Estimates for Green’s Functions of Elliptic Equations in Non-Divergence Form with Continuous Coefficients

Seick Kimand Sungjin Lee

Department of Mathematics, Yonsei University, 50 Yonsei-ro,Seodaemun-gu, Seoul 03722, Republic of Korea

Abstract. We present a new method for the existence and pointwise estimates of a Green’s function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green’s function for constant coefficients equations.

Key words: Green’s function, Elliptic equations in nondivergence form, Dini mean oscillation coefficients.

1 Introduction and main results

We consider a second-order elliptic operatorLin non-divergence form

where the coefficient A:=(aij) are symmetric and satisfy the uniform ellipticity condition. Namely,

for some positive constantsλand Λ in a domain ??Rnwithn≥3. Here and below,we use the usual summation convention over repeated indices.

In this article, we are concerned with construction and pointwise estimates for the Green’s functionG(x,y) of the non-divergent operatorLin ?. In a recent article [15], it is shown that if the coefficients A is of Dini mean oscillation and the domain ? is bounded and hasC2,αboundary, then the Green’s function exists and satisfies the pointwise bound

Before proceeding further, let us introduce the definition of Dini mean oscillation.Forx∈Rnandr>0, we denote byB(x,r) the Euclidean ball with radiusrcentered atx, and write ?(x,r):=?∩B(x,r). We denote

It is clear that if A is Dini continuous,then A is of Dini mean oscillation. Also if A is of Dini mean oscillation, then A is uniformly continuous in ? with its modulus of continuity controlled byωA; see [15, Appendix]. However, a function of Dini mean oscillation is not necessarily Dini continuous; see [7] for an example.

The main result of[15]is interesting because unlike the Green’s function for uniformly elliptic operators in divergence form, the Green’s function for non-divergent elliptic operators does not necessarily enjoy the pointwise bound (1.3) even in the case when the coefficient A is uniformly continuous; see[1]. It should be noted that the Dini mean oscillation condition is the weakest assumption in the literature that guarantees the pointwise bound (1.3). The proof in [15] is based on considering approximate Green’s functions (as in [13,14]) and showing that they satisfy specific estimates, as well as a localL∞estimate for solutions to the adjoint equationL?u=0,which is shown in[7,8]. ThisL∞estimate is crucial for the pointwise bound(1.3) and it is where the Dini mean oscillation condition is strongly used; a mere continuity of A is not enough to produce such an estimate. We should recall that the adjoint operatorL?is given by We should also mention that there are many papers in the literature dealing with the existence and estimates of Green’s functions or fundamental solutions of nondivergence form elliptic or parabolic operators with measurable or continuous coefficients; see e.g., [2–4,9,11,12,17].

In this article we give an alternative proof for the existence of Green’s function.More precisely, we construct Green’s function from that of the corresponding constant coefficients operator resulting from “freezing coefficients”. We shall useLptheory for the adjoint operator in this process. We then utilize the localL∞estimates for adjoint solutions established in [7,8] to get the pointwise bound (1.3) for the Green’s function. One prominent advantage of this approach is that it yields a sharp comparison with the Green’s function for constant coefficients operator. In particular, we shall show that

whereG0is the Green’s function of the constant coefficient operatorL0given by

provided that the mean oscillation of A satisfies so-called “double Dini condition”nearx0; that is, we have

for somer0>0. The asymptotic behavior (1.5) is well known for the Green’s functions for elliptic operators in divergence form with continuous coefficients; see [5].However,in the non-divergence form setting,this is a new result and it is one of the novelties in our work. We now present our main theorem.

Theorem 1.1.Let?be a bounded C2,α domain inRn with n ≥3. Assume the coefficientA=(aij)of the operator L in(1.1)satisfies the uniform ellipticity condition(1.2)and is of Dini mean oscillation in?. Then, there exists a unique Green’s function G(x,y)of the operator L in?and it satisfies the pointwise estimate

where C=C(n,λ,Λ,?,ωA). Moreover,if there is some r0>0such that ωA(t,?(x0,r0))satisfies double Dini condition

then we have

where G0is the Green’s function of the constant coefficient operator L0as in(1.6).

Remark 1.1. As stated in [15], pointwise estimates forDxG(x,y) andD2xG(x,y)are also available. They are obtained from (1.3) via localL∞estimates for first and second derivatives of solutions toLu=0 as established in [7,8]. We only treat the case whenn ≥3 in this article and we refer to [6] for two dimensional case. In a separate paper[16],we construct the fundamental solution for parabolic equations in non-divergence form with Dini mean oscillation coefficients and establish Gaussian bounds for the fundamental solution.

2 Preliminary lemmas

In this section, we present some technical lemmas which will be used in the proof of Theorem 1.1. We need to consider the boundary value problem of the form

where a constant C depends on?, p, n, λ,Λ, and ωA.

