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Lp SOLVABILITY OF DIVERGENCE TYPE PARABOLIC ANDELLIPTIC SYSTEMS WITH PARTIALLY BMO COEFFICIENTS

HONGJIE DONG AND DOYOON KIM

Abstract. We prove the H1p,q solvability of second order systems in diver-

gence form with leading coefficients Aαβ measurable in (t, x1) and having smallBMO (bounded mean oscillation) semi-norms in the other variables. In addi-tion, we assume A11 to be measurable in t and have a small BMO semi-normin x. The corresponding results for the Cauchy problem and elliptic systemsare also established. Our results are new even for scalar equations. Using theresults for systems in the whole space, we obtain the solvability of systems on ahalf space and Lipschitz domain with either the Dirichlet boundary conditionor the conormal derivative boundary condition.

Contents

1. Introduction 12. Main results 33. Some estimates for systems with measurable coefficients 84. Systems with simple leading coefficients 145. Lp solvability of divergence systems with partially BMO coefficients 186. Mixed norm estimates 217. Systems on a half space 258. Remarks on systems in Lipschitz domains 27References 29

1. Introductionintro

The objective of this article is to find the least regularity assumption on thecoefficients for the Lp solvability of parabolic and elliptic systems in divergenceform. Many authors have studied the Lp theory of second order parabolic andelliptic equations with discontinuous coefficients. It is of particular interest notonly because of its various important applications in nonlinear equations, but alsodue to its subtle links with the theory of stochastic processes (for instance see [14]).

In [5] and [6], Chiarenza, Frasca and Longo initiated the study of the W 2p -

estimates for elliptic equations with VMO leading coefficients. Their proof is based

1991 Mathematics Subject Classification. 35R05, 35D10, 35K51, 35J45.Key words and phrases. Second-order equations, vanishing mean oscillation, partially BMO

coefficients, Sobolev spaces, mixed norms.H. Dong was partially supported by a start-up funding from the Division of Applied Mathe-

matics of Brown University, NSF grant number DMS-0635607 from IAS and NSF grant numberDMS-0800129.

1

2 H. DONG AND D. KIM

on certain estimates of Calderon-Zygmund theorem and the Coifman-Rochberg-Weiss commutator theorem. Regarding other developments in this direction, werefer the readers to Bramanti and Cerutti [2], Haller-Dintelmann, Heck and Hieber[10], Di Fazio [7], and references therein.

In this paper we consider parabolic operators in divergence form

Pu = −ut + Dα

(AαβDβu + Aαu

)+ BαDαu + Cu (1.1) eq0617_01

acting on (column) vector-valued functions u = (u1, · · · , um)T given either on Rd+1

or on a cylindrical domain in Rd+1. Here, we have used the notations Dαu = uxα,

Dαβu = uxαxβ, (α, β = 1, . . . , d) and the usual summation convention over repeated

indices is assumed. The coefficients Aαβ , Aα, Bα, and C are m×m matrix-valuedfunctions given on Rd+1; i.e., Aαβ = [Aαβ

ij (t, x)]m×m, etc. When the coefficientsAαβ , Aα, Bα and Cα are independent of t, we also define and consider ellipticoperators in divergence form

Lu = Dα(AαβDβu + Aαu) + BαDαu + Cu (1.2) eq3.27

acting on vector valued functions u = (u1, · · · , um)T given either on Rd or on itssubset.

In a recent interesting paper [15], Krylov treated in a unified way the Lp solv-ability of both divergence and non-divergence form parabolic equations with lead-ing coefficients measurable in the time variable and VMO in the spatial variables,which is denoted as VMOx. Unlike the arguments in [5, 6, 10], the proofs in [15] relymainly on pointwise estimates of sharp functions of spatial derivatives of solutions.It is worth noting that although the results in [15] are claimed for equations withVMO coefficients, the proofs there only require aij to have small mean oscillationsin small cylinders (or balls). This result was later extended in [16] to equations inmixed-norm Sobolev spaces.

The theory of elliptic and parabolic equations with partially VMO coefficientswas originated in Kim and Krylov [13]. In [13], the authors proved the W 2

p solvabil-ity of elliptic equations in non-divergence form with leading coefficients measurablein a fixed direction and VMO in the others. See also Krylov [17] and Kim [11, 12]for generalizations and extensions.

In contrast to scalar equations, until quite recently, there are relatively few re-sults of Lp theory for parabolic systems with discontinuous coefficients; see [18, 19,10, 1, 8] and reference therein. On the other hand, quite naturally, many evolution-ary equations arising from physical and economical problems are coupled systemsinstead of scalar equations, such as the Navier-Stokes equations. For higher orderparabolic systems in non-divergence form, the solvability in mixed norm spaceswith Ap Muckenhoupt weight is proved [10] assuming that the leading coefficientsare bounded, time-independent, and VMO in the spatial variables. Another setof papers concerning the Lp estimate of non-divergence type parabolic systemswith discontinuous coefficients are [18, 19], where the authors established the in-terior regularity of solutions in Lp spaces and Sobolev-Morrey spaces, when thecoefficients are VMO in both spatial and time variables. For both divergence andnon-divergence cases, recently in [8] we extended the results in [16] from scalarequations to systems with VMOx coefficients.

Here we shall extend the results in [16, 3, 8] to a much wider class of leadingcoefficients, i.e. partially BMO coefficients with small semi-norms. In particular,we assume the coefficients Aαβ are measurable in both t and x1, and have small

PARABOLIC AND ELLIPTIC SYSTEMS 3

BMO semi-norms in the other variables, except A11, which is measurable in t andhas a small BMO semi-norm in x (see Assumption 2.1 for a more precise definition).This class of coefficients are also more general than those in [10, 18, 19].

We establish solvability of divergence form parabolic systems in mixed normSobolev spaces H1

q,p (cf. Theorem 2.2), as well as the W 1p solvability for divergence

form elliptic systems (Theorem 2.3), generalizing the corresponding results in [3, 16,8]. We remark that our results are new even for scalar equations. As an application,we also obtain the solvability of divergence form parabolic/elliptic systems on a halfspace and on a Lipschitz domain with BMOx coefficients. As pointed out in [16],the interest in results concerning equations in mixed Sobolev norm spaces arises,for example, when one wants to get better regularity of traces of solutions for eachtime slide.

Our approach is developed upon the aforementioned method from [15] and [16].There are several obstacles in our case, which we explain below. Since Aαβ , αβ > 1are merely measurable in x1, we are not able to estimate the sharp function of thefull gradient Du as in [15, 16, 8]. Roughly speaking, one needs to bound Dx1u byDx′u. For this purpose, our idea is to break the ‘symmetry’ of the coordinates sothat t and x1 are distinguished from x′ ∈ Rd−1; see Lemma 3.3. Also note thatthe De Giorgi-Moser-Nash Holder estimate is not available for general parabolicand elliptic systems. To get around this, another ingredient of the proof is that inLemma 4.2 we use a bootstrap argument to establish an interior Holder estimate ofDx′u, from which a pointwise sharp function estimate of Dx′u follows; see Lemma5.2. These estimates, together with the Hardy-Littlewood maximal function the-orem and the Fefferman-Stein theorem on sharp functions, enable us to establishthe aforementioned solvability theorems.

The article is organized as follows. The main results about systems in the wholespace or on a half space, Theorem 2.2, 2.3, 2.5, 2.7, and 2.8 are stated in the nextsection after we define the functional spaces. Section 3 is devoted to some estimatesfor parabolic systems with measurable coefficients. The most crucial one is Lemma3.5, in which we estimate the Sobolev norm of Dx1u by those of Dx′u. Then inSection 4, we give an estimate for the L2-oscillation of Dx′u after establishing itsinterior Holder estimate. Using the L2-oscillation estimate and Lemma 3.5, weprove in Section 5 the H1

p solvability of systems with partially BMO coefficients,from which we derive Theorem 2.2 when p = q and Theorem 2.3. In Section 6we study the solvability in mixed norm Sobolev spaces to complete the proof ofTheorem 2.2 for arbitrary p, q ∈ (1,∞). Finally, the results for systems on a halfand a bounded domain are covered in Section 7 and Section 8, respectively.

2. Main resultssecmain

We begin the section by introducing some notation. Throughout the paper,we always assume that 1 < p, q < ∞ unless explicitly specified otherwise. ByN(d, p, · · · ) we mean that N is a constant depending only on the prescribed quan-tities d, p, · · · . A typical point in Rd is denoted by x = (x1, x2, · · · , xd). Very oftenwe also write x = (x1, x

′) ∈ Rd, x′ ∈ Rd−1. Let Du be the whole collection of Dαu,α = 1, · · · , d, or one of them depending on the context. By Dx′u we mean one ofDαu, α = 2, · · · , d, or the whole collection of them. For a (matrix-valued) function

4 H. DONG AND D. KIM

g(t, x) in Rd+1, we set

(g)O =1|O|

∫

Og(t, x) dx dt = –

∫

Og(t, x) dx dt,

where O is an open subset in Rd+1 and |O| is the d + 1-dimensional Lebesguemeasure of O.

Let

Br(x) = y ∈ Rd : |x− y| < r, B′r(x

′) = y′ ∈ Rd−1 : |x′ − y′| < r,Γr(x) = (x1 − r, x1 + r)×B′

r(x′),

Qr(t, x) = (t− r2, t)×Br(x), Λr(t, x) = (t− r2, t)× Γr(x).

Let Q =Qr(t, x) : (t, x) ∈ Rd+1, r ∈ (0,∞)

. For a function g defined on Rd+1,

we denote its (parabolic) maximal and sharp function, respectively, by

Mg(t, x) = supQ∈Q:(t,x)∈Q

–∫

Q

|g(s, y)| dy ds,

g#(t, x) = supQ∈Q:(t,x)∈Q

–∫

Q

|g(s, y)− (g)Q| dy ds.

Next we state assumptions on the coefficients of the operator (1.1) and (1.2).If the elliptic operator is considered, we always assume that the coefficients areindependent of t. Throughout the paper, we assume that there are constants K > 0and δ ∈ (0, 1) such that the m×m matrices Aαβ , Aα, Bα, and C satisfy

|Aαβij (t, x)| ≤ δ−1, |Aα

ij(t, x)| ≤ K, |Bαij(t, x)| ≤ K, |Cij(t, x)| ≤ K,

d∑

α,β=1

m∑

i,j=1

Aαβij (t, x)ξi

αξjβ ≥ δ

m∑

i=1

d∑α=1

|ξiα|2

for all (t, x) ∈ Rd+1 and ξα = (ξiα) ∈ Rm, α = 1, · · · , d.

To state another assumption on the coefficients Aαβ we introduce the followingnotations. Set Br = Br(0), |Br| to be the d-dimensional volume of Br(0), and |B′

r|to be the d− 1-dimensional volume of B′

r(0). Denote(Aαβ

)B′r(x′) (t, x1) = –

∫

B′r(x′)Aαβ(t, x1, y

′) dy′,

(Aαβ

)Br(x)

(t) = –∫

Br(x)

Aαβ(t, y) dy,

[Aαβ(t, x1, ·)

]B′r(x′) = –

∫

B′r(x′)

∣∣∣Aαβ(t, x1, y′)− (

Aαβ)B′r(x′) (t, x1)

∣∣∣ dy′,

[Aαβ(t, ·)]

Br(x)= –

∫

Br(x)

∣∣∣Aαβ(t, y)− (Aαβ

)Br(x)

(t)∣∣∣ dy.

