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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 A 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 + D(ADu + Au

    )+ BDu + Cu (1.1) eq0617_01

    acting on (column) vector-valued functions u = (u1, , um)T given either on Rd+1or on a cylindrical domain in Rd+1. Here, we have used the notations Du = ux ,Du = uxx , (, = 1, . . . , d) and the usual summation convention over repeatedindices is assumed. The coefficients A , A, B, and C are mm matrix-valuedfunctions given on Rd+1; i.e., A = [Aij (t, x)]mm, etc. When the coefficientsA , A, B and C are independent of t, we also define and consider ellipticoperators in divergence form

    Lu = D(ADu + Au) + BDu + Cu (1.2) eq3.27acting 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 2p 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 A 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 H1q,p (cf. Theorem 2.2), as well as the W 1p solvability for divergenceform 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 A , > 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 byDxu. For this purpose, our idea is to break the symmetry of the coordinates sothat t and x1 are distinguished from x Rd1; 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 ofDxu, from which a pointwise sharp function estimate of Dxu 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 Dxu. Then inSection 4, we give an estimate for the L2-oscillation of Dxu after establishing itsinterior Holder estimate. Using the L2-oscillation estimate and Lemma 3.5, weprove in Section 5 the H1p 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 Rd1. Let Du be the whole collection of Du, = 1, , d, or one of them depending on the context. By Dxu we mean one ofDu, = 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}, Br(x) = {y Rd1 : |x y| < r},r(x) = (x1 r, x1 + r)Br(x),

    Qr(t, x) = (t r2, t)Br(x), r(t, x) = (t r2, t) r(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) = supQQ:(t,x)Q

    Q

    |g(s, y)| dy ds,

    g#(t, x) = supQQ:(t,x)Q

    Q

    |g(s, y) (g)Q| dy ds.

    Next we state assumptions on the coefficients