Proof.See [10, Lemma 2].

The proof of next lemma is implicitly given in the proof of [7, Theorem 1.10].However, an estimate like (2.3) does not appear explicitly in the literature and we provide a proof in the Appendix for reader’s convenience. It should be emphasized that the lemma asserts that to get a localL∞estimate of the solutionu,only a local information on the data g is needed.

Lemma 2.2.Let R0>0andg=(gij)be of Dini mean oscillation in B(x0,R0).Suppose u is an L2solution ofThe next lemma is an extension of Lemma 2.2 up to theC2,αboundary.

Lemma 2.3.Let?be a bounded C2,α domain. Assume thatg=(gij)are of Dini mean oscillation in?. Let u∈L2(?)be the solution of the adjoint problem

Proof.By flattening the boundary, it suffices to get an estimate in half balls that corresponds to (2.3). It is obtained by replicating the proof of [8, Lemma 2.26] in the same fashion as (2.3) is derived. We leave the details to the readers.

3 Proof of Theorem 1.1

The organization of the proof is as follows. In Section 3.1, we first construct the Green’s functionG?(x,y) for the adjoint operator. In Section 3.2 and 3.3 it will be shown that the adjoint Green’s function has the pointwise bound|G?(x,y)|≤C|x?y|2?n. In Section 3.4, we show thatG(x,y)=G?(y,x) becomes the Green’s function and thus it also has the pointwise bound|G(x,y)|≤C|x?y|2?n. Finally, in Section 3.5 we establish the asymptotic formula (1.8).

3.1 Construction of adjoint Green’s function

Letx0∈? be fixed and denote

LetG0(x,y)be the Green’s function forL0in ?. SinceL0is an elliptic operator with constant coefficients, the existence ofG0as well as the following pointwise bound is well known.

We shall now constructG?(·,x0), Green’s function forL?in ? with a pole atx0.Formally, we would have

which lead us to consider the problem

where we denote

where we use the fact thatG0is the Green’s function for the operatorL0. Therefore,by (3.5), we have

3.2 Pointwise estimates of adjoint Green’s functions

In this section, we shall establish

Letvbe as in (3.2) and define g1and g2by

whereζis a smooth function on Rnsuch that

where C=C(n,λ,Λ,?).

The above lemma, the proof of which is given in Section 3.3, yields that (takeη=ζwithδ=r)

whereC=C(n,λ,Λ,ωA,?). Since

andy0∈?{x0}is arbitrary,the desired estimate(3.8)follows from(3.1)and(3.20).

3.3 Proof of Lemma 3.2

Plugging (3.24) into (3.23), we obtain

The lemma is proved by taking supremum over∈?(y0,2r).

It remains to prove the claim (3.25). Notice that we can choose a sequence of pointsx1,x2,···,xNin ?(x0,7r) withxN=in such a way that each line segment[xi?1,xi]lies in ? and|xi?1?xi|≤tfori=1,···,N. Moreover, there exists a constantC=C(?) independent oftandrsuch that

Then (3.25) follows from the above inequality and (3.27). □

3.4 Construction and symmetry relation for Green’s function

In this section, we shall prove that The fucntionG(x,y) given by

is the Green’s function for the operatorLin ?. Then in light of (3.8), we see that the Green’s functionG(x,y) has the pointwise bound (1.3).

To establish (3.30), first observe thatG?(·,x0) satisfies

We conclude from (3.31) and (3.8) that for anyx,y∈? withx/=y, we have

which coincides with [15, Lemma 2.11]. With the above key estimate at hand, we can replicate the same argument as in[15]and construct the Green’s functionG(x,y)for the operatorLout of the family{G?(x,y)}. In particular, there is a sequence{?j}→0 such that (see [15, (2.24)])

Then, by using the continuity ofG?(·,x0) away fromx0, we derive from (3.31) the desired identity (3.30).

3.5 Asymptotic behavior near a pole

In this section, we assume that condition (1.7) holds for somer0>0. The estimate(3.20) leaves a room that we might be able to get an asymptotic behavior

Using (3.35) together with H¨older’s inequalities, we have

sincey0∈?{x0}and?are arbitrary. SinceG0is symmetric, we obtain (1.8) from the above and (3.30).

Appendix: Proof of Lemma 2.2

Let us consider the quantity

Sinceq∈R is arbitrary, by using (A.1), we thus obtain

whereC=C(κ)=C(n,λ,Λ) and we used [7, Lemma 2.7]. A similar computation holds forωg(r,x), and thus we obtain

taking the average overy∈Bκr(x) and then taking the square, we obtain

Acknowledgements

S. Kim is partially supported by National Research Foundation of Korea (NRF)Grant No. NRF-2019R1A2C2002724 and No. NRF-20151009350.