Then we define

oscx′(Aαβ ,Λr(t, x)

)=

12r3

∫ t

t−r2

∫ x1+r

x1−r

[Aαβ(s, y1, ·)

]B′r(x′) dy1 ds,

oscx

(Aαβ , Qr(t, x)

)=

1r2

∫ t

t−r2

[Aαβ(s, ·)]

Br(x)ds.

PARABOLIC AND ELLIPTIC SYSTEMS 5

Finally we set

A#R = sup

(t,x)∈Rd+1supr≤R

oscx

(A11, Qr(t, x)

)+

∑

αβ>1

oscx′(Aαβ , Λr(t, x)

) .

In the case that Aαβ are independent of t, that is, coefficients for the ellipticoperator, we set

oscx′(Aαβ , Γr(x)

)=

12r

∫ x1+r

x1−r

[Aαβ(y1, ·)

]B′r(x′) dy1,

oscx

(Aαβ , Br(x)

)=

[Aαβ

]Br(x)

,

A#R = sup

(t,x)∈Rd+1supr≤R

oscx

(A11, Br(x)

)+

∑

αβ>1

oscx′(Aαβ , Γr(x)

) .

Our next assumption contains a parameter γ > 0, which will be specified later.

assumption0617_02 Assumption 2.1 (γ). There is an R0 ∈ (0, 1] such that A#R0≤ γ.

The following Sobolev spaces will be used in the article. Let Ω be a subsetin Rd. For elliptic equations, we use the usual Sobolev space W 1

p (Ω). For para-bolic equations, we consider Sobolev spaces with mixed norms. By a mixed norm‖u‖Lq,p((S,T )×Ω) we mean

‖u‖Lq,p((S,T )×Ω) =

(∫ T

S

(∫

Ω

|u(t, x)|p dx

)q/p

dt

)1/q

,

where −∞ ≤ S < T ≤ ∞.If Ω = Rd, we denote

H1q,p((S, T )× Rd) = (1−∆)1/2W 1,2

q,p ((S, T )× Rd),

where

W 1,2q,p ((S, T )× Rd) =

u : u, ut, Du, D2u ∈ Lq,p((S, T )× Rd)

.

We also introduce

H−1q,p((S, T )× Rd) = (1−∆)1/2Lq,p((S, T )× Rd). (2.1) eq10.39pm

As usual,H1

p = H1p,p, H−1

p = H−1p,p.

For any T ∈ (−∞,∞], we denote

RT = (−∞, T ), Rd+1T = RT × Rd.

Thus, for example, we have Lq,p(Rd+1T ) = Lq,p((−∞, T )×Rd). Especially, if T = ∞,

to simplify notation, we write Lq,p instead of Lq,p(R×Rd) or Lq,p(Rd+1∞ ). Similarly,

C∞0 means C∞0 (Rd+1), which is the collection of infinitely differential functions withcompact support in Rd+1.

One important property of the spaces given above is that (see [15])

‖u‖H1q,p(Rd+1

T )∼= ‖ut‖H−1

q,p(Rd+1T ) + ‖u‖Lq,p(Rd+1

T ) + ‖Du‖Lq,p(Rd+1T ).

6 H. DONG AND D. KIM

If Ω ( Rd, we denote H−1q,p((S, T )×Ω) to be the space consisting of all functions

f satisfying

inf‖g‖Lq,p((S,T )×Ω) + ‖h‖Lq,p((S,T )×Ω) | f = div g + h

< ∞.

It is easy to see that H−1q,p((S, T ) × Ω) is a Banach space. Naturally, for any f ∈

H−1q,p((S, T )× Ω), we define the norm

‖f‖H−1q,p((S,T )×Ω) = inf

‖g‖Lq,p((S,T )×Ω) + ‖h‖Lq,p((S,T )×Ω) | f = div g + h

. (2.2) eq10.36pm

Furthermore, when Ω = Rd the norm defined in (2.2) is equivalent to that definedin (2.1). We also define

H1q,p((S, T )× Ω) =

u : u,Du ∈ Lq,p((S, T )× Ω), ut ∈ H−1

q,p((S, T )× Ω)

.

As above, for T ∈ (−∞,∞], we denote ΩT = RT × Ω.For T > 0 we define H1

q,p((0, T ) × Ω) to be the subspace of H1q,p((0, T ) × Ω)

consisting of functions satisfying uχt≥0 ∈ H1q,p((−∞, T )× Ω).

Now we state our main results. By f and g below and throughout the paper wemean

f =(f1, · · · , fm

)tr,

g = (g1, · · · , gd) =[gi

α

]m×d

, gα = (g1α, · · · , gm

α )tr, α = 1, · · · , d.

The first two results are about parabolic and elliptic systems in the whole space.

thm2.3 Theorem 2.2. Let p, q ∈ (1,∞) and T ∈ (0,∞]. Then there is a constant γ =γ(d,m, p, q, δ) > 0 such that under Assumption 2.1 (γ) the following assertionshold.(i) For any f, g ∈ Lq,p((0, T )×Rd), there exists a unique solution u ∈ H1

q,p((0, T )×Rd) of

Pu = div g + f (2.3) eq2.38d

in (0, T )× Rd. Moreover, we have

‖u‖H1q,p((0,T )×Ω) ≤ N

(‖g‖Lq,p((0,T )×Ω) + ‖f‖Lq,p((0,T )×Ω)

),

where N depends only on d, m, p, q, δ, K, T , and R0.(ii) If Aα = Bα = C = 0, A11 = A11(t), and Aαβ = Aαβ(t, x1), αβ > 1, then onecan take N to be independent of T .

It is understood that u ∈ H1q,p((0, T )×Rd) satisfies (2.3) if and only if we have,

when Ω = Rd,∫ T

0

∫

Ω

(u · ht −AαβDβu ·Dαh−Aαu ·Dαh + BαDαu · h + Cu · h)

dx dt

=∫ T

0

∫

Ω

(−gα ·Dαh + f · h) dx dt +∫

Ω

u(T, ·) · h(T, ·) dx (2.4) eq27.3.58pm

for any h ∈ H1q′,p′((0, T )× Rd), where q′ and p′ satisfy

1/q + 1/q′ = 1, 1/p + 1/p′ = 1. (2.5) eq4.10.05

PARABOLIC AND ELLIPTIC SYSTEMS 7

mainthm2 Theorem 2.3. Let p ∈ (1,∞). Then there is a constant γ = γ(d,m, p, δ) > 0 suchthat under Assumption 2.1 (γ) the following assertions hold.(i) Assume u ∈ W 1

p (Rd), f, g ∈ Lp(Rd), and

Lu− λu = div g + f (2.6) eq10.07pm

in Rd. Then there exist nonnegative λ0 and positive N , depending only on d, m, p,δ, K, and R0, such that

√λ‖Du‖Lp(Rd) + λ‖u‖Lp(Rd) ≤ N

√λ‖g‖Lp(Rd) + N‖f‖Lp(Rd), (2.7) eq10.09pm

provided that λ ≥ λ0.(ii) For any λ > λ0 and f, g ∈ Lp(Rd), there exists a unique solution u ∈ W 1

p (Rd)of (2.6) in Rd.

rem2.5 Remark 2.4. One interesting implication of Theorem 2.2 and 2.3 is that underAssumption 2.1 (γ) solutions of parabolic (elliptic) systems are Holder continuousif q, p > d + 2 (p > d, respectively). Unlike scalar equations, for which we have thewell-known De Giorgi-Moser-Nash Holder estimate, generally this property is notpossessed by solutions to systems (see, for instance, [20]).

Next we state a few solvability results regarding parabolic and elliptic systems ona half space with either the Dirichlet boundary condition or the conormal derivativeboundary condition. First we state two theorems concerning parabolic systems.

mainthm3 Theorem 2.5. Let Ω = Rd+ := x ∈ Rd |x1 > 0, p, q ∈ (1,∞) and T ∈ (0,∞).

There is a constant γ = γ(d,m, p, q, δ) > 0 such that under Assumption 2.1 (γ)the following assertions hold. For any f, g ∈ Lq,p((0, T )× Ω) there exists a uniquesolution u ∈ H1

q,p((0, T )× Ω) of Pu = div g + f in (0, T )× Ω

u = 0 on (0, T )× ∂Ω . (2.8) eq10.59pm

Moreover, it holds that

‖u‖H1q,p((0,T )×Ω) ≤ N

(‖g‖Lq,p((0,T )×Ω) + ‖f‖Lq,p((0,T )×Ω)

), (2.9) eq10.59am

where N = N(d,m, p, q, δ,K, R0, T ) > 0.

It is understood that u ∈ H1q,p((0, T ) × Ω) satisfies (2.8) if and only if u = 0

on (0, T ) × ∂Ω and u satisfies (2.4) for any h ∈ H1q′,p′((0, T ) × Ω) vanishing on

(0, T )× ∂Ω, where p′ and q′ satisfy (2.5).

conPara Theorem 2.6. Let Ω = Rd+, p, q ∈ (1,∞) and T ∈ (0,∞). There is a constant γ =

γ(d,m, p, q, δ) > 0 such that under Assumption 2.1 (γ) the following assertions hold.For any f, g ∈ Lq,p((0, T )× Ω) there exists a unique solution u ∈ H1

q,p((0, T )× Ω)of Pu = div g + f in (0, T )× Ω

A1βDβu + A1u = g1 on (0, T )× ∂Ω , (2.10) eq2008061902

Moreover, (2.9) holds.

Solutions of (2.10) are understood in the weak sense. More precisely, we sayu ∈ H1

q,p((0, T )×Ω) satisfies (2.10) if we have (2.4) for any h ∈ H1q′,p′((0, T )×Ω),

where p′ and q′ satisfy (2.5).Now we present two theorems about elliptic systems on a half space.

8 H. DONG AND D. KIM

mainthm4 Theorem 2.7. Let Ω = Rd+, p ∈ (1,∞). There is a positive constant γ = γ(d,m, p, δ)

such that under Assumption 2.1 (γ) the following assertions hold.(i) Assume u ∈ W 1

p (Ω), u = 0 on ∂Ω, f, g ∈ Lp(Ω) and

Lu− λu = div g + f (2.11) eq11.01pm

in Ω. Then there exist nonnegative λ0 and positive N , depending only on d, m, p,δ, K, and R0, such that

√λ‖Du‖Lp(Ω) + λ‖u‖Lp(Ω) ≤ N

√λ‖g‖Lp(Ω) + N‖f‖Lp(Ω), (2.12) eq11.09pm

provided that λ ≥ λ0.(ii) For any λ > λ0 and f, g ∈ Lp(Ω), there exists a unique solution u ∈ W 1

p (Ω) of(2.11) in Ω and u|∂Ω = 0.

conEllip Theorem 2.8. Let Ω = Rd+, p ∈ (1,∞). There is a positive constant γ = γ(d,m, p, δ)

such that under Assumption 2.1 (γ) the following assertions hold.(i) Assume u ∈ W 1

p (Ω), f, g ∈ Lp(Ω) and Lu− λu = div g + f in Ω

A1βDβu + A1u = g1 on ∂Ω . (2.13) eq2008101702

Then there exist nonnegative λ0 and positive N , depending only on d, m, p, δ, K,and R0, such that (2.12) holds provided that λ ≥ λ0.(ii) For any λ > λ0 and f, g ∈ Lp(Ω), there exists a unique solution u ∈ W 1

p (Ω) of(2.13) in Ω.

