Mean Oscillation Gradient Estimates for Elliptic Systems in Divergence Form with VMO Coefficients

Abstract

We consider gradient estimates for H¹ solutions of linear elliptic systems in divergence form $\partial _{\alpha }(A_{ij}^{\alpha \beta } \partial _{\beta } u^{j}) = 0$. It is known that the Dini continuity of coefficient matrix $A = (A_{ij}^{\alpha \beta })$ is essential for the differentiability of solutions. We prove the following results:

(a) If A satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the L² mean oscillation ω_A,2 of A satisfies

$ X_{A,2} := \limsup\limits_{r\rightarrow 0} r {{\int \limits }_{r}^{2}} \frac {\omega _{A,2}(t)}{t^{2}} \exp \left (C_{*} {{\int \limits }_{t}^{R}} \frac {\omega _{A,2}(s)}{s} ds\right ) dt < \infty , $

where C_∗ is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO.

(b) If X_A,2 = 0, then ∇u ∈ V MO.

(c) Finally, examples satisfying X_A,2 = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when $\nabla u \in L^{\infty } \cap VMO$.

1 Introduction

Let n ≥ 2, N ≥ 1 and consider the elliptic system for u = (u¹,…,u^N)

$$ \partial_{\alpha}(A_{ij}^{\alpha\beta} \partial_{\beta} u^{j}) = 0 \quad \text{ in } B_{4}, \quad i = 1, \ldots, N, $$

(1.1)

where B₄ is the ball in $\mathbb R^{n}$ of radius four and centered at the origin, and the coefficient matrix $A = (A_{ij}^{\alpha \beta })$ is assumed to be bounded and measurable in $\bar B_{4}$ and to satisfy, for some positive constants λ and Λ,

$$ \begin{array}{@{}rcl@{}} |A(x)| &\leq& {\varLambda} \quad\text{ for a.e. } \quad x \in B_{4}, \end{array} $$

(1.2)

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{2}} A_{ij}^{\alpha\beta} \partial_{\beta} \varphi^{j} \partial_{\alpha} \varphi^{i} dx & \geq& \lambda \|\nabla \varphi\|_{L^{2}(B_{4})}^{2} \text{ for all } \varphi \in {H_{0}^{1}}(B_{4}). \end{array} $$

(1.3)

It is well known that if the coefficient matrix A belongs to $C^{0,\alpha }_{\text {loc}}(B_{4})$ then every solution u ∈ H¹(B₄) of (1.1) belongs to $C^{1,\alpha }_{\text {loc}}(B_{2})$; see, e.g., Giaquinta [13, Theorem 3.2] where the result is attributed to Campanato [7] and Morrey [24]. It was conjectured by Serrin [26] that the assumption u ∈ H¹(B₄) can be relaxed to u ∈ W^1,1(B₄). This has been settled in the affirmative by Brezis [2, 3]. (See Hager and Ross [14] for the relaxation from u ∈ H¹(B₄) to u ∈ W^1,p(B₄) for some 1 < p < 2.) Moreover, in [2, 3], it was shown that if A satisfies the Dini condition

$$ {{\int}_{0}^{2}} \frac{\bar\omega_{A}(t)}{t} dt < \infty, \quad \text{ where } \bar\omega_{A}(r) := \sup\limits_{x, y \in B_{2},|x - y| < r} |A(x) - A(y)|, $$

(1.4)

then every solution u ∈ W^1,1(B₄) of (1.1) belongs to C¹(B₂). For related works on the differentiability of weak solutions under suitable conditions on $\bar \omega _{A}$, see also [15, 22, 23].

Differentiability of weak solutions under weaker Dini conditions involving integral mean oscillation of A has also been studied. For 0 < r ≤ 2, let

$$ \begin{array}{@{}rcl@{}} \bar\varphi_{A}(r) &:=& \sup\limits_{x \in B_{2}} \left\{\frac{1}{|B_{r}(x)|}{\int}_{B_{r}(x)} |A(y) - A(x)|^{2} dy\right\}^{1/2},\\ \omega_{A}(r) &:=& \sup\limits_{x \in B_{2}} \frac{1}{|B_{r}(x)|}{\int}_{B_{r}(x)} |A(y) - (A)_{B_{r}(x)}| dy, \\ (A)_{B_{r}(x)} &:=& \frac{1}{|B_{r}(x)|}{\int}_{B_{r}(x)} A(y) dy, \quad 0 < r \leq 2. \end{array} $$

In Li [21] it was shown that if

$$ {{\int}_{0}^{2}} \frac{\bar\varphi_{A}(t)}{t} dt < \infty, $$

(1.5)

then every solution u ∈ H¹(B₄) of (1.1) belongs to C¹(B₂). In Dong and Kim [12] (see also [9]), this conclusion was shown to remain valid under the weaker condition that

$$ \begin{array}{@{}rcl@{}} {{\int}_{0}^{2}} \frac{\omega_{A}(t)}{t} dt < \infty. \end{array} $$

(1.6)

(Note that the finiteness of ${{\int \limits }_{0}^{2}} \frac {\omega _{A}(t)}{t} dt$ or ${{\int \limits }_{0}^{2}} \frac {\bar \varphi _{A}(t)}{t} dt$ implies that A is continuous.) For related works for nonlinear elliptic equations, see, e.g., [19, 25, 28] and the references therein.

The Dini condition (1.4) and its integral variants (1.5) and (1.6) are phenomenologically sharp for the differentiablity of weak solutions of (1.1). In Jin, Maz’ya and van Schaftingen [17], examples of continuous coefficient matrices A with moduli of continuity $\bar \omega _{A}(t) \sim \frac {1}{|\ln t|}$ as $t \rightarrow 0$ were given showing the following phenomena:

∙ There exists a solution u ∈ W^1,1(B₄) of (1.1) such that u ∈ W^1,p(B₄) for all $p \in [1,\infty )$, and ∇u ∈ BMO_loc(B₄) but $\nabla u \notin L^{\infty }_{\text {loc}}(B_{2})$ and ∇u∉V MO_loc(B₂)¹.

∙ There exists a solution u ∈ W^1,1(B₄) of (1.1) such that u ∈ W^1,p(B₄) for all $p \in [1,\infty )$ but ∇u∉BMO_loc(B₂).

In this paper, we consider mean oscillation estimates for ∇u when A slightly fails the Dini conditions (1.4), (1.5) and (1.6). For $1 \leq p < \infty$, let $\omega _{A,p}: (0, 2] \rightarrow [0,\infty )$ denote the L^p mean oscillation of A

$$ \begin{array}{@{}rcl@{}} \omega_{A,p}(r) &= \sup\limits_{x \in B_{2}} \left\{\frac{1}{|B_{r}(x)|}{\int}_{B_{r}(x)} |A(y) - (A)_{B_{r}(x)}|^{p} dy\right\}^{1/p}. \end{array} $$

It is clear that ω_A,1 = ω_A, $\omega _{A,2} \leq \bar \varphi _{A}$, ω_A,p is non-decreasing in p, and $\omega _{A,p} \leq \bar \omega _{A}$ for all $p \in [1,\infty )$.

We now state our first result.

Theorem 1.1

Let $A = (A_{ij}^{\alpha \beta })$ satisfy (1.2) and (1.3). There exists a constant C_∗ > 0, depending only on n, N, Λ and λ such that if

$$ X_{A,2} := \limsup\limits_{r\rightarrow 0} r {{\int}_{r}^{2}} \frac{\omega_{A,2}(t)}{t^{2}} \exp\Big(C_{*} {{\int}_{t}^{2}} \frac{\omega_{A,2}(s)}{s} ds\Big) dt < \infty, $$

(1.7)

then every solution u ∈ H¹(B₄) of (1.1) satisfies ∇u ∈ BMO_loc(B₂). Moreover, if

$$ X_{A,2} = 0, $$

(1.8)

then every solution u ∈ H¹(B₄) of (1.1) satisfies ∇u ∈ V MO_loc(B₂).

