Regularity for

Divergence Operators on Distribution Spaces

Robert Haller-Dintelmann and Joachim Rehberg

Abstract. We show that elliptic second-order operators A of divergence type

fulﬁll maximal parabolic regularity on distribution spaces, even if the underly-

ing domain is highly non-smooth, the coeﬃcients of A are discontinuous and A

is complemented with mixed boundary conditions. Applications to quasilinear

parabolic equations with non-smooth data are presented.

Mathematics Subject Classiﬁcation (2000). Primary 35A05, 35B65; Secondary

35K15, 35K20.

Keywords. Maximal parabolic regularity, quasilinear parabolic equations,

mixed Dirichlet-Neumann conditions.

1. Introduction

It is the aim of this paper to provide an abridged version of our work [40]. The goal

is to provide a text with only few proofs and with a considerable reduction of the

sophisticated technicalities. In particular, we present a direct way to carry over

maximal parabolic regularity from L

spaces to the distribution spaces, avoiding

the Dore-Venni argument. So our hope is to produce a more readable text for

colleagues who are only interested in the principal ideas and results of [40]. On

the other hand, due to discussions with K. Gr¨oger, we succeeded in eliminating

the crucial supposition in [40] that the local bi-Lipschitz charts for the boundary

of the domain have to be volume preserving – at least what concerns maximal

parabolic regularity. Thus the conditions get a lot easier to control in examples

from beyond the class of strong Lipschitz domains. Naturally, the proof of this is

pointed out below, see Section 4.

Our motivation was to ﬁnd a concept which allows us to treat nonlinear

parabolic equations of the formal type





−∇·G(u)μ∇u = R(t, u),

u(T

)=u

(1.1)

314 R. Haller-Dintelmann and J. Rehberg

combined with mixed, nonlinear boundary conditions:

ν ·G(u)μ∇u + b(u)=g on Γ and u =0on∂Ω \Γ, (1.2)

where Γ is a suitable open subset of ∂Ω.

The main feature is here – in contrast to [43] – that inhomogeneous Neumann

conditions and the appearance of distributional right-hand sides (e.g., surface den-

sities) should be admissible. Thus, one has to consider the equations in suitably

chosen distribution spaces. The concept to solve (1.1) is to apply a theorem of

Pr¨uss (see [51], see also [15]) which bases on maximal parabolic regularity. This

has the advantage that right-hand sides are admissible which depend discontinu-

ously on time, which is desirable in many applications. Pursuing this idea, one has,

of course, to prove that the occurring elliptic operators satisfy maximal parabolic

regularity on the chosen distribution spaces.

In fact, we show that, under very mild conditions on the domain Ω, the

Dirichlet boundary part ∂Ω \Γ and the coeﬃcient function, elliptic divergence op-

erators with real, symmetric L

∞

-coeﬃcients satisfy maximal parabolic regularity

on a huge variety of spaces, among which are Sobolev, Besov and Lizorkin-Triebel

spaces, provided that the diﬀerentiability index is between 0 and −1(cf.Theo-

rem 5.18). We consider this as the ﬁrst main result of this work, also interesting

in itself. Up to now, the only existing results for mixed boundary conditions in

distribution spaces (apart from the Hilbert space situation) are, to our knowledge,

that of Gr¨oger [36] and the recent one of Griepentrog [31]. Concerning the Dirichlet

case, compare [10] and references therein.

Let us point out some ideas, which will give a certain guideline for the paper:

In principle, our strategy for proving maximal parabolic regularity for diver-

gence operators on H

−1,q

was to show an analog of the central result of [9], this

time in case of mixed boundary conditions, namely that



−∇·μ∇ +1



−1/2

: L

→ H

1,q

(1.3)

provides a topological isomorphism for suitable q. This would give the possibil-

ity of carrying over the maximal parabolic regularity, known for L

, to the dual

of H

1,q



, because, roughly spoken, (−∇ · μ∇ +1)

−1/2

commutes with the corre-

sponding parabolic solution operator. Unfortunately, we were only able to prove

the continuity of (1.3) within the range q ∈ ]1, 2], due to a result of Duong and

Intosh [22], see also [49], but did not succeed in proving the continuity of the

inverse in general.

It turns out, however, that (1.3) provides a topological isomorphism, if Ω∪Γ

is the image under a bi-Lipschitz mapping of one of Gr¨oger’s model sets [35],

describing the geometric conﬁguration in neighborhoods of boundary points of Ω.

Thus, in these cases one may carry over the maximal parabolic regularity from L

to H

−1,q

. Knowing this, we localize the linear parabolic problem, use the ‘local’

maximal parabolic information and interpret this again in the global context at

the end. Interpolation with the L

result then yields maximal parabolic regularity

on the corresponding interpolation spaces.