Like before, solutions of (2.13) are understood in the weak sense. More precisely,we say u ∈ W 1

p (Ω) satisfies (2.13) if we have∫

Ω

(−AαβDβu ·Dαh−Aαu ·Dαh + BαDαu · h + (C − λ)u · h)dx

=∫

Ω

(−gα ·Dαh + f · h) dx

for any h ∈ W 1p′(Ω), where p′ satisfies 1/p + 1/p′ = 1.

As a consequence of Theorem 2.5-2.8, by using a partition of unity and the tech-nique of flattening the boundary, we obtain the solvability of elliptic and parabolicsystems with BMOx leading coefficients on a Lipschitz domain Ω with a small Lip-schitz constant. The exact statements of these results are given in section 8. Notethat results of this type were obtained recently in [3] for scalar equations, and in[4] for elliptic systems without lower order terms. Let us remark that the maindifference of the methods is that here the boundary estimate is obtained almostimmediately from the estimate in the whole space.

3. Some estimates for systems with measurable coefficientssec3

In this section, we set

P0u = −ut + Dα

(AαβDβu

),

where, unless stated otherwise, we do not impose any regularity assumptions onAαβ .

The following result is the classical L2 solvability.

PARABOLIC AND ELLIPTIC SYSTEMS 9

theorem001 Proposition 3.1. Let u ∈ H12(R

d+1T ) and P0u − λu = div g + f in Rd+1

T , whereT ∈ (−∞,∞], λ > 0, and f, g ∈ L2(Rd+1

T ). Then there exists N = N(d,m, δ) suchthat

‖ut‖H−12 (Rd+1

T ) +√

λ‖Du‖L2(Rd+1T ) + λ‖u‖L2(Rd+1

T )

≤ N(max

√λ, 1‖g‖L2(Rd+1

T ) + maxλ−1/2, 1‖f‖L2(Rd+1T )

)(3.1) eq6.48

and √λ‖Du‖L2(Rd+1

T ) + λ‖u‖L2(Rd+1T ) ≤ N

(√λ‖g‖L2(Rd+1

T ) + ‖f‖L2(Rd+1T )

).

Especially, if λ = 0 and f = 0, we have

‖Du‖L2(Rd+1T ) ≤ N‖g‖L2(Rd+1

T ).

Furthermore, for any λ > 0 and f, g ∈ L2(Rd+1T ), there exists a unique u ∈

H12(R

d+1T ) satisfying P0u− λu = div g + f and the estimate (3.1).

Due to the method of continuity, the proof of the above theorem follows from thea priori estimate (3.1), the proof of which is, as is well known, done by integrationby parts. More precisely, if T = ∞, using integration by parts we easily get

√λ‖Du‖L2 + λ‖u‖L2 ≤ N

(√λ‖g‖L2 + ‖f‖L2

).

Then we observe that

ut = div g + f + λu−Dα

(AαβDβu

),

‖ut‖H−12≤ N (‖g‖L2 + ‖f‖L2 + λ‖u‖L2 + ‖Du‖L2) . (3.2) eq002

The case T < ∞ is proved by the following standard argument. Using thesolvability result when T = ∞, we find w ∈ H1

p satisfying, in Rd+1,

P0w− λw = div g + f, where g = It<Tg, f = It<T f.

Then we see that the estimate (3.1) (with T < ∞) holds true if u is replaced by w.Thus we only need to show that u = w in Rd+1

T . Take a function v ∈ H12 such that

v = u in Rd+1T . Since (P0 − λ) v = (P0 − λ)u = (P0 − λ)w in Rd+1

T ,

(P0 − λ) (v−w) = div g1 + f1, where g1 = f1 = 0 if t < T.

From this equation, if P0 is replaced by −∂/∂t+∆, we see that v−w = 0 whenevert < T . Using this, the estimate for T = ∞, and the method of continuity, we seethat v − w = 0 for t < T with P0 as it is. Since v = u if t < T , we conclude thatu = w in Rd+1

T .Using the above global L2-estimate, we obtain a local L2-estimate as follows.

lem10.44 Corollary 3.2. Let r > 0 and ν > 1. Assume that u ∈ C∞0 and P0u = div g + fin Qνr, where f, g ∈ L2(Qνr). Then there exists a constant N = N(d,m, δ, ν) suchthat

‖Du‖L2(Qr) ≤ N(‖g‖L2(Qνr)+ r‖f‖L2(Qνr)

+ r−1‖u‖L2(Qνr)). (3.3) eq10.48

Proof. We remark that (3.3) can be proved in a standard way by multiplying bothsides of the equation by ζu, where ζ is a suitable cut-off function, and then inte-grating by parts. Here we present another proof, which is more flexible and alsoworks if one replaces the L2 norm by an Lp norm, provided that the correspondingglobal estimate is available (see Corollary 3.4 and 6.3).

10 H. DONG AND D. KIM

Set R = νr and

r0 = r, rn = r + (R− r)n∑

k=1

12k

, Qn = (−r2n, 0)×Brn

, n = 1, 2, · · · ,

Then we find ζn(t, x) ∈ C∞0 such that

ζn =

1 on Qn

0 on Rd+1 \ (−r2n+1, r

2n+1)×Brn+1

and

|(ζn)x| ≤ N2n

R− r, |(ζn)t| ≤ N

22n

(R− r)2, |(ζn)xx| ≤ N

22n

(R− r)2.

Observe that, for λn > 0,

(P0 − λn)(ζnu) = div gn + fn in Rd+1,

where

gn = (gnα) , gnα = ζngα +d∑

β=1

(Dβζn)Aαβu,

fn = ζnf−d∑

α=1

(Dαζn) gα +d∑

α,β=1

(Dαζn)AαβDβu− (Dtζn)u− λnζnu.

Then by Proposition 3.1

‖D (ζnu) ‖L2(Rd+10 ) ≤ N

(‖gn‖L2(Rd+1

0 ) + λ−1/2n ‖fn‖L2(Rd+1

0 )

)

≤ N

(2n

R− r+ λ−1/2

n

22n

(R− r)2+ λ1/2

n

)‖u‖L2(QR) + Nλ−1/2

n ‖f‖L2(QR)

+N

(1 + λ−1/2

n

2n

R− r

)‖g‖L2(QR) + Nλ−1/2

n

2n

R− r‖D (ζn+1u) ‖L2(Rd+1

0 ).

Set

An = ‖D (ζnu) ‖L2(Rd+10 ), B = ‖u‖L2(QR), C = ‖g‖L2(QR), D = ‖f‖L2(QR).

Then

An ≤ N

(2n

R− r+ λ−1/2

n

22n

(R− r)2+ λ1/2

n

)B

+N

(1 + λ−1/2

n

2n

R− r

)C + Nλ−1/2

n D + Nλ−1/2n

2n

R− rAn+1.

By multiplying both sides by εn and summing up with respect to n, we have∞∑

n=0

εnAn ≤ NB∞∑

n=0

(2n

R− r+ λ−1/2

n

22n

(R− r)2+ λ1/2

n

)εn

+NC∞∑

n=0

(1 + λ−1/2

n

2n

R− r

)εn + ND

∞∑n=0

λ−1/2n εn + N

∞∑n=0

λ−1/2n

(2ε)n

R− rAn+1.

Now set ε = 2−2 and √λn =

N

(R− r)ε2n,

PARABOLIC AND ELLIPTIC SYSTEMS 11

where N is the constant in the last term of the above inequality. Then∞∑

n=0

(2n

R− r+ λ−1/2

n

22n

(R− r)2+ λ1/2

n

)εn =

N + 2−2 + 22N2

N(R− r)

∞∑n=0

2−n,

∞∑n=0

(1 + λ−1/2

n

2n

R− r

)εn =

(1 +

2−2

N

) ∞∑n=0

2−2n,

∞∑n=0

λ−1/2n εn =

R− r

4N

∞∑n=0

2−3n,

N

∞∑n=0

λ−1/2n

(2ε)n

R− rAn+1 =

∞∑n=0

εn+1An+1 =∞∑

n=1

εnAn.

Therefore,∞∑

n=0

εnAn ≤ N(R− r)−1B + NC + N(R− r)D +∞∑

n=1

εnAn. (3.4) eq_001

On the other hand∞∑

n=0

εnAn < ∞

becauseAn ≤ N2n(R− r)−1‖u‖L2(QR) + N‖Du‖L2(QR).

Then the inequality (3.4) implies that

A0 ≤ N(R− r)−1B + NC + N(R− r)D

= N(R− r)−1‖u‖L2(QR) + N‖g‖L2(QR) + N(R− r)‖f‖L2(QR).

Finally, note that‖Du‖L2(Qr) ≤ A0.

The lemma is proved. ¤

We shall prove that, for a solution u to the parabolic system P0u = div g, theLp-norm of Du is estimated by that of Dx′u and g if the first coefficient matrix A11

is a function of only t ∈ R. Recall that by Dx′u we mean Dαu, where α = 2, · · · , d.Note that we don’t require any regularity assumptions on the coefficient matricesAαβ for αβ > 1.

lem1 Lemma 3.3. Let T ∈ (−∞,∞], p ∈ (1,∞), and A11 = A11(t). Assume u ∈H1

p(Rd+1T ) and P0u − λu = div g + f in Rd+1

T , where λ ≥ 0 and f, g ∈ Lp(Rd+1T ).

Then there exists a constant N , depending only on d, m, δ and p, such that√

λ‖Du‖Lp(Rd+1T ) + λ‖u‖Lp(Rd+1

T )

≤ N(√

λ‖Dx′u‖Lp(Rd+1T ) +

√λ‖g‖Lp(Rd+1

T ) + ‖f‖Lp(Rd+1T )

).

Especially, if λ = 0 and f = 0, then

‖Du‖Lp(Rd+1T ) ≤ N

(‖Dx′u‖Lp(Rd+1

T ) + ‖g‖Lp(Rd+1T )

).

12 H. DONG AND D. KIM

Proof. The case when λ = 0 and f = 0 follows by just letting λ 0 after theestimate for λ > 0 is proved.

We use a scaling argument. Let v(t, x1, x′) = u(µ−2t, µ−1x1, x

′) with a suffi-ciently large constant µ to be chosen later. Then v satisfies, in Rd+1

µ2T ,

−vt + Dx1(A11Dx1v) + µ−1

∑α>1

Dα(Aα1Dx1v) + µ−1∑

β>1

Dx1(A1βDβv)

+µ−2∑

α,β>1

Dα(AαβDβv)− µ−2λv = µ−2 div g + µ−2f,

where Aαβ = Aαβ(µ−2t, µ−1x1, x′) and

g = (µg1, g2, · · · , gd)(µ−2t, µ−1x1, x

′), f = f(µ−2t, µ−1x1, x′).

Denote P0v = −vt + Dx1(A11v) + ∆d−1v, where ∆d−1v =

∑dα=2 D2

αv. Then wehave

P0v− µ−2λv = div g + f (3.5) eq3.17

in Rd+1µ2T , where

f = µ−2f, g1 = µ−2g1 − µ−1∑

β>1

A1βDβv,

gα = µ−2gα − µ−1Aα1Dx1v− µ−2∑

β>1

AαβDβv + Dαv, α ≥ 2.