Note that condition (1.7) implies that $\omega _{A,2}(t) \rightarrow 0$ as $t \rightarrow 0$, i.e., A ∈ V MO_loc(B₂).

Remark 1.2

Let $1 < p < \infty$. Theorem 1.1 remains valid if ω_A,2 is replaced by ω_A,p and the regularity assumption u ∈ H¹(B₄) is replaced by u ∈ W^1,p(B₄), where the constant C_∗ is now allowed to depend also on p. For p ≥ 2, this follows from the inequality ω_A,2 ≤ ω_A,p for those p. For 1 < p < 2, see Proposition 3.2.

It is clear that if ω_A,2 satisfies (1.5), then it satisfies (1.8) (and hence (1.7)). The following lemma gives examples which satisfy (1.8) but not necessarily (1.5).

Lemma 1.3

If $\limsup _{t \rightarrow 0} \omega _{A,2}(t) \ln \frac {1}{t} < \frac {1}{C_{*}}$, then X_A,2 = 0. If $\liminf _{t \rightarrow 0} \omega _{A,2}(t) \ln \frac {1}{t}$ $> \frac {1}{C_{*}}$, then $X_{A,2} = \infty$.

We note that, in case $\omega _{A,2}(t) \ln \frac {1}{t} \rightarrow 0$ as $t \rightarrow 0$, the BMO regularity of ∇u was proved by Acquistapace [1]. (See also [16].)

By Lemma 1.3, an explicit example of ω_A,2 satisfying (1.8) (for any constant C_∗) but not (1.5) is

$$\omega_{A,2}(t) \sim \frac{1}{\ln \frac{64}{t} (\ln \ln \frac{64}{t})^{\beta}}, \quad \beta \in (0,1].$$

In addition, unlike (1.5) or (1.6), (1.8) does not imply that A is continuous, e.g.,

$$A_{ij}^{\alpha\beta}(x) = \left( 2 + \sin \ln \ln \ln \frac{64}{|x|}\right) \delta_{ij} \delta^{\alpha\beta}.$$

(This can be checked using the fact that the function $s \mapsto {\sin \limits } s$ is Lipschitz on $\mathbb R$ and the fact that the function $x \mapsto \ell (x) := \ln \ln \ln \frac {64}{|x|}$ has L² mean oscillation $\omega _{\ell ,2}(t) \sim \frac {1}{\ln \frac {64}{t} \ln \ln \frac {64}{t}}$.)

When A is merely of vanishing mean oscillation, we note that

Remark 1.4

Let $A = (A_{ij}^{\alpha \beta })$ belong to V MO(B₄) and satisfy (1.2) and (1.3). Then every solution $u \in W^{1,\infty }(B_{4})$ of (1.1) satisfies ∇u ∈ V MO(B₂). See [12, equation (2.14)]. See also Section 2 for a different proof.

The obtained regularity in Theorem 1.1 and Remark 1.4 appears sharp. As in [17], counterexamples can be produced to show that, under (1.8),

∙ Solutions of (1.1) may not have bounded gradients (though their gradients are of vanishing mean oscillation by Theorem 1.1),

∙ $W^{1,\infty }$ solutions of (1.1) may not be differentiable (though their gradients are of vanishing mean oscillation by Remark 1.4).

Proposition 1.5

There exist a coefficient matrix $A = (A_{ij}^{\alpha \beta }) \in C(\bar B_{4})$ satisfying (1.2), (1.3) and (1.8) and a solution u ∈ H¹(B₄) of (1.1) such that ∇u ∈ V MO(B₄) but $\nabla u \notin L^{\infty }_{\text {loc}}(B_{2})$.

Proposition 1.6

There exist a coefficient matrix $A = (A_{ij}^{\alpha \beta })\in C(\bar B_{4})$ satisfying (1.2), (1.3) and (1.8) and a solution u ∈ H¹(B₄) of (1.1) such that $\nabla u \in L^{\infty }(B_{4}) \cap VMO(B_{4})$ but ∇u∉C(B₂).

Theorem 1.1 is a consequence of the following proposition on the mean oscillation of the gradient ∇u in terms of the L² mean oscillation ω_A,2 of A.

Proposition 1.7

Let $A= (A_{ij}^{\alpha \beta })$ satisfy (1.2) and (1.3). Then there exists a constant C_∗ > 0, depending only on n, N, Λ and λ such that for every u ∈ H¹(B₄) satisfying (1.1) and for 0 < r ≤ R/4 ≤ 1/2, there hold

$$ {\int}_{B_{r}} |\nabla u|^{2} dx \leq \frac{C_{*}r^{n}}{R^{n}} \exp\left( 2C_{*} {\int}_{2r}^{R} \frac{\omega_{A,2}(t)}{t} dt\right) {\int}_{B_{R}} |\nabla u|^{2} dx, $$

(1.9)

and

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{r}} |\nabla u - (\nabla u)_{r}|^{2} dx &\leq& \frac{C_{*}r^{n+2}}{R^{n}} {\int}_{B_{R}} |\nabla u|^{2} dx \times\\ &&\times \left\{{\int}_{2r}^{R} \frac{\omega_{A,2}(t)}{t^{2}} \exp\left( C_{*} {{\int}_{t}^{R}} \frac{\omega_{A,2}(s)}{s} ds\right) dt\right\}^{2}, \end{array} $$

(1.10)

where $(\nabla u)_{r} = \frac {1}{|B_{r}|} {\int \limits }_{B_{r}} \nabla u dx$ for 0 < r ≤ 2.

Moreover, if $u \in W^{1,\infty }(B_{4})$, then, for 0 < r ≤ R/4 ≤ 1/2,

$$ {\int}_{B_{r}} |\nabla u - (\nabla u)_{r}|^{2} dx \leq \frac{C_{*}r^{n+2}}{R^{n}}\left\{{\int}_{2r}^{R} \frac{\omega_{A,2}(t)}{t^{2}} dt\right\}^{2} \sup_{B_{R}} |\nabla u|^{2}. $$

(1.11)

Remark 1.8

Let 1 < p < 2. Under an additional assumption that $[A]_{BMO(B_{4})}$ is sufficiently small, the estimates in Proposition 1.7 hold if ω_A,2 is replaced by ω_A,p and the regularity assumption u ∈ H¹(B₄) is replaced by u ∈ W^1,p(B₄). We do not know if this smallness assumption can be dropped except for p close to 2. See Proposition 3.2.

Remark 1.9

It would be interesting to see if Theorem 1.1 and Proposition 1.7 remain valid if ω_A,2 is replaced by ω_A,1. In view of [9, 12], it is plausible that the answer is affirmative, but this is not clear from the techniques used in the present paper.

Remark 1.10

In the above, to keep things simple, we chose to state our results for the homogeneous system (1.1) without lower order terms. They can be generalized for non-homogeneous systems or to allow for lower order terms. See, e.g., Proposition 3.1.

2 Proofs of the Main Results

Proof of Lemma 1.3

We claim: For δ ∈ (0,1) and $a \in (0,\infty )$, the limit

$$L_{a} = \limsup\limits_{r \rightarrow 0} r {\int}_{r}^{\delta} \frac{1}{t^{2}} \left( \ln\frac{1}{t}\right)^{a - 1} dt$$

satisfies $L_{a} = \infty$ if a > 1, L_a = 1 if a = 1 and $L_{a} \leq (\ln \frac {1}{\delta })^{a-1}$ if a < 1.