Maximal Parabolic Regularity for Divergence Operators 315

Let us explicitly mention that the concept of Gr¨oger’s regular sets, where the

domain itself is a Lipschitz domain, seems adequate to us, because it covers many

realistic geometries that fail to be domains with Lipschitz boundary. One striking

example are the two crossing beams, cf. [40, Subsection 7.3].

The strategy for proving that (1.1), (1.2) admit a unique local solution is as

follows. We reformulate (1.1) by adding the distributional terms, corresponding to

the boundary condition (1.2) to the right-hand side of (1.1). Assuming additionally

that the elliptic operator −∇ · μ∇ +1 : H

1,q

→ H

−1,q

provides a topological

isomorphism for a q larger than the space dimension d, the above-mentioned result

of Pr¨uss for abstract quasilinear equations applies to the resulting quasilinear

parabolic equation. The detailed discussion how to assure all requirements of [51],

including the adequate choice of the Banach space, is presented in Section 6. Let

us further emphasize that the presented setting allows for coeﬃcient functions

that really jump at hetero interfaces of the material and permits mixed boundary

conditions, as well as domains which do not possess a Lipschitz boundary. It is

well known that this is highly desirable when modelling real world problems. One

further advantage is that nonlinear, nonlocal boundary conditions are admissible

in our concept, despite the fact that the data is highly non-smooth, compare [2]. It

is remarkable that, irrespective of the discontinuous right-hand sides, the solution

is H¨older continuous simultaneously in space and time, see Corollary 6.17 below.

In Section 7 we give examples for geometries, Dirichlet boundary parts and

coeﬃcients in three dimensions for which our additional supposition, the isomorphy

−∇·μ∇+1:H

1,q

→ H

−1,q

really holds for a q>d.

Finally, some concluding remarks are given in Section 8.

2. Notation and general assumptions

Throughout this article the following assumptions are valid.

• Ω ⊆ R

is a bounded Lipschitz domain (cf. Assumption 3.1) and Γ is an open

subset of ∂Ω.

• The coeﬃcient function μ is a Lebesgue measurable, bounded function on Ω

taking its values in the set of real, symmetric, positive deﬁnite d×d matrices,

satisfying the usual ellipticity condition.

Remark 2.1. Concerning the notions ‘Lipschitz domain’ and ‘domain with Lip-

schitz boundary’ (synonymous: strongly Lipschitz domain) we follow the termi-

nology of Grisvard [34].

Remark 2.2. Since the requirement ‘Lipschitz domain’ does not become apparent

explicitly in the subsequent considerations, let us brieﬂy comment on this: it as-

sures the existence of a continuous extension operator E : L

(Ω) → L

)whose

restriction to H

1,2

(Ω) maps this space continuously into H

1,2

). This property

is fundamental for nearly all harmonic analysis techniques applied below, see [49,

Ch. 6.3] and [7].

316 R. Haller-Dintelmann and J. Rehberg

For ς ∈ ]0, 1] and 1 <q<∞ we deﬁne H

ς,q

(Ω) as the closure of

∞

(Ω) := {ψ|

: ψ ∈ C

∞

), supp(ψ) ∩ (∂Ω \Γ) = ∅}

in the Sobolev space H

ς,q

(Ω). Concerning the dual of H

ς,q

(Ω), we have to dis-

tinguish between the space of linear and the space of anti-linear forms on this

space. We deﬁne H

−ς,q

(Ω) as the space of continuous, linear forms on H

ς,q



(Ω)

and

−ς,q

(Ω) as the space of anti-linear forms on H

ς,q



(Ω) if 1/q +1/q



=1.Note

that L

spaces may be viewed as part of

−ς,q

for suitable ς,q via the identiﬁcation

of an element f ∈ L

with the anti-linear form H

ς,q



 ψ →



fψ dx.

If misunderstandings are not to be expected, we drop the Ω in the nota-

tion of spaces, i.e., function spaces without an explicitly given domain are to be

understood as function spaces on Ω.

By K we denote the open unit cube ]−1, 1[

in R

,byK

−

the lower half-cube

K ∩{x:x

< 0},byΣ=K ∩{x:x

=0} the upper plate of K

−

and by Σ

the

left half of Σ, i.e., Σ

=Σ∩{x:x

d−1

< 0}.

Throughout the paper we will use x, y,... for vectors in R

If B is a closed operator on a Banach space X,thenwedenotebydom

(B)

the domain of this operator. L(X,Y ) denotes the space of linear, continuous op-

erators from X into Y ;ifX = Y , then we abbreviate L(X). Furthermore, we will

write ·, ·



for the dual pairing of elements of X and the space X



of anti-linear

forms on X.