Now we apply, for instance, Theorem 7.1 ∗ of [8] on (3.5) to get√

µ−2λ‖Dv‖Lp(Rd+1µ2T

) + µ−2λ‖v‖Lp(Rd+1µ2T

) ≤ N

(√µ−2λ‖g‖Lp(Rd+1

µ2T) + ‖f‖Lp(Rd+1

µ2T)

)

≤ Nµ−2√

µ−2λ‖gα‖Lp(Rd+1µ2T

) + N√

µ−2λ‖Dx′v‖Lp(Rd+1µ2T

)

+Nµ−2‖f‖Lp(Rd+1µ2T

) + Nµ−1√

µ−2λ‖Dv‖Lp(Rd+1µ2T

),

where N = N(d,m, δ, p). By choosing µ = 2N , where N is the constant in the lastterm above, we arrive at

√µ−2λ‖Dv‖Lp(Rd+1

µ2T) + 2µ−2λ‖v‖Lp(Rd+1

µ2T)

≤ 2N

(µ−2

√µ−2λ‖gα‖Lp(Rd+1

µ2T) +

√µ−2λ‖Dx′v‖Lp(Rd+1

µ2T) + µ−2‖f‖Lp(Rd+1

µ2T)

).

Returning to u, we obtain

µ−1√

µ−2λ‖Dx1u‖Lp(Rd+1T ) +

√µ−2λ‖Dx′u‖Lp(Rd+1

T ) + 2µ−2λ‖u‖Lp(Rd+1T )

≤ Nµ−1√

µ−2λ‖g1‖Lp(Rd+1T ) + Nµ−2

√µ−2λ

∑

α≥2

‖gα‖Lp(Rd+1T )

+N√

µ−2λ‖Dx′u‖Lp(Rd+1T ) + Nµ−2‖f‖Lp(Rd+1

T ).

∗The right-hand side of the estimate in this theorem is, in fact, a constant times (√

λ+1)‖div g+

f‖H−1p (Rd+1

T), which can be replaced by

√λ‖g‖

Lp(Rd+1T

)+ ‖f‖

Lp(Rd+1T

)(see Corollary 7.5 in the

same paper). Moreover, in that theorem λ must be a number such that λ ≥ λ0, where λ0 is notnecessarily zero, but in our case since the coefficients of the operator P0 depends only on t, usinga scaling argument we obtain the estimate for any λ ≥ 0.

PARABOLIC AND ELLIPTIC SYSTEMS 13

Multiplying both sides of the above inequality by µ2(µ = 2N) leads us to√

λ‖Dx1u‖Lp(Rd+1T ) + µ

√λ‖Dx′u‖Lp(Rd+1

T ) + λ‖u‖Lp(Rd+1T )

≤ N√

λ‖g1‖Lp(Rd+1T ) + Nµ−1

√λ

∑

α≥2

‖gα‖Lp(Rd+1T )

+N√

λµ‖Dx′u‖Lp(Rd+1T ) + N‖f‖Lp(Rd+1

T ).

This proves the estimate for λ > 0 in the lemma. The lemma is proved. ¤

We also need a local version of Lemma 3.3. As Corollary 3.2 is proved by usingthe global estimate in Proposition 3.1, the lemma below follows from the globalestimate in Lemma 3.3. Thus we simply repeat the proof of Corollary 3.2 with pin place of 2.

lem1.4 Corollary 3.4. Let r > 0, ν, p ∈ (1,∞), and A11 = A11(t). Assume u ∈ C∞0and P0u = div g in Qνr, where g ∈ Lp(Qνr). Then there exists a constant N =N(d,m, δ, p, ν) such that

‖Du‖Lp(Qr) ≤ N(‖Dx′u‖Lp(Qνr) + ‖g‖Lp(Qνr) + r−1‖u‖Lp(Qνr)).

Now we prove the same type of estimate as in Lemma 3.3 when A11 is measurablein t ∈ R and has a small BMO semi-norm in x ∈ Rd. Again, all the other coefficientsare just measurable and bounded.

lem3 Lemma 3.5. Let p ∈ (1,∞) and g ∈ Lp. Then there is a constant γ = γ(d, m, p, δ) >0 such that if A11 satisfies Assumption 2.1 (γ), the following assertion holds.

There exist µ ∈ [1,∞) and N , depending only on d, m, p and δ, such that, foru ∈ C∞0 vanishing outside Qµ−1R0 , we have

‖Du‖Lp ≤ N(‖Dx′u‖Lp + ‖g‖Lp

), (3.6) eq3.02b

provided that P0u = div g.

Proof. As in the proof of Lemma 3.3, we define v(t, x1, x′), A, g, g, P0 in the same

way, that is,

P0v = −vt + Dx1

(A11Dx1v

)+ ∆d−1v,

P0v = div g,

Clearly v is supported in QR0 . It is easy to check that A11 satisfies Assumption2.1 (2µ4γ) with the same R0 as A11. Then by Lemma 7.3 of [8] we have

(|Dv− (Dv)Qr(t,x)|q)Qr(t,x)

≤ Nκ−q (|Dv|q)Qκr(t,x)

+Nκd+2((|g|q)Qκr(t,x) + (µ4γ)1/σ (|Dv|qτ )1/τ

Qκr(t,x)

)

for any r ∈ (0,∞), κ ≥ 4, and (t, x) ∈ Rd+1, where q ∈ (1,∞), τ ∈ (1,∞),1/τ + 1/σ = 1, and N = N(d,m, p, δ, τ). Take q > 1 and τ > 1 such thatqτ ∈ (1, p). We get from the above inequality that

(Dv)#(t, x) ≤ Nκ(d+2)/q (M(|g|q)(t, x))1/q + Nκ−1 (M(|Dv|q)(t, x))1/q

+Nκ(d+2)/q(µ4γ)1/(qσ)(M(|Dv|qτ )(t, x)

)1/(qτ)

14 H. DONG AND D. KIM

for all κ ≥ 4, (t, x) ∈ Rd+1, where N = N(d,m, p, δ). By applying the Fefferman-Stein theorem on sharp functions and the Hardy-Littlewood maximal function the-orem to the above inequality we obtain

‖Dv‖Lp≤ Nκ(d+2)/q‖g‖Lp

+ N(κ−1 + κ(d+2)/q(µ4γ)1/(qσ)

)‖Dv‖Lp

,

where κ ≥ 4. Note

‖g‖Lp≤ µ−2‖g‖Lp

+ Nµ−1‖Dv‖Lp+ N(µ−2 + 1)‖Dx′v‖Lp

.

Hence from the above two inequalities,

‖Dv‖Lp≤ N

(‖g‖Lp+ ‖Dx′v‖Lp

)

+N1

[κ−1 + κ(d+2)/q

((µ4γ)1/(qσ) + µ−1

)]‖Dx1v‖Lp ,

where N = N(d,m, p, δ, κ, µ) and N1 = N1(d,m, p, δ), i.e., N1 is independent of κand µ. Now choose κ sufficiently large, then µ sufficiently large, and γ sufficientlysmall such that

κ−1 + κ(d+2)/q((µ4γ)1/(qσ) + µ−1

)≤ 1

2N1.

Then‖Dv‖Lp ≤ N

(‖g‖Lp + ‖Dx′v‖Lp

).

By returning to u and g, we have the desired inequality. ¤

4. Systems with simple leading coefficientssec4

Throughout this section we set

Pu = −ut + Dα

(AαβDβ

), (4.1) 20071120_01

where the coefficient A11 = A11(t), and Aαβ = Aαβ(t, x1) whenever (α, β) 6= (1, 1).The main purpose of this section is to estimate the L2-oscillations of Dx′u (Propo-sition 4.5), which will be used in the next section.

A proof of the following Sobolev type inequality can be found, for example, inLemma 8.1 of [16].

lem2.2 Lemma 4.1. Let r ∈ (0,∞), q ∈ (1, p], and assume that

1q− 1

p≤ 1

d + 2.

Let ζ ∈ C∞0 be such that ζ = 1 in Qr. Then for any function u such that uζ ∈H1

q(Rd+10 ), we have u ∈ Lp(Qr) and

‖u‖Lp(Qr) ≤ N‖uζ‖H1q(Rd+1

0 ),

where N = N(r, d,m, p, q, ζ).

Now we are ready to state and prove the following Holder estimate of Dx′u byusing a bootstrap argument. We note that for scalar equations similar estimate withcertain exponents γ follows immediately from the De Giorgi-Moser-Nash estimate.

As usual, for 0 < γ < 1,

‖u‖Cγ/2,γ(Ω) = |u|0,Ω + [u]γ/2,γ,Ω,

PARABOLIC AND ELLIPTIC SYSTEMS 15

where

|u|0,Ω = sup(t,x)∈Ω

|u(t, x)|, [u]γ/2,γ,Ω = sup(t,x),(s,y)∈Ω(t,x)6=(s,y)

|u(t, x)− u(s, y)||t− s|γ/2 + |x− y|γ .

importantlem Lemma 4.2. Assume u ∈ C∞loc and Pu = 0 in Q2. Then

‖u‖Cγ/2,γ(Q1) + ‖Dx′u‖Cγ/2,γ(Q1) ≤ N(d,m, δ, γ)‖u‖L2(Q2)

for any γ ∈ (0, 1).

Proof. We will prove the lemma by a bootstrap argument. Take an increasingsequence pj ∈ [2,∞), j = 0, 1, · · · , n, where n depends only on d, such that

p0 = 2, pn = (d + 2)/(1− γ),1pj− 1

pj+1<

1d + 2

, j = 0, · · · , n− 1.

Then take a sequence of cylinders

Q(j)(:= Q(j)(0, 0)), j = −n− 1, · · · , 0, 1, · · · , 3n + 2,

such that Q(3n+2) = Q1, Q(−n−1) ( Q2, and

Q(j) ( Q(j−1), j = −n, · · · , 0, 1, · · · , 3n + 2.

We also come up with a sequence of suitable cutoff functions ζ(j), j = 1, 2, · · · , 3n+2, such that ζ(j) = 1 on Q(j) and ζ(j) vanishes outside the closure of Q(j−1) ∪(−Q(j−1)).

First we consider the following equation in Rd+10 satisfied by ζ(3j+2)u, j =

0, 1, · · · , n, (P − 1)(ζ(3j+2)u) = div g(j) + f(j),

where

g(j)α =

d∑

β=1

Dβζ(3j+2)Aαβu, α = 1, · · · , d,

f(j) =d∑

α,β=1

Dαζ(3j+2)AαβDβu−Dtζ(3j+2)u− ζ(3j+2)u.

Then by Lemma 3.3 (also recall the definition of H1p(R

d+1T ) and the inequality (3.2)

with p and Rd+10 in place of 2 and Rd+1 respectively)

‖ζ(3j+2)u‖H1pj

(Rd+10 ) ≤ N

(‖Du‖Lpj

(Q(3j+1)) + ‖u‖Lpj(Q(3j+1))

).

By Corollary 3.4 applied to Pu = 0 in Q(3j), we have

‖Du‖Lpj(Q(3j+1)) ≤ N

(‖Dx′u‖Lpj

(Q(3j)) + ‖u‖Lpj(Q(3j))

).

Hence from the above two inequalities we obtain, for j = 0, 1, · · · , n,

‖ζ(3j+2)u‖H1pj

(Rd+10 ) ≤ N

(‖Dx′u‖Lpj

(Q(3j)) + ‖u‖Lpj(Q(3j))

). (4.2) eq004

Now we prove that, for j = 0, 1, · · · , n,

‖Dx′u‖Lpj(Q(3j)) + ‖u‖Lpj

(Q(3j)) ≤ N‖u‖L2(Q(−j−1)). (4.3) eq003

This inequality holds true if j = 0 (p0 = 2) due to Corollary 3.2 applied to Pu = 0in Q(−1), so we prove the inequality (4.3) for j + 1 assuming that it is satisfied for

16 H. DONG AND D. KIM

0, 1, · · · , j, where 0 ≤ j ≤ n − 1. Observe that by Lemma 4.1 and the inequality(4.2)

‖u‖Lpj+1 (Q(3j+3)) ≤ ‖ζ(3j+2)u‖H1pj

(Rd+10 ) ≤ N

(‖Dx′u‖Lpj

(Q(3j)) + ‖u‖Lpj(Q(3j))

).