When a = 1, the claim is clear. By integrating by parts, we have

$$ {\int}_{r}^{\delta} \frac{1}{t^{2}} \left( \ln\frac{1}{t}\right)^{a - 1} dt = - \frac{1}{t} \left( \ln\frac{1}{t}\right)^{a - 1}\Bigg|_{r}^{\delta} - (a-1) {\int}_{r}^{\delta} \frac{1}{t^{2}} \left( \ln \frac{1}{t}\right)^{a-2} dt. $$

(2.1)

If a < 1, we see from (2.1) that

$$ \begin{array}{@{}rcl@{}} L_{a} &=& |a - 1| \limsup\limits_{r \rightarrow 0} r{\int}_{r}^{\delta} \frac{1}{t^{2}} \left( \ln\frac{1}{t}\right)^{a - 2} dt \leq |a - 1| \limsup\limits_{r \rightarrow 0} {\int}_{r}^{\delta} \frac{1}{t} \left( \ln\frac{1}{t}\right)^{a - 2} dt\\ &=& \limsup\limits_{r \rightarrow 0} \left( \ln\frac{1}{t}\right)^{a - 1}\Big|_{r}^{\delta} = \left( \ln\frac{1}{\delta}\right)^{a - 1}. \end{array} $$

To prove the claim in the case a > 1, we may assume without loss of generality that a < 2. Note that (2.1) implies

$$L_{a} + (a-1) L_{a-1} = \limsup\limits_{r \rightarrow 0 } r \left\{- \frac{1}{t} \left( \ln\frac{1}{t}\right)^{a - 1}\Bigg|_{r}^{\delta}\right\} = \infty.$$

As L_a− 1 is finite (as 1 < a < 2), we thus have that $L_{a} = \infty$. The claim is proved.

We now apply the claim to obtain the desired conclusions. Consider first the case that $\limsup _{t \rightarrow 0} \omega _{A,2}(t) \ln \frac {1}{t} < \frac {1}{C_{*}}$. Then there exist $\varepsilon \in (0,\frac {1}{C_{*}})$ and δ ∈ (0,1) so that $\omega _{A,2}(t) \leq \varepsilon (\ln \frac {1}{t})^{-1}$ in (0,δ). For $\hat \delta \in (0,\delta )$, we compute

$$ \begin{array}{@{}rcl@{}} X_{A,2} &=& \limsup\limits_{r\rightarrow 0} r {\int}_{r}^{\hat\delta} \frac{\omega_{A,2}(t)}{t^{2}} \exp\left( C_{*} {{\int}_{t}^{2}} \frac{\omega_{A,2}(s)}{s} ds\right) dt\\ &\leq &\varepsilon \left( \ln \frac{1}{\hat\delta}\right)^{-C_{*}\varepsilon} \exp\left( C_{*} {\int}_{\hat\delta}^{2} \frac{\omega_{A,2}(s)}{s} ds\right) \limsup\limits_{r\rightarrow 0} r {\int}_{r}^{\hat\delta} \frac{1}{t^{2}}\left( \ln \frac{1}{t}\right)^{C_{*} \varepsilon-1} dt. \end{array} $$

As C_∗ε < 1, we can apply the claim to obtain

$$ \begin{array}{@{}rcl@{}} X_{A,2} &\leq& \varepsilon \left( \ln \frac{1}{\hat\delta}\right)^{-1} \exp\left( C_{*} {\int}_{\hat\delta}^{2} \frac{\omega_{A,2}(s)}{s} ds\right)\\ &\leq& \varepsilon \left( \ln \frac{1}{\hat\delta}\right)^{-1 + C_{*} \varepsilon} \left( \ln \frac{1}{\delta}\right)^{-C_{*}\varepsilon} \exp\left( C_{*} {\int}_{\delta}^{2} \frac{\omega_{A,2}(s)}{s} ds\right). \end{array} $$

Sending $\hat \delta \rightarrow 0$, we obtain that X_A,2 = 0.

Consider next the case that $\liminf _{t \rightarrow 0} \omega _{A,2}(t) \ln \frac {1}{t} > \frac {1}{C_{*}}$. Then there exist $b > \frac {1}{C_{*}}$ and δ ∈ (0,1) so that $\omega _{A,2}(t) \geq b (\ln \frac {1}{t})^{-1}$ in (0,δ). We then have

$$ \begin{array}{@{}rcl@{}} X_{A,2} &=& \limsup\limits_{r\rightarrow 0} r {\int}_{r}^{\delta} \frac{\omega_{A,2}(t)}{t^{2}} \exp\left( C_{*} {{\int}_{t}^{2}} \frac{\omega_{A,2}(s)}{s} ds\right) dt\\ &\geq& b \left( \ln \frac{1}{\delta}\right)^{-C_{*}b} \exp\left( C_{*} {\int}_{\delta}^{2} \frac{\omega_{A,2}(s)}{s} ds\right) \limsup\limits_{r\rightarrow 0} r {\int}_{r}^{\delta} \frac{1}{t^{2}}\left( \ln \frac{1}{t}\right)^{C_{*} b - 1} dt. \end{array} $$

As C_∗b > 1, we deduce from the claim that $X_{A,2} = \infty$ as desired. □

Proofs of Theorem 1.1 and Remark 1.4

The results follow immediately from Proposition 1.7. □

In order to prove Proposition 1.7, we need the following estimate for harmonic replacements. (Compare [5, Lemma 3.5], [20, Lemma 3.1].)

Lemma 2.1

Let $A, \bar A$ satisfy (1.2) and (1.3) with $\bar A$ being constant in B₄ and $f = (f_{i}^{\alpha }) \in L^{2}(B_{4})$. Let R ∈ (0,2) and suppose u,h ∈ H¹(B_2R) satisfy

$$ \begin{array}{@{}rcl@{}} \partial_{\alpha}(A{}_{ij}^{\alpha\beta} \partial_{\beta} u^{j}) &=& \partial_{\alpha} f_{i}^{\alpha}\quad \text{ in } B_{2R}, \quad i = 1, \ldots, N,\\ \partial_{\alpha}(\bar A{}_{ij}^{\alpha\beta} \partial_{\beta} h^{j}) &=& 0 \quad \text{ in } B_{2R}, \quad i = 1, \ldots, N,\\ u & =& h \quad \text{ on } \partial B_{2R}. \end{array} $$

Then there exists a constant C > 0 depending only on n,N,Λ and λ such that

$$\|\nabla (u-h)\|_{L^{2}(B_{3R/2})} \leq C \Big[\|f\|_{L^{2}(B_{2R})} + R^{-n/2} \|A - \bar A\|_{L^{2}(B_{2R})} \|\nabla u\|_{L^{2}(B_{2R})}\Big].$$

Proof

In the proof, C denotes a generic positive constant which depends only on n, N, Λ and λ. Using that $\bar A$ is constant, we have by standard elliptic estimates that

$$\|\nabla h\|_{L^{\infty}(B_{7R/4})} \leq CR^{-n/2} \|\nabla h\|_{L^{2}(B_{2R})} \leq CR^{-n/2}\|\nabla u\|_{L^{2}(B_{2R})}.$$

Observing that

$$\partial_{\alpha}(A{}_{ij}^{\alpha\beta} \partial_{\beta} (u - h)^{j}) = \partial_{\alpha} (f_{i}^{\alpha} + (\bar A - A)_{ij}^{\alpha\beta} \partial_{\beta} h^{j}) \quad \text{ in } B_{2R}, \quad i = 1, \ldots, N,$$

we deduce that

$$ \begin{array}{@{}rcl@{}} \|\nabla (u - h)\|_{L^{2}(B_{3R/2})} &\leq& C\Big[\|f\|_{L^{2}(B_{7R/4})} + \|A - \bar A\|_{L^{2}(B_{7R/4})}\|\nabla h\|_{L^{\infty}(B_{7R/4})}\\ &&\qquad + R^{-(n+2)/2}\|u - h\|_{L^{1}(B_{7R/4})}\Big]\\ &\leq& C\Big[\|f\|_{L^{2}(B_{2R})} + R^{-n/2} \|A - \bar A\|_{L^{2}(B_{2R})}\|\nabla u\|_{L^{2}(B_{2R})} \\ &&\qquad + R^{-(n+2)/2} \|u - h\|_{L^{1}(B_{2R})}\Big]. \end{array} $$