Finally, the letter c denotes a generic constant, not always of the same value.

3. Preliminaries

In this section we will properly deﬁne the elliptic divergence operator and after-

wards collect properties of the L

realizations of this operator which will be needed

in the subsequent sections.

Let us ﬁrst recall the concept of regular sets Ω ∪ Γ, introduced by Gr¨oger

in his pioneering paper [35], which will provide us with an adequate geometric

framework for all that follows.

Assumption 3.1. For any point x ∈ ∂Ω there is an open neighborhood Υ

x and a bi-Lipschitz mapping φ

from Υ

into R

, such that φ

(x) = 0 and



(Ω ∪Γ) ∩Υ



= K

−

or K

−

∪ ΣorK

−

∪ Σ

Remark 3.2. It is not hard to see that every Lipschitz domain and also its closure

is regular in the sense of Gr¨oger, the corresponding model sets are then K

−

∪Σ, respectively, see [34, Ch. 1.2]. In two and three space dimensions one can

give the following simplifying characterization for a set Ω ∪Γtoberegularinthe

sense of Gr¨oger, i.e., to satisfy Assumption 3.1, see [39]:

If Ω ⊆ R

is a bounded Lipschitz domain and Γ ⊆ ∂Ω is relatively open,

then Ω ∪ Γ is regular in the sense of Gr¨oger, iﬀ ∂Ω \ Γ is the ﬁnite union of

(non-degenerate) closed arc pieces.

Maximal Parabolic Regularity for Divergence Operators 317

In R

the following characterization can be proved:

If Ω ⊆ R

is a bounded Lipschitz domain and Γ ⊆ ∂Ω is relatively open, then

Ω∪Γ is regular in the sense of Gr¨oger, iﬀ the following two conditions are satisﬁed:

i) ∂Ω \ Γ is the closure of its interior (within ∂Ω).

ii) For any x ∈ Γ ∩ (∂Ω \ Γ)thereisanopenneighborhoodU of x and a bi-

Lipschitz mapping κ : U∩Γ ∩ (∂Ω \ Γ) → ]−1, 1[.

Following [27, Ch. 3.3.4 C], for every Lipschitz hypersurface H⊆Ωonecan

introduce a surface measure σ on H.ThisisinparticulartrueforH = ∂Ω, see

also [41]. Having this at hand, one can prove the following trace theorem.

Proposition 3.3. Assume q ∈ ]1, ∞[ and θ ∈ ]

, 1].LetΠ be a Lipschitz hyper-

surface in Ω and let be any measure on Π which is absolutely continuous with

respect to the surface measure σ. If the corresponding Radon-Nikod´ym derivative

is essentially bounded (with respect to σ), then the trace operator Tr is continuous

from H

θ,q

(Ω) to L

(Π, ).

Later we will repeatedly need the following interpolation result.

Proposition 3.4. Let Ω and Γ satisfy Assumption 3.1 and let θ ∈ ]0, 1[. Then for

∈ ]1, ∞[ and

1−θ

one has

1,q

]

and

−1,q

]

. (3.1)

We deﬁne the operator A : H

1,2

→

−1,2

Aψ, ϕ

−1,2



μ∇ψ ·∇ϕ dx +



κ ψ ϕ dσ, ψ, ϕ ∈ H

1,2

, (3.2)

where κ ∈ L

∞

(Γ, dσ). Note that in view of Proposition 3.3 the form in (3.2) is

well deﬁned.

In the special case κ = 0, we write more suggestively −∇ ·μ∇ instead of A.

The L

realization of A, i.e., the maximal restriction of A to the space L

,we

denote by the same symbol A; clearly this is identical with the operator which is

induced by the form on the right-hand side of (3.2). If B is a self-adjoint operator

on L

, then by the L

realization of B we mean its restriction to L

if p>2and

the L

closure of B if p ∈ [1, 2[.

First, we collect some basic facts on the operator −∇ ·μ∇.

Proposition 3.5.

i) The operator ∇·μ∇ generates an analytic semigroup on

−1,2

ii) The operator −∇ · μ∇ is self-adjoint on L

and bounded by 0 from below.

The restriction of −A to L

is densely deﬁned and generates an analytic

semigroup there.

iii) If λ>0 then the operator (−∇·μ∇+λ)

1/2

: H

1,2

→ L

provides a topological

isomorphism; in other words: the domain of (−∇ · μ∇ + λ)

1/2

on L

is the

form domain H

1,2