From this and the induction hypothesis we obtain

‖u‖Lpj+1 (Q(3j+3))) ≤ N‖u‖L2(Q(−j−1)). (4.4) eq005

This along with the fact that Dx′u satisfies the same equation as u implies

‖Dx′u‖Lpj+1 (Q(3j+3))) ≤ N‖Dx′u‖L2(Q(−j−1)). (4.5) eq006

Note that by Corollary 3.2

‖Dx′u‖L2(Q(−j−1)) ≤ N‖u‖L2(Q(−j−2)).

Combining this inequality as well as (4.5) and (4.4) we prove (4.3) with j + 1 inplace of j. Thus the inequality (4.3) is proved for all j = 0, 1, · · · , n.

Considering the inequalities (4.2) and (4.3) when j = n, we arrive at

‖ζ(3n+2)u‖H1pn

(Rd+10 ) ≤ N‖u‖L2(Q(−n−1)).

Due to the classical Sobolev embedding theorem of parabolic type, this inequalitygives

‖u‖Cγ/2,γ(Q1) ≤ N‖u‖L2(Q(−n−1))

since γ = 1− (d+2)p−1n . Using again the fact that Dx′u satisfies the same equation

in Q2, we have

‖Dx′u‖Cγ/2,γ(Q1) ≤ N‖Dx′u‖L2(Q(−n−1)) ≤ N‖u‖L2(Q2), (4.6) eq11.49b

where the last inequality is due to Corollary 3.2. This completes the proof. ¤

Based on the above estimate we prove that the Holder norm of a solution to, insome sense, a harmonic equation is estimated by the L2-norm of its derivatives.

lemma0802 Lemma 4.3. Let κ ≥ 2, r > 0, and λ ≥ 0. Assume u ∈ C∞loc and Pu− λu = 0 inQκr. Then, for any γ ∈ (0, 1), we have

[Dx′u]γ/2,γ,Qr≤ N(κr)−γ(|Du|2 + λ|u|2)1/2

Qκr,

where N = N(d,m, δ, γ).

Proof. First we deal with the case λ = 0. To use a scaling argument we set

Aαβ(t, x) = Aαβ((κr/2)2t, (κr/2)x), v(t, x) = u((κr/2)2t, (κr/2)x).

Then v satisfies, in Q2,

vt + Dα

(AαβDβv

)= 0.

Now we apply Lemma 4.2 to Dx′v, which also satisfies the above equation, (or justthe first inequality in (4.6)) to get

[Dx′v]γ/2,γ,Q1≤ N‖Dv‖L2(Q2).

Returning back to u, we obtain

[Dx′u]γ/2,γ,Qr≤ [Dx′u]γ/2,γ,Qκr/2

≤ N(κr)−γ(|Du|2)1/2Qκr

.

PARABOLIC AND ELLIPTIC SYSTEMS 17

For general λ > 0, we use the idea by S. Agmon. Denote by z = (x, y) a pointin Rd+1, where x ∈ Rd, y ∈ R. Then we introduce u(t, z) and Qr given by

u(t, z) = u(t, x, y) = u(t, x) cos(√

λy),

Qr = (−r2, 0)× |z| < r : z ∈ Rd+1.Observe that

[Dx′u]γ/2,γ,Qr≤ [Dx′ u]γ/2,γ,Qr

.

On the other hand, due to the fact that Pu− λu = 0 in Qκr, we have

Pu + Dy (Dyu) = 0 in Qκr.

Then by applying the above result to u we obtain

[Dx′ u]γ/2,γ,Qr≤ N(κr)−γ

(|Dzu|2)1/2

Qκr. (4.7) eq0804

Notice that Dzu is the collection consisting of

cos(√

λy)Dxu, −√

λ sin(√

λy)u.

Thus the right-hand side of (4.7) is not greater than the right-hand side of theinequality in the lemma. The lemma is proved. ¤

cor1.52 Corollary 4.4. Let κ ≥ 2, r > 0, and λ ≥ 0. Assume u ∈ C∞loc and Pu− λu = 0in Qκr. Then for any γ ∈ (0, 1) we have(

|Dx′u− (Dx′u)Qr|2

)Qr

≤ Nκ−2γ(|Du|2 + λ|u|2)

Qκr,

where N = N(d,m, δ, γ).

Proof. The estimate follows immediately from Lemma 4.3. ¤By applying Corollary 4.4 and the L2 solvability, we establish the following

sharp function estimate. We emphasize that contrary to the arguments in [16], thisestimate alone is not sufficient for proving the main theorems. However, combiningit with Lemma 3.5 enables us to prove Theorem 2.2 with p = q in the next section.

thm12.14 Proposition 4.5. Let κ ≥ 4, r > 0, u ∈ C∞loc, and g ∈ L2,loc. Assume thatPu = div g in Qκr. Then there exists a constant N depending only on d, m, and δsuch that(|Dx′u− (Dx′u)Qr |2

)Qr≤ Nκ−1

(|Du|2)Qκr

+ Nκd+2(|g|2)

Qκr. (4.8) eq12.34

Proof. We can certainly assume that u and g have compact supports. In addition,we assume that Aαβ , u, and g are infinitely differentiable. Indeed, if not, we takethe standard mollifications and prove the estimate for the mollifications. Then wetake the limits because the concerned constants are independent of the smoothnessof the functions involved.

Take ζ ∈ C∞0 such that

ζ = 1 on Qκr/2, ζ = 0 outside (−(κr)2, (κr)2)×Bκr.

By Proposition 3.1, for any fixed λ > 0 there exists a unique v ∈ H12(R

d+10 ) satis-

fying (P − λ)v = div((1− ζ)g).

Let w = u− v so that w ∈ H12(R

d+10 ) satisfies(P − λ

)w = div(ζg)− λu.

18 H. DONG AND D. KIM

Note (1−ζ)g = 0 in Qκr/2, i.e.,(P − λ

)v = 0 in Qκr/2, and κ/2 ≥ 2. Also note that

by the classical result v is in fact infinitely differentiable because the coefficients ofthe operator as well as (1− ζ)g are smooth. Thus by Corollary 4.4 with γ = 1/2(

|Dx′v− (Dx′v)Qr|2

)Qr

≤ Nκ−1(|Dv|2 + λ|v|2)

Qκr. (4.9) eq1003

On the other hand, by Proposition 3.1 we have√λ‖Dw‖L2(Rd+1

0 ) + λ‖w‖L2(Rd+10 ) ≤ N

√λ‖ζg‖L2(Rd+1

0 ) + Nλ‖u‖L2(Rd+10 ).

In particular,‖Dw‖L2(Qr) ≤ N‖g‖L2(Qκr) + Nλ‖u‖L2(Rd+1

0 ), (4.10) eq1001

‖Dw‖L2(Qκr) +√

λ‖w‖L2(Qκr) ≤ N‖g‖L2(Qκr) + Nλ‖u‖L2(Rd+10 ). (4.11) eq1004

The left-hand side of (4.8) is bounded by a constant times(|Dx′v− (Dx′v)Qr |2

)Qr

+(|Dx′w|2

)Qr

≤ Nκ−1(|Dv|2 + λ|v|2)

Qκr+ Nκd+2

(|g|2)Qκr

+ Nλr−d−2‖u‖2L2(Rd+1

0 ),

where the inequality is due to (4.9) and (4.10). By (4.11)(|Dv|2 + λ|v|2)

Qκr≤ N

(|Du|2 + λ|u|2)Qκr

+ N(|Dw|2 + λ|w|2)

Qκr

≤ N(|Du|2 + λ|u|2)

Qκr+ N

(|g|2)Qκr

+ Nλ(κr)−d−2‖u‖2L2(Rd+1

0 ).

Therefore, (|Dx′u− (Dx′u)Qr |2)Qr

≤ Nκ−1(|Du|2 + λ|u|2)

Qκr+ Nκd+2

(|g|2)Qκr

+ Nλr−d−2‖u‖2L2(Rd+1

0 ).

To finish the proof, we let λ 0. ¤

5. Lp solvability of divergence systems with partially BMOcoefficients

sec5

We prove in this section Theorem 2.2 when p = q and Theorem 2.3. In fact, weconcentrate on proving Theorem 5.1 below, from which the above two results canbe derived easily. Throughout the section, by P we mean the operator in Theorem2.2, the coefficients of which satisfy Assumption 2.1. That is, A11 is measurable in tand has a small BMO semi-norm in x ∈ Rd and Aαβ , (α, β) 6= (1, 1), are measurablein (t, x1) and have small BMO semi-norms in x′ ∈ Rd−1.

thm3.1 Theorem 5.1. Let p ∈ (1,∞) and T ∈ (−∞,∞]. Then there is a constant γ =γ(d,m, p, δ) > 0 such that under Assumption 2.1 (γ) the following assertions hold.(i) Assume u ∈ H1

p(Rd+1T ), f, g ∈ Lp. There exist positive λ0 and N , depending

only on d, m, p, δ, K, and R0, such that

‖ut‖H−1p (Rd+1

T ) +√

λ‖Du‖Lp(Rd+1T ) + λ‖u‖Lp(Rd+1

T )

≤ N√

λ‖g‖Lp(Rd+1T ) + N‖f‖Lp(Rd+1

T ), (5.1) eq2.37

provided that λ ≥ λ0 and, in Rd+1T ,

Pu− λu = div g + f. (5.2) eq2.38

(ii) For any λ ≥ λ0 and f, g ∈ Lp(Rd+1T ), there exists a unique solution u ∈

H1p(R

d+1T ) of (5.2) satisfying (5.1).

PARABOLIC AND ELLIPTIC SYSTEMS 19

(iii) If Aα = Bα = C = 0, A11 = A11(t), and Aαβ = Aαβ(t, x1), αβ > 1, then thesolvability result in (ii) holds true for all λ > 0. Moreover, we have an estimate asin (3.1) for all λ > 0 with p in place of 2.

We have established almost all necessary steps toward proving this theorem inthe previous sections, so we need only a few more observations before we present acomplete proof of the theorem.

lem2.49 Lemma 5.2. Let Aα = Bα = C = 0, τ, σ ∈ (1,∞), 1/τ+1/σ = 1, and R ∈ (0,∞).Assume u ∈ C∞loc vanishing outside QR and Pu = div g, where g ∈ L2. Then thereexists a positive constant N , depending only on d, m, δ, and τ , such that(|Dx′u− (Dx′u)Qr(s,y)|2

)Qr(s,y)

≤ Nκ−1(|Du|2)

Qκr(s,y)

+ Nκd+2((|g|2)Qκr(s,y) + (A#

R)1/σ(|Du|2τ )1/τQκr(s,y)

), (5.3) eq2.55

for any r ∈ (0,∞), κ ≥ 4 and (s, y) ∈ Rd+1.

Proof. It follows immediately from Proposition 4.5 by using the technique of freez-ing coefficients; cf. Theorem 5.3 of [15] or Lemma 7.3 of [8]. ¤

Below, as in the proof of Lemma 3.5, we obtain an Lp-estimate of solutions byusing pointwise estimates of sharp functions of their derivatives.

lem3.05 Lemma 5.3. Let p ∈ (2,∞), Aα = Bα = C = 0, and µ be the constant fromLemma 3.5. Then there is a constant γ = γ(d,m, p, δ) > 0 such that under As-sumption 2.1 (γ) the following assertion holds. There exists a positive constant N ,depending only on d, m, p, and δ, such that, for u ∈ C∞0 vanishing outside Qµ−1R0

and g ∈ Lp satisfying Pu = div g, we have

‖Du‖Lp ≤ N‖g‖Lp .