(2.2)

To estimate $\|u - h\|_{L^{1}(B_{2R})}$, fix some t > 0 and consider an auxiliary equation

$$ \begin{array}{@{}rcl@{}} \partial_{\beta}\left( \bar A_{ij}^{\alpha\beta} \partial_{\alpha} \phi^{i}\right) &=& \frac{(u-h)^{j}}{\sqrt{|u-h|^{2} + t^{2}}} \text{ in } B_{2R}, \quad j = 1, \ldots, N,\\ \phi & =& 0 \quad \text{ on } \partial B_{2R}. \end{array} $$

Testing the above against u − h, we obtain

$$ {\int}_{B_{2R}} \frac{|u-h|^{2}}{\sqrt{|u-h|^{2} + t^{2}}} dx = {\int}_{B_{2R}} \bar A_{ij}^{\alpha\beta} \partial_{\alpha} \phi^{i} \partial_{\beta}(u - h)^{j} dx. $$

(2.3)

As u − h satisfies

$$\partial_{\alpha}\left( \bar A{}_{ij}^{\alpha\beta} \partial_{\beta} (u - h)^{j}\right) = \partial_{\alpha} \left( f_{i}^{\alpha} + (\bar A - A)_{ij}^{\alpha\beta} \partial_{\beta} u^{j}\right) \quad \text{ in } B_{2R}, \quad i = 1, \ldots, N,$$

we have

$$ {\int}_{B_{2R}} \bar A_{ij}^{\alpha\beta} \partial_{\beta}(u - h)^{j}\partial_{\alpha} \phi^{i} dx = {\int}_{B_{2R}} \left( f_{i}^{\alpha} + (\bar A - A)_{ij}^{\alpha\beta} \partial_{\beta} u^{j}\right)\partial_{\alpha} \phi^{i} dx. $$

(2.4)

Inserting (2.4) into (2.3) and noting that $\|\nabla \phi \|_{L^{\infty }(B_{2R})} \leq CR$ (as $|\partial _{\beta }(\bar A_{ij}^{\alpha \beta } \partial _{\alpha } \phi ^{i}) | \leq 1$), we arrive at

$${\int}_{B_{2R}} \frac{|u-h|^{2}}{\sqrt{|u-h|^{2} + t^{2}}} dx \leq C\Big[R^{(n+2)/2} \|f\|_{L^{2}(B_{2R})} + R \|A - \bar A\|_{L^{2}(B_{2R})}\|\nabla u\|_{L^{2}(B_{2R})} \Big].$$

Noting that the constant C is independent of t, we may send $t \rightarrow 0$ to obtain

$$ \|u-h\|_{L^{1}(B_{2R})} \leq CR^{(n+2)/2} \Big[\|f\|_{L^{2}(B_{2R})} + R^{-n/2}\|A - \bar A\|_{L^{2}(B_{2R})}\|\nabla u\|_{L^{2}(B_{2R})} \Big]. $$

(2.5)

The conclusion follows from (2.2) and (2.5). □

Proof of Proposition 1.7

We only need to give the proof for a fixed R, say R = 2. Our proof is inspired by that of [21].

In the proof, C denotes a generic positive constant which depends only on n, N, Λ and λ. In particular it is independent of the parameter k which will appear below. Also, we will simply write ω instead of ω_A,2.

Proof of (1.9): For k ≥ 0, let R_k = 4^−k, $\bar A_{k} = (A)_{B_{2R_{k}}}$ and $h_{k} \in H^{1}(B_{2R_{k}})$ be the solution to

$$ \begin{array}{@{}rcl@{}} \partial_{\alpha}\left( (\bar A_{k})_{ij}^{\alpha\beta} \partial_{\beta} h_{k}^{j}\right) &=& 0\quad \text{ in } B_{2R_{k}}, \quad i = 1, \ldots, N,\\ h_{k} &=& u \quad \text{ on } \partial B_{2R_{k}}. \end{array} $$

Let $a_{k} = R_{k}^{-n/2}\|\nabla (u - h_{k})\|_{L^{2}(B_{R_{k}})}$ and $b_{k} = \|\nabla h_{k}\|_{L^{\infty }(B_{R_{k}})}$.

Note that, by triangle inequality, we have

$$ \|\nabla u\|_{L^{2}(B_{R_{k}})} \leq R_{k}^{n/2}(a_{k} + b_{k}). $$

(2.6)

By elliptic estimates for h_k, we have

$$ \begin{array}{@{}rcl@{}} \|\nabla h_{k}\|_{L^{2}(B_{2R_{k}})} & \leq &C \|\nabla u\|_{L^{2}(B_{2R_{k}})}, \end{array} $$

(2.7)

$$ \begin{array}{@{}rcl@{}} \|\nabla h_{k}\|_{L^{\infty}(B_{3R_{k}/2})} &\leq &C R_{k}^{-n/2}\|\nabla u\|_{L^{2}(B_{2R_{k}})}, \end{array} $$

(2.8)

$$ \begin{array}{@{}rcl@{}} \|\nabla^{2} h_{k}\|_{L^{\infty}(B_{3R_{k}/2})} + R_{k} \|\nabla^{3} h_{k}\|_{L^{\infty}(B_{3R_{k}/2})} & \leq& C R_{k}^{-(n+2)/2}\|\nabla u\|_{L^{2}(B_{2R_{k}})}. \end{array} $$

(2.9)

By Lemma 2.1,

$$ \|\nabla (u - h_{k})\|_{L^{2}(B_{3R_{k}/2})} \leq C\omega(2R_{k}) \|\nabla u\|_{L^{2}(B_{2R_{k}})}. $$

(2.10)

By (2.10) and (2.7),

$$R_{k}^{n/2}(a_{k} + b_{k}) \leq C\|\nabla u\|_{L^{2}(B_{2R_{k}})}.$$

By (2.6) and (2.10), we have

$$ \begin{array}{@{}rcl@{}} \|\nabla (u - h_{k+1})\|_{L^{2}(B_{R_{k+1}})} &\leq& C\omega(2R_{k}) \|\nabla u\|_{L^{2}(B_{R_{k}})} \\ &\leq& C\omega(2R_{k}) R_{k}^{n/2}(a_{k} + b_{k}). \end{array} $$

Hence

$$ a_{k+1} \leq C\omega(2R_{k}) (a_{k} + b_{k}). $$

(2.11)

Next, we have by (2.10) that

$$ \begin{array}{@{}rcl@{}} &&\|\nabla (h_{k+1} - h_{k})\|_{L^{2}(B_{3R_{k+1}/2})} \\ &\leq& \|\nabla (u - h_{k+1})\|_{L^{2}(B_{3R_{k+1}/2})} + \|\nabla (u - h_{k})\|_{L^{2}(B_{3R_{k+1}/2})} \\ &\leq& C\omega(2R_{k}) \|\nabla u\|_{L^{2}(B_{R_{k}})} \\ &\leq& C\omega(2R_{k}) R_{k}^{n/2}(a_{k} + b_{k}). \end{array} $$

Note that h_k+ 1 − h_k satisfies

$$ \begin{array}{@{}rcl@{}} \partial_{\alpha}\left( (\bar A_{k})_{ij}^{\alpha\beta} \partial_{\beta} (h_{k+1} - h_{k})^{j}\right) &= \partial_{\alpha}\left( (\bar A_{k} - \bar A_{k+1})_{ij}^{\alpha\beta} \partial_{\beta} h_{k+1}^{j}\right) \text{ in } B_{2R_{k+1}},~ i = 1, \ldots, N, \end{array} $$

As $\bar A_{k}$ is constant and

$$|\bar A_{k} - \bar A_{k+1}| \leq C\omega(2R_{k}),$$

we have by standard estimates for elliptic equations with constant coefficients and (2.8) and (2.9) (applied to h_k+ 1) that

$$ \begin{array}{@{}rcl@{}} \|\nabla (h_{k+1} - h_{k})\|_{L^{\infty}(B_{R_{k+1}})} &\leq& C\omega(2R_{k}) (a_{k} + b_{k}), \end{array} $$

(2.12)

$$ \begin{array}{@{}rcl@{}} R_{k+1}\|\nabla^{2} (h_{k+1} - h_{k})\|_{L^{\infty}(B_{R_{k+1}})} &\leq& C\omega(2R_{k}) (a_{k} + b_{k}). \end{array} $$

(2.13)

By (2.12),

$$ b_{k+1} \leq b_{k} + C\omega(2R_{k}) (a_{k} + b_{k}). $$

(2.14)

By (2.11) and (2.14), we have

$$a_{k+1} + b_{k+1} \leq (1 + C\omega(2R_{k})) (a_{k} + b_{k}).$$

We deduce that

$$ \begin{array}{@{}rcl@{}} a_{k} + b_{k} &\leq& \prod\limits_{j=0}^{k} (1 + C\omega(2R_{j})) (a_{0} + b_{0}) \leq C\exp\left( C \sum\limits_{j=0}^{k}\omega(2R_{j})\right) \|\nabla u\|_{L^{2}(B_{2})}\\ &\leq& C\exp\left( C {\varOmega}(2R_{k})\right) \|\nabla u\|_{L^{2}(B_{2})}, \end{array} $$

(2.15)

where

$${\varOmega}(t) := {{\int}_{t}^{2}} \frac{\omega(s)}{s} ds,$$

and where we have used the fact that ω(t) ≤ Cω(s) whenever 0 < t ≤ s ≤ 4t. We have thus shown that

$${\int}_{B_{R_{k}}} |\nabla u|^{2} dx \leq C {R_{k}^{n}} \exp\big(C {\varOmega}(2R_{k}) \big) {\int}_{B_{2}} |\nabla u|^{2} dx \text{ for } k \geq 0.$$

Estimate (1.9) is readily seen.

Proof of (1.10): We write

$$h_{R_{k}} = \sum\limits_{j=0}^{k} w_{j}, \text{ where } w_{0} = h_{R_{0}} \text{ and } w_{j} = h_{R_{j}} - h_{R_{j-1}} \text{ for } j \geq 1.$$

Using the estimate $\|\nabla ^{2} h_{R_{0}}\|_{L^{\infty }(B_{1})} \leq C\|\nabla u\|_{L^{2}(B_{2})}$ together with (2.13) and (2.15), we have

$$ \begin{array}{@{}rcl@{}} |\nabla h_{R_{k}}(x) - \nabla h_{R_{k}}(0)| &\leq& C |x|\sum\limits_{j=0}^{k} \frac{ \omega(2R_{j}) }{R_{j}} \exp\big(C{\varOmega}(2R_{j}) \big) \|\nabla u\|_{L^{2}(B_{2})}\\ &\leq& C |x| {\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} \exp\big(C{\varOmega}(t) \big) dt \|\nabla u\|_{L^{2}(B_{2})}, \end{array} $$

(2.16)

where we have again used the fact that ω(t) ≤ Cω(s) whenever 0 < t ≤ s ≤ 4t. This implies

$$ \begin{array}{@{}rcl@{}} &&\|\nabla h_{R_{k}} - \nabla h_{R_{k}}(0)\|_{L^{2}(B_{R_{k}})}\\ &\leq& C R_{k}^{(n+2)/2} {\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} \exp\big(C {\varOmega}(t)\big) dt \|\nabla u\|_{L^{2}(B_{2})}. \end{array} $$

(2.17)

Combining (2.17) with (2.10) and (2.15), we get

$$ \begin{array}{@{}rcl@{}} \|\nabla u - (\nabla u)_{R_{k}}\|_{L^{2}(B_{R_{k}})} &\leq& \|\nabla u - \nabla h_{R_{k}}(0)\|_{L^{2}(B_{R_{k}})}\\ &\leq& \|\nabla (u - h_{R_{k}})\|_{L^{2}(B_{R_{k}})} + \|\nabla h_{R_{k}} - \nabla h_{R_{k}}(0)\|_{L^{2}(B_{R_{k}})}\\ &\leq& C R_{k}^{(n+2)/2} {\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} \exp\big(C {\varOmega}(t)\big) dt \|\nabla u\|_{L^{2}(B_{2})}\\ &\quad& + C R_{k}^{n/2} \omega(2R_{k}) \exp\big(C {\varOmega}(2R_{k})\big) \|\nabla u\|_{L^{2}(B_{2})}. \end{array} $$

(2.18)

As ω(2R_k) ≤ Cω(t) whenever 2R_k ≤ t ≤ 4R_k, we have

$$ {\int}_{2R_{k}}^{4R_{k}} \frac{ \omega(t)}{t^{2}} \exp\big(C {\varOmega}(t)\big) dt \geq \frac{\omega(2R_{k})}{CR_{k}} \exp\big(C {\varOmega}(2R_{k})\big).$$

Using this in (2.18), we deduce that for k ≥ 1 that

$$ \begin{array}{@{}rcl@{}} \|\nabla u - (\nabla u)_{R_{k}}\|_{L^{2}(B_{R_{k}})} &\leq C R_{k}^{(n+2)/2} {\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} \exp\big(C {\varOmega}(t)\big) dt \|\nabla u\|_{L^{2}(B_{2})}. \end{array} $$

Estimate (1.10) follows.

Proof of (1.11): We adjust the proof of (1.10) exploiting the fact that $\nabla u \in L^{\infty }(B_{2})$. First, using the fact that $a_{k} + b_{k} \leq C \|\nabla u\|_{L^{\infty }(B_{2})}$ in (2.13) we get instead of (2.16) the stronger estimate

$$ \begin{array}{@{}rcl@{}} |\nabla h_{R_{k}}(x) - \nabla h_{R_{k}}(0)| &\leq C |x| {\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} dt \|\nabla u\|_{L^{\infty}(B_{2})}, \end{array} $$

and so

$$ \|\nabla h_{R_{k}} - \nabla h_{R_{k}}(0)\|_{L^{2}(B_{R_{k}})}\\ \leq C R_{k}^{(n+2)/2} {\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} dt \|\nabla u\|_{L^{\infty}(B_{2})}. $$

(2.19)

Combining (2.19) with (2.10), we get for k ≥ 1 that

$$ \begin{array}{@{}rcl@{}} \|\nabla u - (\nabla u)_{R_{k}}\|_{L^{2}(B_{R_{k}})} &\leq& \|\nabla u - \nabla h_{R_{k}}(0)\|_{L^{2}(B_{R_{k}})}\\ &\leq& \|\nabla (u - h_{R_{k}})\|_{L^{2}(B_{R_{k}})} + \|\nabla h_{R_{k}} - \nabla h_{R_{k}}(0)\|_{L^{2}(B_{R_{k}})}\\ &\leq& C R_{k}^{(n+2)/2} {\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} dt \|\nabla u\|_{L^{\infty}(B_{2})}\\ &\quad& + C R_{k}^{n/2} \omega(2R_{k}) \|\nabla u\|_{L^{\infty}(B_{2})}\\ &\leq& C R_{k}^{(n+2)/2} {\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} dt \|\nabla u\|_{L^{\infty}(B_{2})}. \end{array} $$

Estimate (1.11) follows. □

Remark 2.2

If the Dini condition (1.4) or (1.5) holds, it can be seen from (2.12) that {∇h_k(0)} converges to some $P \in \mathbb R^{N \times n}$, from which it follows that

$$\lim\limits_{r\rightarrow 0} r^{-n/2}\|\nabla u - P \|_{L^{2}(B_{r})} = 0,$$

yielding the continuity of ∇u at the origin. We have thus recovered the results on the continuous differentiability of H¹ solutions of Brezis [2, 3] and Li [21].