Proof. Choose a τ ∈ (1,∞) such that p > 2τ > 2 and set

A(t, x) = M(|g|2)(t, x), B(t, x) = M(|Du|2)(t, x), C(t, x) = M(|Du|2τ )(t, x).

Observe that(|g|2)

Qκr(s,y)≤ A(t, x) for all (t, x) ∈ Qr(s, y). Similar inequalities

hold for B and C. From this and (5.3) it follows that, for any (t, x) ∈ Rd+1 andQ ∈ Q such that (t, x) ∈ Q,

(|Dx′u− (Dx′u)Q|2)Q≤ Nκd+2A(t, x) + Nκ−1B(t, x) + Nκd+2γ1/σC(t, x)1/τ

for κ ≥ 4. Take the supremum of the left-hand side of the above inequality over allQ ∈ Q containing (t, x). Also observe that

(|Dx′u− (Dx′u)Q|)2Q ≤ (|Dx′u− (Dx′u)Q|2)Q

.

Then we obtain((Dx′u)#(t, x)

)2 ≤ Nκd+2A(t, x) + Nκ−1B(t, x) + Nκd+2γ1/σC(t, x)1/τ

for κ ≥ 4. Apply the Fefferman-Stein theorem and the Hardy-Littlewood maximalfunction theorem on the above inequality. More precisely,

‖Dx′u‖Lp ≤ N‖ (Dx′u)# ‖Lp ≤ Nκ(d+2)/2‖M(|g|2)‖1/2Lp/2

+Nκ−1/2‖M(|Du|2)‖1/2Lp/2

+ Nκ(d+2)/2γ1/(2σ)‖M(|Du|2τ )‖1/(2τ)Lp/(2τ)

≤ Nκ(d+2)/2‖g‖Lp + N(κ−1/2 + κ(d+2)/2γ1/(2σ)

)‖Du‖Lp ,

20 H. DONG AND D. KIM

where the last inequality is possible due to p > 2τ > 2. This and (3.6) yield

‖Du‖Lp≤ Nκ(d+2)/2‖g‖Lp

+ N(κ−1/2 + κ(d+2)/2γ1/(2σ)

)‖Du‖Lp

if γ is smaller than the one in Lemma 3.5. We then take a sufficiently large κ anda smaller γ so that

N(κ−1/2 + κ(d+2)/2γ1/(2σ)

)≤ 1/2.

The lemma is proved. ¤

By again using the idea of Agmon, we have the following corollary.

cor3.21 Corollary 5.4. Let p ∈ (2,∞), f, g ∈ Lp, u ∈ C∞0 , and µ be the constant fromLemma 3.5. Then there is a constant γ = γ(d,m, p, δ) > 0 such that under As-sumption 2.1 (γ) the following assertion holds. There exist positive constants λ0

and N , depending only on d, m, p, δ, K, and R0, such that if u ∈ C∞0 vanishesoutside Qµ−1R0 , we have

√λ‖Du‖Lp + λ‖u‖Lp ≤ N(

√λ‖g‖Lp + ‖f‖Lp),

provided that λ ≥ λ0 and Pu− λu = div g + f.

Proof of Theorem 5.1. Due to the method of continuity, the first two assertionsfollow from the a priori estimate (5.1). Because of Proposition 3.1 and the dualityargument we can assume that p ∈ (2,∞). Moreover, by the argument followingProposition 3.1 we also assume that T = ∞. Now the estimate (5.1) when Aα =Bα = C = 0 is a consequence of Corollary 5.4 by applying the standard partitionof unity technique. For general Aα, Bα and C, we move lower order terms to theright-hand side and get

‖ut‖H−1p (Rd+1

T ) +√

λ‖Du‖Lp(Rd+1T ) + λ‖u‖Lp(Rd+1

T )

≤ N(√

λ‖u‖Lp(Rd+1T ) + ‖Du‖Lp(Rd+1

T ) +√

λ‖g‖Lp(Rd+1T ) + ‖f‖Lp(Rd+1

T )

).

Then we take sufficiently large λ0 to obtain (5.1).The last assertion is proved using the a priori estimate and a scaling argument.

¤

By considering e−(λ0+1)tu instead of u, it is easy to see that Theorem 5.1 impliesTheorem 2.2 when p = q. (see, for example, the proof of Theorem 2.1 in [15]). Weend this section by presenting a proof of Theorem 2.3.

Proof of Theorem 2.3. Note that it suffices to verify the apriori estimate (2.7). Letλ0 be the constant in Theorem 5.1. Take a λ ≥ λ0. Let η be a smooth functionon R supported on [−2, 2] and η(t) = 1 on [−1, 1]. For a fixed T > 0, denotev(t, x) = η(t/T )u(x). Then it is clear that v ∈ H1

p and

(P − λ)v(t, x) = −T−1η′(t/T )u(x) + η(t/T )(L − λ)u(x).

Thanks to Theorem 5.1, we have

λ‖v‖Lp +√

λ‖Dv‖Lp ≤ N√

λ‖η(t/T )g‖Lp + N‖η(t/T )f− T−1η′(t/T )u‖Lp .

PARABOLIC AND ELLIPTIC SYSTEMS 21

This combined with the triangle inequality gives

‖η(t/T )‖Lp(R)

(λ‖u‖Lp(Rd) +

√λ‖Du‖Lp(Rd)

)

≤ N‖η(t/T )‖Lp(R)(√

λ‖g‖Lp(Rd) + ‖f‖Lp(Rd)) + NT−1‖η′(t/T )‖Lp(RT )‖u‖Lp(Rd).

Therefore,

‖η‖Lp(R)

(λ‖u‖Lp(Rd) +

√λ‖Du‖Lp(Rd)

)

≤ N‖η‖Lp(R)(√

λ‖g‖Lp(Rd) + ‖f‖Lp(Rd)) + NT−1‖η′‖Lp(R)‖u‖Lp(Rd).

Letting T →∞ yields (2.7). The theorem is proved. ¤

6. Mixed norm estimatesmixe_Sec

In this section, we study the solvability of parabolic systems in mixed normSobolev spaces and complete the proof of Theorem 2.2 for any p, q ∈ (1,∞). AsTheorem 2.2 for p = q is a direct consequence of Theorem 5.1, Theorem 2.2 for themixed norm case is implied by the following Lq,p solvability theorem.

mixed_thm Theorem 6.1. Let p, q ∈ (1,∞) and T ∈ (−∞,∞]. There exists a positive con-stant γ = γ(d, m, p, q, δ) such that under Assumption 2.1(γ) the following hold.(i) Assume u ∈ H1

q,p(Rd+1T ), f, g ∈ Lq,p(Rd+1

T ). There exist positive λ0 and N ,depending only on d, m, p, q, δ, K, and R0, such that

‖ut‖H−1q,p(Rd+1

T ) +√

λ‖Du‖Lq,p(Rd+1T ) + λ‖u‖Lq,p(Rd+1

T )

≤ N√

λ‖g‖Lq,p(Rd+1T ) + N‖f‖Lq,p(Rd+1

T ), (6.1) eq10.03pm

provided that λ ≥ λ0 and, in Rd+1T ,

Pu− λu = div g + f. (6.2) eq2.38b

(ii) For any λ ≥ λ0 and f, g ∈ Lq,p(Rd+1T ), there exists a unique solution u ∈

H1q,p(R

d+1T ) of (6.2) satisfying (6.1).

(iii) If Aα = Bα = C = 0, A11 = A11(t), and Aαβ = Aαβ(t, x1), αβ > 1, then thesolvability result in (ii) holds true for all λ > 0. Moreover, we have an estimate asin (3.1) for all λ > 0 with Lq,p(Rd+1

T ) in place of L2(Rd+1T ).

The case p = q is proved in Theorem 5.1. We only have to consider the caseq > p because the opposite case follows from the duality argument.

Recall the definition of P in (4.1). The following local estimate is derived fromTheorem 5.1 (iii) as Corollary 3.2 is derived from Proposition 3.1.

lem5.44 Corollary 6.2. Let r ∈ (0,∞) and ν ∈ (1,∞). Assume u ∈ C∞0 and Pu = div g+fin Qνr, where f, g ∈ Lp(Qνr). Then there exists a constant N = N(d,m, δ, ν, p)such that

‖Du‖Lp(Qr) ≤ N(‖g‖Lp(Qνr)

+ r‖f‖Lp(Qνr)+ r−1‖u‖Lp(Qνr)

).

By using Corollary 6.2 in place of Corollary 3.2, we then obtain a generalizationof Lemma 4.2.

22 H. DONG AND D. KIM

lem4.3 Lemma 6.3. Let p ∈ (1,∞). Assume u ∈ C∞loc and Pu = 0 in Q2. Then for anyγ ∈ (0, 1), we have

‖u‖Cγ/2,γ(Q1) + ‖Dx′u‖Cγ/2,γ(Q1) ≤ N‖u‖Lp(Q2),

where N = N(d,m, δ, γ, p).

In the sequel, we set γ = 1/2. Along the line of the proofs in Section 4, we geta counterpart of Proposition 4.5.

thm10.12 Proposition 6.4. Let p ∈ (1,∞), κ ∈ [4,∞), r ∈ (0,∞), u ∈ C∞loc and g ∈ Lp,loc.Assume that Pu = div g in Qκr. Then there exists a constant N depending onlyon d,m, δ, p such that

(|Dx′u− (Dx′u)Qr|p)Qr

≤ Nκ−p/2 (|Du|p)Qκr+ Nκd+2 (|g|p)Qκr

.

To prove Theorem 6.1, we will also need the following lemma.

lemma7.2 Lemma 6.5. Let p, q ∈ (1,∞), Aα = Bα = 0 and C = 0. There exists γ =γ(d,m, p, q, δ) > 0 such that, under Assumption 2.1 (γ), for any r ∈ (0, R0] andu ∈ H1

q,loc satisfying Pu = 0 in Q2r, we have Du ∈ Lp(Qr) and

(|Du|p)1/pQr

≤ N(|Du|q)1/qQ2r

, (6.3) eq081030_01

where N depends only on d, m, p, q, and δ.

Proof. The same types of results are proved in Corollary 8.4 of [16] for scalar equa-tions and in Lemma 8.2 of [8] for systems with the differences that the leadingcoefficients are assumed to be in VMOx and the constant N depends also on theregularity assumption on the coefficients in those papers, i.e. the modulus of con-tinuity ω or R0. The key ingredient of the proof of Corollary 8.4 in [16] is the Lp

solvability of equations with VMOx coefficients. Since the Lp solvability (Theorem5.1) is already established with coefficients in our class, we can just reproduce theproof with almost no change.

To see that the constant N is independent of R0, we use a scaling argument.Indeed, if the inequality (6.3) holds when r = R0 = 1, then, for u ∈ H1

q,loc satisfyingPu = 0 in Q2r, we consider u(t, x) := u(r2t, rx), which satisfies

Pu := −ut + Dα

(AαβDβu

)= 0

in Q2, where Aαβ(t, x) = Aαβ(r2t, rx). The coefficients Aαβ carry the same con-stant δ as Aαβ . Moreover, Aαβ satisfy Assumption 2.1 (γ) with 1 in place of R0.Then by applying the estimate (6.3) with r = R0 = 1 to the equation Pu = 0 inQ2, we have

(|Du|p)1/pQ1

≤ N (|Du|q)1/qQ2

.