Proof of Proposition 1.5

We take N = 1 and drop the indices i, j in the expression of A (so that A = (A^αβ)). Following [17, Lemma 2.1], we make the ansatz that

$$ \begin{array}{@{}rcl@{}} A^{\alpha\beta}(x) &=& \delta^{\alpha\beta} + a(|x|) \left( \delta^{\alpha\beta} - \frac{x^{\alpha} x^{\beta}}{|x|^{2}}\right),\\ u(x) &=& x^{1} v(|x|). \end{array} $$

Then

$$ \begin{array}{@{}rcl@{}} \partial_{\alpha} (A^{\alpha\beta}\partial_{\beta} u) = x^{1} \left( v^{\prime\prime}(|x|) + \frac{n+1}{|x|} v^{\prime}(|x|) - \frac{n-1}{|x|^{2}} a(|x|) v(|x|)\right). \end{array} $$

Selecting now

$$ \begin{array}{@{}rcl@{}} a(r) &=& - \frac{1 + n \ln \frac{64}{r}}{(n-1) (\ln \frac{64}{r})^{2} \ln \ln \frac{64}{r}},\\ v(r) &=& \ln \ln \frac{64}{r}, \end{array} $$

we see that A is continuous in $\bar B_{4}$, satisfies (1.2) and (1.3) and u is an H¹ solution of (1.1). The matrix A admits a modulus of continuity $\bar \omega _{A}(t) \sim \frac {1}{\ln \frac {64}{t} \ln \ln \frac {64}{t}}$ as $t \rightarrow 0$ and so (1.8) holds. It is readily seen that u ∈ W^1,p(B₄) for all $p \in [1,\infty )$, ∇u ∈ V MO(B₄) but $\nabla u \notin L^{\infty }_{\text {loc}}(B_{2})$. □

Proof of Proposition 1.6

Instead of the choice in the proof of Proposition 1.5, we now choose

$$ \begin{array}{@{}rcl@{}} a(r) &=& - \frac{\sin \ln \ln \ln \frac{64}{r} + \cos \ln \ln \ln \frac{64}{r}(1+ \ln \frac{64}{r} + n \ln \frac{64}{r} \ln \ln \frac{64}{r}) }{(n-1) (\ln \frac{64}{r})^{2} (\ln \ln \frac{64}{r})^{2} (2 + \sin\ln \ln \frac{64}{r})},\\ v(r) &=& 2 + \sin\ln \ln \ln \frac{64}{r}. \end{array} $$

It is readily checked that A is continuous in $\bar B_{4}$, satisfies (1.2), (1.3) and (1.8) and u is an H¹ solution of (1.1), $\nabla u \in L^{\infty }(B_{4}) \cap VMO(B_{4})$ but ∇u∉C(B₂). □

3 Some Extensions

As announced in the introduction, Proposition 1.7 can be adapted for non-homogeneous systems with or without lower order terms. To illustrate, consider for example the system

$$ \partial_{\alpha}(A_{ij}^{\alpha\beta} \partial_{\beta} u^{j}) = \partial_{\alpha} f^{\alpha}_{i} \quad \text{ in } B_{4}, \quad i = 1, \ldots, N. $$

(3.1)

Let $\omega _{f,2}:(0,2] \rightarrow [0,\infty )$ denote the L² mean oscillation of f

$$ \omega_{f,2}(r) = \sup_{x \in B_{2}} \left\{\frac{1}{|B_{r}(x)|}{\int}_{B_{r}(x)} |f(y) - (f)_{B_{r}(x)}|^{2} dy\right\}^{1/2}. $$

We have

Proposition 3.1

Let $A= (A_{ij}^{\alpha \beta })$ satisfy (1.2) and (1.3). Then there exists a constant C_∗ > 0, depending only on n, N, Λ and λ such that for every u ∈ H¹(B₄) satisfying (3.1) with f ∈ L²(B₄) and for 0 < r ≤ R/4 ≤ 1/2, there hold

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{r}} |\nabla u|^{2} dx &\leq &\frac{C_{*}r^{n}}{R^{n}} \exp\left( 2C_{*} {\int}_{2r}^{R} \frac{\omega_{A,2}(t)}{t} dt\right) {\int}_{B_{R}} |\nabla u|^{2} dx\\ && + \frac{C_{*} r^{n}}{R^{n}} \left\{{\int}_{2r}^{2} \exp\left( C_{*}{\int}_{2r}^{t} \frac{\omega_{A,2}(s)}{s} ds\right) \frac{\omega_{f,2}(t)}{t} dt\right\}^{2} \end{array} $$

(3.2)

and

$$ \begin{array}{@{}rcl@{}} &&{\int}_{B_{r}} |\nabla u - (\nabla u)_{r}|^{2} dx \\ &\leq& \frac{C_{*} r^{n+2}}{R^{n}} \Bigg\{ {\int}_{2r}^{2} \frac{ \omega_{A,2}(t)}{t^{2}} \Bigg[\exp\left( C_{*} {{\int}_{t}^{2}} \frac{\omega_{A,2}(s)}{s} ds\right) \|\nabla u\|_{L^{2}(B_{2})}\\ && + {{\int}_{t}^{2}} \exp\left( C_{*}{\int}_{t}^{\tau} \frac{\omega_{A,2}(s)}{s} ds\right) \frac{\omega_{f,2}(\tau)}{\tau} d\tau \Bigg]dt + {\int}_{2r}^{2} \frac{\omega_{f,2}(t)}{t^{2}} dt\Bigg\}^{2}. \end{array} $$

(3.3)

Moreover, if $u \in W^{1,\infty }(B_{4})$, then, for 0 < r ≤ R/4 ≤ 1/2,

$$ {\int}_{B_{r}} |\nabla u - (\nabla u)_{r}|^{2} dx \leq \frac{C_{*}r^{n+2}}{R^{n}} \left\{{\int}_{2r}^{R} \frac{ \omega_{A,2}(t)}{t^{2}} dt \|\nabla u\|_{L^{\infty}(B_{R})} + {\int}_{2r}^{R} \frac{ \omega_{f,2}(t)}{t^{2}} dt\right\}^{2}. $$

(3.4)

Proof of Proposition 3.1

The proof is a simple adjustment of that of Proposition 1.7. We will only point out the main changes.

Proof of (3.2): Define h_k, a_k, b_k as in the proof of Proposition 1.7. Proceeding as before but paying attention to the application of Lemma 2.1 which now involves an inhomogeneous right hand side, we consecutively obtain the following estimates:

$$ \begin{array}{@{}rcl@{}} &&\|\nabla u\|_{L^{2}(B_{R_{k}})} \leq R_{k}^{n/2}(a_{k} + b_{k}),\\ &&\|\nabla (u - h_{k})\|_{L^{2}(B_{3R_{k}/2})} \leq C\omega(2R_{k}) \|\nabla u\|_{L^{2}(B_{2R_{k}})} + C\omega_{f,2}(2R_{k})R_{k}^{n/2},\\ &&R_{k}^{n/2}(a_{k} + b_{k}) \leq C\|\nabla u\|_{L^{2}(B_{2R_{k}})} + C\omega_{f,2}(2R_{k})R_{k}^{n/2},\\ &&a_{k+1} \leq C\omega(2R_{k}) (a_{k} + b_{k}) + C\omega_{f,2}(2R_{k}),\\ && R_{k+1}\|\nabla^{2} (h_{k+1} - h_{k})\|_{L^{\infty}(B_{R_{k+1}})} \leq C\omega(2R_{k}) (a_{k} + b_{k}) + C\omega_{f,2}(2R_{k}),\\ &&b_{k+1} \leq b_{k} + C\omega(2R_{k}) (a_{k} + b_{k}) + C\omega_{f,2}(2R_{k}),\\ &&a_{k+1} + b_{k+1} \leq (1 + C\omega(2R_{k})) (a_{k} + b_{k}) + C\omega_{f,2}(2R_{k}). \end{array} $$

Hence, instead of (2.15), we now get

$$ \begin{array}{@{}rcl@{}} a_{k} + b_{k} &\leq& C\exp\big(C {\varOmega}(2R_{k})\big) \|\nabla u\|_{L^{2}(B_{2})}\\ && + C {\int}_{2R_{k}}^{2} \exp\big(C[{\varOmega}(2R_{k}) - {\varOmega}(t)] \big) \frac{\omega_{f,2}(t)}{t} dt, \end{array} $$

where we recall the notation ${\varOmega }(t) = {{\int \limits }_{t}^{2}} \frac {\omega (s)}{s} ds$. Estimate (3.2) follows.

Proof of (3.3): Instead of (2.16) and (2.17), we now have

$$ \begin{array}{@{}rcl@{}} &&|\nabla h_{R_{k}}(x) - \nabla h_{R_{k}}(0)|\\ &\leq& C |x| \sum\limits_{j=0}^{k} \frac{\omega_{f,2}(2R_{j})}{R_{j}} + C |x| \sum\limits_{j=0}^{k} \frac{ \omega(2R_{j}) }{R_{j}} \Big[ \exp\big(C {\varOmega}(2R_{j})\big) \|\nabla u\|_{L^{2}(B_{2})} \\ && + {\int}_{2R_{j}}^{2} \big(C[{\varOmega}(2R_{j}) - {\varOmega}(t)] \big) \frac{\omega_{f,2}(t)}{t} dt\Big] \\ &\leq& C|x| {\int}_{2R_{k}}^{2} \frac{\omega_{f,2}(t)}{t^{2}} dt + C |x| {\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} \Big[\exp\big(C {\varOmega}(t) \big) \|\nabla u\|_{L^{2}(B_{2})} \\ && + {{\int}_{t}^{2}} \exp\big(C [{\varOmega}(t) - {\varOmega}(\tau)] \big) \frac{\omega_{f,2}(\tau)}{\tau} d\tau \Big]dt, \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} &&\|\nabla h_{R_{k}} - \nabla h_{R_{k}}(0)\|_{L^{2}(B_{R_{k}})} \\ &\leq& CR_{k}^{(n+2)/2} {\int}_{2R_{k}}^{2} \frac{\omega_{f,2}(t)}{t^{2}} dt + CR_{k}^{(n+2)/2}{\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} \Big[\exp\big(C {\varOmega}(t) \big) \|\nabla u\|_{L^{2}(B_{2})} \\ && + {{\int}_{t}^{2}} \exp\big(C [{\varOmega}(t) - {\varOmega}(\tau)] \big) \frac{\omega_{f,2}(\tau)}{\tau} d\tau \Big]dt. \end{array} $$

It then follows that

$$ \begin{array}{@{}rcl@{}} &&\|\nabla u - (\nabla u)_{R_{k}}\|_{L^{2}(B_{R_{k}})}\\ &\leq& \|\nabla u - \nabla h_{R_{k}}(0)\|_{L^{2}(B_{R_{k}})}\\ &\leq& \|\nabla (u - h_{R_{k}})\|_{L^{2}(B_{R_{k}})} + \|\nabla h_{R_{k}} - \nabla h_{R_{k}}(0)\|_{L^{2}(B_{R_{k}})}\\ &\leq& CR_{k}^{(n+2)/2} {\int}_{2R_{k}}^{2} \frac{\omega_{f,2}(t)}{t^{2}} dt + CR_{k}^{(n+2)/2}{\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} \Big[\exp\big(C {\varOmega}(t) \big) \|\nabla u\|_{L^{2}(B_{2})} \\ && + {{\int}_{t}^{2}} \exp\big(C [{\varOmega}(t) - {\varOmega}(\tau)] \big) \frac{\omega_{f,2}(\tau)}{\tau} d\tau \Big]dt\\ && + CR_{k}^{n/2} \omega_{f,2}(2R_{k})\\ && + C R_{k}^{n/2} \omega(2R_{k}) \exp\big(C {\varOmega}(2R_{k}) \big) \|\nabla u\|_{L^{2}(B_{2})}\\ && + CR_{k}^{n/2} \omega(2R_{k}) {\int}_{2R_{k}}^{2} \exp\big(C[{\varOmega}(2R_{k}) - {\varOmega}(t)]\big) \frac{\omega_{f,2}(t)}{t} dt. \end{array} $$

Noting that, on the right hand side, the third term can be absorbed into the first term, and the fourth and fifth terms can be absorbed into the second term, we arrive at estimate (3.3).

Proof of (3.4): Using $\nabla u \in L^{\infty }(B_{2})$, we obtain this time that

$$ \begin{array}{@{}rcl@{}} |\nabla h_{R_{k}}(x) - \nabla h_{R_{k}}(0)| &\leq C |x| \left\{{\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} dt \|\nabla u\|_{L^{\infty}(B_{2})} + {\int}_{2R_{k}}^{2} \frac{ \omega_{f,2}(t)}{t^{2}} dt\right\}, \end{array} $$

and so

$$ \begin{array}{@{}rcl@{}} &&\|\nabla h_{R_{k}} - \nabla h_{R_{k}}(0)\|_{L^{2}(B_{R_{k}})}\\ &\leq& C R_{k}^{(n+2)/2} \left\{{\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} dt \|\nabla u\|_{L^{\infty}(B_{2})} + {\int}_{2R_{k}}^{2} \frac{ \omega_{f,2}(t)}{t^{2}} dt\right\}. \end{array} $$

It then follows that

$$ \begin{array}{@{}rcl@{}} \|\nabla u - (\nabla u)_{R_{k}}\|_{L^{2}(B_{R_{k}})} &\leq& \|\nabla u - \nabla h_{R_{k}}(0)\|_{L^{2}(B_{R_{k}})}\\ &\leq& \|\nabla (u - h_{R_{k}})\|_{L^{2}(B_{R_{k}})} + \|\nabla h_{R_{k}} - \nabla h_{R_{k}}(0)\|_{L^{2}(B_{R_{k}})}\\ &\leq& C R_{k}^{(n+2)/2} \left\{{\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} dt \|\nabla u\|_{L^{\infty}(B_{2})} + {\int}_{2R_{k}}^{2} \frac{ \omega_{f,2}(t)}{t^{2}} dt\right\}\\ &\quad& + C R_{k}^{n/2} \omega(2R_{k}) \|\nabla u\|_{L^{\infty}(B_{2})} + CR_{k}^{n/2} \omega_{f,2}(2R_{k})\\ &\leq& C R_{k}^{(n+2)/2} \left\{{\int}_{2R_{k}}^{2} \frac{ \omega(t)}{t^{2}} dt \|\nabla u\|_{L^{\infty}(B_{2})} + {\int}_{2R_{k}}^{2} \frac{ \omega_{f,2}(t)}{t^{2}} dt\right\}. \end{array} $$

Estimate (3.4) is readily seen. □

Finally, we briefly touch on the validity of Theorem 1.1 when ω_A,2 is replaced by ω_A,p for 1 < p < 2. For this, we only need the following L^p version of Proposition 1.7.