Returning back to u proves (6.3) for r ∈ (0, R0]. ¤

thm4.55 Proposition 6.6. Let Aα = Bα = C = 0 and κ ≥ 8. There exists γ0 =γ0(d,m, δ, p) > 0 such that, under Assumption 2.1 (γ), γ ∈ (0, γ0], the follow-ing holds. If g ∈ Lp,loc, u ∈ C∞loc, and Pu = div g, for any r ∈ (0, R0/κ], wehave

(|Dx′u− (Dx′u)Qr |p)Qr≤ Nκd+2(|g|p)Qκr + N

(κ−p/2 + κd+2γ1/2

)(|Du|p)Qκr ,

where N depends only on d, m, p, and δ.

PARABOLIC AND ELLIPTIC SYSTEMS 23

Proof. We omit the details and only refer the reader to the proof of Lemma 4.1 [16].Note that in the proof, we use Proposition 6.4 and Lemma 6.5 instead of Theorem7.1 and Corollary 8.4 of [16]. ¤

The next corollary follows from Proposition 6.6.

cor7.3 Corollary 6.7. Under the assumptions of Proposition 6.6, for any κ ∈ [8,∞) andr ∈ (0, R0/κ] we have

–∫

(−r2,0)

–∫

(−r2,0)

|ϕ(t)− ϕ(s)|p dt ds

≤ Nκd+2 –∫

(−(κr)2,0)

ψ(t)p dt + N(κ−p/2 + κd+2γ1/2) –∫

(−(κr)2,0)

φ(t)p dt, (6.4) eq12.47pm

where

ϕ(t) = ‖Dx′u(t, ·)‖Lp(Rd), ψ(t) = ‖g(t, ·)‖Lp(Rd), φ(t) = ‖Dxu(t, ·)‖Lp(Rd).

Proof. By the triangle inequality, the left-hand side of (6.4) is less than

I := –∫

(−r2,0)

–∫

(−r2,0)

∫

Rd

|Dx′u(t, x)−Dx′u(s, x)|p dx dt ds

= –∫

(−r2,0)

–∫

(−r2,0)

∫

Rd

–∫

Br(x)

|Dx′u(t, y)−Dx′u(s, y)|p dy dx dt ds

≤ N

∫

Rd

(|Dx′u− (Dx′u)Qr(0,x)|p)Qr(0,x)

dx.

Due to Proposition 6.6, we get

I ≤ Nκd+2

∫

Rd

(|g|p)Qκr(0,x) dx + N(κ−p/2 + κd+2γ1/2

) ∫

Rd

(|Du|p)Qκr(0,x) dx.

(6.5) eq5.33pm

Note that the right-hand side of (6.4) is exactly the right-hand side of (6.5). Thecorollary is proved. ¤

The following lemma is an Lq,p version of Lemma 3.5.

lem3b Lemma 6.8. Let 1 < p < q < ∞, g ∈ Lq,p and Aα = Bα = C = 0. Then there is aconstant γ1 = γ1(d,m, p, q, δ) > 0 such that under Assumption 2.1 (γ), γ ∈ (0, γ1],the following assertion holds. There exist µ ∈ [1,∞), N , and R1, depending onlyon d, m, p, q, and δ, such that, for u ∈ C∞0 vanishing outside (−µ−2R2

0R1, 0)×Rd,we have

‖Du‖Lq,p ≤ N(‖Dx′u‖Lq,p + ‖g‖Lq,p

), (6.6) eq3.02c

provided that P0u = div g.

Proof. We proceed as in the proof of Lemma 3.5, but using Corollary 8.7 of [8]. ¤

thm7.4 Proposition 6.9. Let 1 < p < q < ∞, and Aα = Bα = C = 0. Also let µ bethe constant in Lemma 6.8. Then there is a constant γ = γ(d, m, p, q, δ) > 0 suchthat, under Assumption 2.1 (γ), the following assertion holds. There exist positiveconstants N and R, depending only on d, m, δ, p, and q, such that, for u ∈ C∞0vanishing outside (−µ−2R2

0R, 0) × Rd and satisfying Pu = div g, where g ∈ Lq,p,we have

‖Du‖Lq,p ≤ N‖g‖Lq,p . (6.7) eq12.54pm

24 H. DONG AND D. KIM

Proof. We first show that it suffices to prove the case R0 = 1. Suppose the theoremholds when R0 = 1. For coefficients Aαβ satisfying Assumption 2.1 (γ) with R0 ∈(0, 1], and for u ∈ C∞0 vanishing outside (−µ−2R2

0R, 0) × Rd and satisfying Pu =div g, g ∈ Lq,p, we consider u(t, x) := u(R2

0t, R0x) and g(t, x) := R0g(R20t, R0x).

Note that u vanishes outside (−µ−2R, 0)× Rd and

Pu := −ut + Dα

(AαβDβu

)= div g,

where Aαβ(t, x) = Aαβ(R20t, R0x). Since Aαβ satisfy Assumption 2.1 (γ) with

R0 = 1, by the theorem for the case R0 = 1, we have the inequality (6.7). Turningu and g back into u and g proves the theorem for arbitrary R0 ∈ (0, 1].

Now we set R0 = 1 and define ϕ, ψ and φ as in the proof of Corollary 6.7. LetR ∈ (0, R1] and γ ≤ minγ0, γ1 be numbers to be chosen later, where γ0 is fromProposition 6.6, and R1 as well as γ1 are from Lemma 6.8. Assume that u vanishesoutside (−µ−2R, 0)× Rd. We claim

ϕ# ≤ N((κ2µ−2R)(1−

1p ) + κ−

12 + κ

d+2p γ

12p

)(M(φp))

1p + Nκ

d+2p (M(ψp))

1p . (6.8) eq2.41pm

Indeed, fix a point t0 ∈ R and consider an interval (S, T ) ⊂ R containing t0.When T − S ≤ κ−2, by a shift of the origin we get from Corollary 6.7 that (|ϕ −(ϕ)(S,T )|)(S,T ) is less than the right-hand side of (6.8) at t0. When T − S > κ−2,we then have

(|ϕ− (ϕ)(S,T )|)(S,T ) ≤ 2 –∫

(S,T )

χ(−µ2R,0)|ϕ(t)| dt

≤ 2

(–∫

(S,T )

χ(−µ2R,0) dt

)1−1/p (–∫

(S,T )

|ϕ(t)|p dt

)1/p

≤ N(κ2µ−2R)(1−1/p) (M(ϕp)(t0))1/p

.

By taking supremum over all intervals (S, T ) containing t0, we obtain (6.8) at pointt0. Since t0 ∈ R is arbitrary, the claim is proved.

It follows from the Fefferman-Stein theorem and the Hardy-Littlewood theorem(recall that q > p) that

‖Dx′u‖Lq,p ≤ N((κ2µ−2R)(1−

1p ) + κ−

12 + κ

d+2p γ

12p

)‖Du‖Lq,p + Nκ

d+2p ‖g‖Lq,p .

(6.9) eq2.51pm

Since γ ≤ γ1 and R ≤ R1, Lemma 6.8 is applicable. From (6.6) and (6.9) we thenget

‖Du‖Lq,p ≤ N((κ2µ−2R)(1−

1p ) + κ−

12 + κ

d+2p γ

12p

)‖Du‖Lq,p + Nκ

d+2p ‖g‖Lq,p .

To complete the proof of Proposition 6.9, it suffices to take κ sufficiently large, γsufficient small, and R sufficiently small such that

N((κ2µ−2R)(1−

1p ) + κ−

12 + κ

d+2p γ

12p

)≤ 1/2.

¤

Proof of Theorem 6.1. The case p = q is already proved in Theorem 5.1. By usinga duality argument it suffices to consider the case p < q. In this case, the resultfollows from Proposition 6.9 along with a partition of unity with respect to the timevariable. (see Section 3 of [16]). ¤

PARABOLIC AND ELLIPTIC SYSTEMS 25

7. Systems on a half spacesec7

The objective of this section is to establish the following solvability results, whichimply Theorem 2.5-2.8 as Theorem 6.1 (or Theorem 5.1) implies Theorem 2.2 and2.3.

half_mixed_thm Proposition 7.1. Let p, q ∈ (1,∞), Ω = Rd+ and T ∈ (−∞,∞]. Then there is a

constant γ = γ(d,m, p, q, δ) > 0 such that under Assumption 2.1 (γ) the followingassertions hold.(i) Assume u ∈ H1

q,p(ΩT ), f, g ∈ Lq,p(ΩT ). There exist positive λ0 and N , depend-ing only on d, m, p, q, δ, K, and R0, such that

‖ut‖H−1q,p(ΩT ) +

√λ‖Du‖Lq,p(ΩT ) + λ‖u‖Lq,p(ΩT )

≤ N√

λ‖g‖Lq,p(ΩT ) + N‖f‖Lq,p(ΩT ), (7.1) eq1.19pm

provided that λ ≥ λ0 and, Pu− λu = div g + f in ΩT

u = 0 on (−∞, T )× ∂Ω . (7.2) eq1.20pm

(ii) For any λ ≥ λ0 and f, g ∈ Lq,p(ΩT ), there exists a unique solution u ∈H1

q,p(ΩT ) of (7.2) satisfying (7.1).

thm6.2 Proposition 7.2. The assertions of Proposition 7.1 hold true if (7.2) is replacedby Pu− λu = div g + f in ΩT

A1βDβu + A1u = g1 on (−∞, T )× ∂Ω . (7.3) eq1.21pm

We shall use the idea of odd/even extensions. For this purpose, we need thefollowing lemma.

lem7.2 Lemma 7.3. Let p, q ∈ (1,∞), −∞ ≤ S < T ≤ ∞.(i) A function u belongs to H1

q,p((S, T )×Rd+) if and only if its even extension u with

respect to x1 belongs to H1q,p((S, T ) × Rd). Moreover, there exists N = N(d) > 0

such that

N−1‖u‖H−1q,p((S,T )×Rd

+) ≤ ‖u‖H−1q,p((S,T )×Rd) ≤ N‖u‖H−1

q,p((S,T )×Rd+), (7.4) eq2.27pm

N−1‖u‖Lq,p((S,T )×Rd+) ≤ ‖u‖Lq,p((S,T )×Rd) ≤ N‖u‖Lq,p((S,T )×Rd

+), (7.5) eq2.27pmb

N−1‖Du‖Lq,p((S,T )×Rd+) ≤ ‖Du‖Lq,p((S,T )×Rd) ≤ N‖Du‖Lq,p((S,T )×Rd

+). (7.6) eq2.27pmc

(ii) A function u belongs to H1q,p((S, T )×Rd

+) and vanishes on (S, T )×∂Rd+ if and

only if its odd extension u with respect to x1 belongs to H1q,p((S, T )×Rd). Moreover,

we have (7.4)-(7.6).

Now we are ready to prove Proposition 7.1 and 7.2.

Proof of Proposition 7.1. Define

Aαβ(t, x) = sgn(x1)Aαβ(t, |x1|, x′) for α = 1, β ≥ 2 or β = 1, α ≥ 2,

Aαβ(t, x) = Aαβ(t, |x1|, x′) otherwise,and

A1(t, x) = sgn(x1)A1(t, |x1|, x′), Aβ(t, x) = Aβ(t, |x1|, x′), β ≥ 2,

B1(t, x) = sgn(x1)B1(t, |x1|, x′), Bβ(t, x) = Bβ(t, |x1|, x′), β ≥ 2,

26 H. DONG AND D. KIM

C(t, x) = C(t, |x1|, x′), f(t, x) = sgn(x1)f(t, |x1|, x′),g1(t, x) = g1(t, |x1|, x′), gβ(t, x) = sgn(x1)gβ(t, |x1|, x′), β ≥ 2.

It is easy to see that if Aαβ , Aα, Bα, C satisfy Assumption 2.1 (γ), then thenew coefficients Aαβ , Aα, Bα, C satisfy Assumption 2.1 (4γ). Moreover, f, g ∈Lq,p(Rd+1

T ). Let P be the divergence form parabolic operator with coefficientsAαβ , Aα, Bα, C.

Due to Theorem 6.1, we can find γ > 0 and λ0 > 0 such that there exists aunique solution u ∈ H1

q,p(Rd+1T ) of

Pu− λu = div g + f in Rd+1T , (7.7) eq24.3.29

provided that λ ≥ λ0. By the definition of the coefficients and the data, we have

Pu(t,−x1, x′)− λu(t,−x1, x

′) = − div g(t, x)− f(t, x) in Rd+1T .

Consequently, −u(t,−x1, x′) is also a solution to (7.7). By the uniqueness of the

solution, we obtain u(t, x) = −u(t,−x1, x′). This implies that, as a function on

Rd+, u has zero trace on the boundary and clearly u satisfies (7.2). This proves the

existence of the solution.On the other hand, it is easy to see that if u ∈ H1

q,p(ΩT ) is a solution to (7.2),then its odd extension with respect to x1 is a solution to (7.7). So the uniquenessfollows from Theorem 6.1. To prove the estimate (7.1), we use (6.1) and Lemma7.3. The theorem is proved.

¤

Proof of Proposition 7.2. We define Aαβ , Aα, Bα and C as in the proof of Propo-sition 7.1. Let P be the divergence form parabolic operator with coefficientsAαβ , Aα, Bα and C. Different from above, we define

f(t, x) = f(t, |x1|, x′),

g1(t, x) = sgn(x1)g1(t, |x1|, x′), gβ(t, x) = gβ(t, |x1|, x′), β ≥ 2.

Recall that Aαβ satisfy Assumption 2.1 (4γ). Moreover, we have f, g ∈ Lq,p(Rd+1T ).

Due to Theorem 6.1, we can find γ > 0 and λ0 > 0 such that there exists a uniquesolution u ∈ H1

q,p(Rd+1T ) of (7.7) provided that λ ≥ λ0. By the definition of the

coefficients and the data, we have

Pu(t,−x1, x′)− λu(t,−x1, x

′) = div g(t, x) + f(t, x) in Rd+1T .

Consequently, u(t,−x1, x′) is also a solution to (7.7). By the uniqueness of the

solution, we obtain u(t, x) = u(t,−x1, x′).

Let p′, q′ be such that 1/p + 1/p′ = 1 and 1/q + 1/q′ = 1. For any h ∈H1

q′,p′((−∞, T )× Rd+), denote h to be its even extension with respect to x1. Since

u satisfies (7.7), integrating by parts gives

∫ T

−∞

∫

Rd

(− ut · h− AαβDβu ·Dαh− Aαu ·Dαh + BαDαu · h

+ (C − λ)u · h)

dx dt =∫ T

−∞

∫

Rd

(−gα · Dαh + f · h

)dx dt. (7.8) eq27.4.50pm

PARABOLIC AND ELLIPTIC SYSTEMS 27

By the definition of Aαβ , Aα, Bα, C, g, and f as well as the evenness of u and h, allterms inside the integrals in (7.8) are even with respect to x1. Thus, (7.8) implies

∫ T

−∞

∫

Rd+

(− ut · h−AαβDβu ·Dαh−Aαu ·Dαh + BαDαu · h

+ (C − λ)u · h)

dx dt =∫ T

−∞

∫

Rd+

(−gα ·Dαh + f · h) dx dt. (7.9) eq27.4.56pm

Since h ∈ H1q′,p′((−∞, T )× Rd

+) is arbitrary, by the definition u solves (7.3). Thisproves the existence of the solution.

For the uniqueness, let v be another solution of (7.3) so that, for any h ∈H1

q′,p′((−∞, T ) × Rd+), the equality (7.9) holds. Let v to be the even extension of

v with respect to x1. Then by the definition of Aαβ , Aα, Bα, C, g, and f, for anyh ∈ H1

q′,p′(Rd+1T ) we have

∫ T

−∞

∫

Rd

(−vt · h− AαβDβ v ·Dαh− Aαv ·Dαh + BαDαv · h + (C − λ)v · h

)dx dt

=

∫ T

−∞

∫

Rd+

(−vt · h−AαβDβv ·Dαh−Aαv ·Dαh + BαDαv · h + (C − λ)v · h

)dx dt

+

∫ T

−∞

∫

Rd+

(−vt · h−AαβDβv · (Dαh)(t,−x1, x′)−Aαv · (Dαh)(t,−x1, x′)

)dx dt

+

∫ T

−∞

∫

Rd+

BαDαv · h(t,−x1, x′) + (C − λ)v · h(t,−x1, x′) dx dt.

Due to (7.9), the sum above is equal to∫ T

−∞

∫

Rd+

−gα ·Dαh + f · h dx dt +

∫ T

−∞

∫

Rd+

−gα · (Dαh)(t,−x1, x′) + f · h(t,−x1, x′) dx dt

=

∫ T

−∞

∫

Rd−gα ·Dαh + f · h dx dt.

This yields that v ∈ H1q,p(R

d+1T ) is a solution of (7.7). By the uniqueness, we get

u = v, which implies that u = v in (−∞, T ) × Rd+. Finally, the estimate (7.1)

follows from (6.1) and Lemma 7.3. The theorem is proved. ¤

8. Remarks on systems in Lipschitz domainsbddom

We present in this section the solvability results of parabolic and elliptic systemson a bounded domain. We assume the leading coefficient matrices Aαβ to bemeasurable in t and have small BMO semi-norms in the spatial variables. Thisassumption guarantees that, if the domain is locally flattened, the coefficients Aαβ

satisfy Assumption 2.1 regardless of the choice of a local coordinate system. Tostate this assumption precisely, set

A#R = sup

(t,x)∈Rd+1supr≤R

∑

α,β=1

oscx

(Aαβ , Qr(t, x)

).

assump1 Assumption 8.1 (γ). There is an R0 ∈ (0, 1] such that A#R0≤ γ.

If elliptic systems are considered, all coefficients involved are functions indepen-dent of t. Moreover, in the definition of A#

R , Qr is replaced by Br.We also impose the condition that the boundary ∂Ω of the domain Ω is locally

the graph of a Lipschitz continuous function with a small Lipschitz constant. More

28 H. DONG AND D. KIM

precisely, we make the following assumption containing a parameter θ ∈ (0, 1],which will be specified.

assump2 Assumption 8.2 (θ). There is a constant ρ0 ∈ (0, 1] such that, for any x0 ∈ ∂Ωand r ∈ (0, ρ0], there exists a Lipschitz function φ: Rd−1 → R such that in somecoordinate system

Ω ∩Br(x0) = x = Br(x0) : x1 > φ(x′)and

supx′,y′∈B′r(x′0),x′ 6=y′

|φ(y′)− φ(x′)||y′ − x′| ≤ θ.

We note that any C1 domain satisfies this assumption for any θ > 0.Since we have solvability results of parabolic and elliptic systems on a half space

with the Dirichlet boundary condition or the conormal derivative boundary condi-tion in Section 7, the standard argument of flattening the boundary as well as apartition of unity imply the results below. We only state the theorems. For proofs,we refer the interested reader to [9].

First we consider parabolic systems on a bounded domain with either the Dirich-let boundary condition or the conormal derivative boundary condition:

Pu = div g + f in (0, T )× Ωu = 0 on (0, T )× ∂Ω , (8.1) eq27.8.51

and Pu = div g + f in (0, T )× ΩnαAαβDβu + nαAαu = nαgα on (0, T )× ∂Ω , (8.2) eq081103_03

where n = (n1, · · · , nd) is the outward normal vector on ∂Ω.

thmA Theorem 8.3. Let p, q ∈ (1,∞) and Ω be a bounded domain. There exist γ =γ(d,m, p, q, δ) and θ = θ(d,m, p, q, δ) such that, under Assumption 8.1 (γ) andAssumption 8.2 (θ), for any f, g = (g1, · · · , gd) ∈ Lq,p(ΩT ) there exists a uniqueu ∈ H1

q,p(ΩT ) satisfying (8.1) or (8.2). Moreover, we have

‖u‖H1q,p(ΩT ) ≤ N‖f‖Lq,p(ΩT ) + N‖g‖Lq,p(ΩT ),

where N = N(d,m, p, q, δ,K, R0, ρ0, T, Ω).

The last two theorems are about the Dirichlet boundary value problem Lu− λu = div g + f in Ω

u = 0 on ∂Ω , (8.3) eq081103_01

and the conormal derivative problem Lu− λu = div g + f in Ω

nαAαβDβu + nαAαu = nαgα on ∂Ω (8.4) eq081103_02

for elliptic systems on a bounded domain, respectively.

thm081103_01 Theorem 8.4. Let p ∈ (1,∞) and Ω be a bounded domain. There exist γ =γ(d,m, p, δ) and θ = θ(d,m, p, δ) such that, under Assumption 8.1 (γ) and As-sumption 8.2 (θ) the following assertions hold.

PARABOLIC AND ELLIPTIC SYSTEMS 29

(i) For any f, g = (g1, · · · , gd) ∈ Lp(Ω), there exists a unique u ∈ W 1p (Ω) satisfying

(8.3) provided that λ ≥ λ0, where λ0 = λ0(d,m, p, δ,K, R0, ρ0, Ω) ≥ 0. Moreover,we have

‖u‖W 1p (Ω) ≤ N‖f‖Lp(Ω) + N‖g‖Lp(Ω), (8.5) eq05.5.16

where N = N(d,m, p, δ,K,R0, ρ0,Ω).(ii) If Aα = Bα = C = 0, we can take any λ ≥ 0 in (i).

Theorem 8.5. Under the same assumptions of Theorem 8.4, the following asser-tions hold.(i) For any f, g = (g1, · · · , gd) ∈ Lp(Ω), there exists a unique u ∈ W 1

p (Ω) satisfying(8.4) provided that λ ≥ λ0, where λ0 = λ0(d,m, p, δ,K, R0, ρ0, Ω) > 0. Moreover,we have (8.5) where N = N(d,m, p, δ,K, R0, ρ0, Ω).(ii) If Aα = Bα = C = 0, the solvability holds for any λ > 0 in (i) and the constantN in (8.5) may depend on λ as well.(iii) If Aα = Bα = C = 0 and f = 0, the solvability holds even when λ = 0. In thiscase the solution is unique up to a constant, and instead of (8.5) we have

‖u− (u)Ω‖W 1p (Ω) ≤ N‖g‖Lp(Ω).

Remark 8.6. As in [9], away from the lateral boundary of the domain, the coef-ficients Aαβ are allowed to be in the partially BMO space defined by Assumption2.1.

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(H. Dong) Division of Applied Mathematics, Brown University, 182 George Street,Providence, RI 02912, USA

E-mail address: Hongjie [email protected]

(D. Kim) Department of Mathematics, University of Southern California, 3620 SouthVermont Avenue, KAP 108, Los Angeles, CA 90089-2532, USA

E-mail address: [email protected]