Proposition 3.2

Let $A= (A_{ij}^{\alpha \beta })$ satisfy (1.2) and (1.3). Let 1 < p < 2. Then there exist constants γ > 0 and C_∗ > 0 depending only on n, N, p, Λ and λ such that, provided [A]_BMO(B₄) < γ, there hold for every u ∈ W^1,p(B₄) satisfying (1.1) and for 0 < r ≤ R/4 ≤ 1/2 that

$$ {\int}_{B_{r}} |\nabla u|^{p} dx \leq \frac{C_{*}r^{n}}{R^{n}} \exp\left( 2C_{*} {\int}_{2r}^{R} \frac{\omega_{A,p}(t)}{t} dt\right) {\int}_{B_{R}} |\nabla u|^{p} dx, $$

and

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{r}} |\nabla u - (\nabla u)_{r}|^{p} dx &\leq& \frac{C_{*}r^{n+2}}{R^{n}} {\int}_{B_{R}} |\nabla u|^{p} dx \times\\ &&\times \left\{{\int}_{2r}^{R} \frac{\omega_{A,p}(t)}{t^{2}} \exp\left( C_{*} {{\int}_{t}^{R}} \frac{\omega_{A,p}(s)}{s} ds\right) dt\right\}^{2}, \end{array} $$

where $(\nabla u)_{r} = \frac {1}{|B_{r}|} {\int \limits }_{B_{r}} \nabla u dx$ for 0 < r ≤ 2.

Moreover, if $u \in W^{1,\infty }(B_{4})$, then, for 0 < r ≤ R/4 ≤ 1/2,

$$ {\int}_{B_{r}} |\nabla u - (\nabla u)_{r}|^{p} dx \leq \frac{C_{*}r^{n+2}}{R^{n}}\left\{{\int}_{2r}^{R} \frac{\omega_{A,p}(t)}{t^{2}} dt\right\}^{2} \sup_{B_{R}} |\nabla u|^{p} . $$

The proof of Proposition 3.2 is the same as that of Proposition 1.7, but now using the following harmonic replacement estimate:

Lemma 3.3

Let 1 < p < 2. Let $A, \bar A$ satisfy (1.2) and (1.3) with $\bar A$ being constant in B₄ and $f = (f_{i}^{\alpha }) \in L^{p^{\prime }}(B_{4})$ with $p^{\prime } = \frac {p}{p-1}$. Let R ∈ (0,1) and suppose u,h ∈ W^1,p(B_4R) satisfy

$$ \begin{array}{@{}rcl@{}} \partial_{\alpha}(A{}_{ij}^{\alpha\beta} \partial_{\beta} u^{j}) &=& \partial_{\alpha} f_{i}^{\alpha}\quad \text{ in } B_{3R}, \quad i = 1, \ldots, N,\\ \partial_{\alpha}(\bar A{}_{ij}^{\alpha\beta} \partial_{\beta} h^{j}) &=& 0 \quad \text{ in } B_{2R}, \quad i = 1, \ldots, N,\\ u & =& h \quad \text{ on } \partial B_{2R}. \end{array} $$

Then there exist constants γ > 0 and C > 0 depending only on n,N,p,Λ and λ such that, provided $[A]_{BMO(B_{4R})} \leq \gamma$,

$$ \begin{array}{@{}rcl@{}} &&\|\nabla (u-h)\|_{L^{p}(B_{3R/2})} \\ &\leq& C \Big[R^{n(1/p - 1/p^{\prime})}\|f\|_{L^{p^{\prime}}(B_{3R})} + R^{-n/p} \|A - \bar A\|_{L^{p}(B_{3R})} \|\nabla u\|_{L^{p}(B_{3R})}\Big]. \end{array} $$

Proof

We amend the proof of Lemma 2.1 using L^p theories for elliptic systems whose leading coefficients have small BMO semi-norm.² In the proof, C denotes a generic positive constant which depends only on n, N, p, Λ and λ.

It is known that (see, e.g., Dong and Kim [10, 11])³, provided $[A]_{BMO(B_{4R})} \leq \gamma$ for some small enough γ depending only on n,N,p,Λ and λ, one has

$$ \|\nabla u\|_{L^{p^{\prime}}(B_{2R})} \leq C\Big[\|f\|_{L^{p^{\prime}}(B_{3R})} + R^{n(1/p^{\prime} - 1/p)}\|\nabla u\|_{L^{p}(B_{3R})}\Big]. $$

(3.5)

Using that $\bar A$ is constant, we have by standard elliptic estimates that

$$\|\nabla h\|_{L^{\infty}(B_{7R/4})} \leq CR^{-n/{p}} \|\nabla h\|_{L^{p}(B_{2R})} \leq CR^{-n/{p}}\|\nabla u\|_{L^{p}(B_{2R})}.$$

Using

$$\partial_{\alpha}\left( A{}_{ij}^{\alpha\beta} \partial_{\beta} (u - h)^{j}\right) = \partial_{\alpha} \left( f_{i}^{\alpha} + (\bar A - A)_{ij}^{\alpha\beta} \partial_{\beta} h^{j}\right) \quad \text{ in } B_{2R}, \quad i = 1, \ldots, N,$$

and once again the fact that $[A]_{BMO(B_{4R})} \leq \gamma$, we have

$$ \begin{array}{@{}rcl@{}} &&\|\nabla (u - h)\|_{L^{p}(B_{3R/2})} \\ &\leq& C\Big[\|f\|_{L^{p}(B_{7R/4})} + \|A - \bar A\|_{L^{p}(B_{7R/4})}\|\nabla h\|_{L^{\infty}(B_{7R/4})}\\ & &+ R^{-(n+p^{\prime})/{p^{\prime}}}\|u - h\|_{L^{1}(B_{7R/4})}\Big]\\ &\leq& C\Big[R^{n(1/p - 1/p^{\prime})}\|f\|_{L^{p^{\prime}}(B_{2R})} + R^{-n/{p}} \|A - \bar A\|_{L^{p}(B_{2R})}\|\nabla u\|_{L^{p}(B_{2R})} \\ && + R^{-(n+{p^{\prime}})/{p^{\prime}}} \|u - h\|_{L^{1}(B_{2R})}\Big]. \end{array} $$

(3.6)

To estimate $\|u - h\|_{L^{1}(B_{2R})}$, recall from the proof of Lemma 2.1 the chain of identities

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{2R}} \frac{|u-h|^{2}}{\sqrt{|u-h|^{2} + t^{2}}} dx &=& {\int}_{B_{2R}} \bar A_{ij}^{\alpha\beta} \partial_{\alpha} \phi^{i} \partial_{\beta}(u - h)^{j} dx\\ &=& {\int}_{B_{2R}} \left( f_{i}^{\alpha} + (\bar A - A)_{ij}^{\alpha\beta} \partial_{\beta} u^{j}\right)\partial_{\alpha} \phi^{i} dx, \end{array} $$

which imply

$${\int}_{B_{2R}} \frac{|u-h|^{2}}{\sqrt{|u-h|^{2} + t^{2}}} dx \leq C\Big[R^{(n+p)/{p}} \|f\|_{L^{p^{\prime}}(B_{2R})} + R \|A - \bar A\|_{L^{p}(B_{2R})}\|\nabla u\|_{L^{p^{\prime}}(B_{2R})} \Big].$$

Noting that the constant C is independent of t, we may send $t \rightarrow 0$ to obtain

$$ \|u-h\|_{L^{1}(B_{2R})} \leq CR^{(n+p)/p} \Big[\|f\|_{L^{p^{\prime}}(B_{2R})} + R^{-n/p}\|A - \bar A\|_{L^{p}(B_{2R})}\|\nabla u\|_{L^{p^{\prime}}(B_{2R})} \Big]. $$

(3.7)

The conclusion follows from (3.5), (3.6) and (3.7). □