On $C^1$, $C^2$, and weak type-(1, 1) estimates for linear elliptic operators: part II

Abstract

We extend and improve the results in Dong and Kim (Commun Partial Differ Equ 42(3):417–435, 2017): showing that weak solutions to full elliptic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain conditions. Similar results are obtained for non-divergence form equations. We extend the weak type-(1, 1) estimates in Dong and Kim (Commun Partial Differ Equ 42(3):417–435, 2017) and Escauriaza (Duke Math J 74(1):177–201, 1994) up to the boundary and derive a Harnack inequality for non-negative adjoint solutions to non-divergence form elliptic equations, when the leading coefficients have Dini mean oscillation.

Appendix

The following lemma is a slight generalization of [7, Lemma 2.1]. For the completeness, we present a proof here.

Lemma 4.1

Let $\Omega \subset \mathbb {R}^n$ be a bounded domain satisfying the condition (1.5) and let T be a bounded linear operator from $L^2(\Omega )$ to $L^2(\Omega )$. Let $\mu \in (0,1)$ be a constant. Suppose that for any $x_0 \in \Omega$ and $0<r< \mu {\text {diam}}\Omega$, we have

$$\begin{aligned} \int _{\Omega {\setminus } B(x_0, cr)} |Tb| \le C \int _{B(x_0, r)\cap \Omega } |b| \end{aligned}$$

(4.1)

whenever $b \in L^2(\Omega )$ is supported in $B(x_0, r)\cap \Omega$, $\int _{\Omega } b =0$, and $c>1$ and $C>0$ are constants. Then for $f \in L^2(\Omega )$ and any $t>0$, we have

$$\begin{aligned} |\{x \in \Omega : |Tf(x)| > t\}| \le \frac{C'}{t} \int _{\Omega } |f|, \end{aligned}$$

(4.2)

where $C'=C'(n, c , C, \mu , \Omega , A_0)$ is a constant.

Proof

To begin with, we note that $\Omega$ equipped with the standard Euclidean metric and the Lebesgue measure (restricted to $\Omega$) is a space of homogeneous type. By [5, Theorem 11], there exists a collection of open subsets (called “cubes”)

$$\begin{aligned} \{Q^k_\alpha \subset \Omega : k \in \mathbb {Z}, \; \alpha \in I_k\}, \end{aligned}$$

with $I_k$ at most countable set and constants $\delta \in (0,1)$, $a_0>0$ and $C_1<\infty$ such that

• (i) $|\Omega {\setminus } \bigcup _\alpha Q^k_\alpha | =0\quad \forall k$.

• (ii) If $\ell \ge k$ then either $Q^\ell _\beta \subset Q^k_\alpha$ or $Q^\ell _\beta \cap Q^k_\alpha =\emptyset$.

• (iii) For each $(k,\alpha )$ and each $\ell < k$ there is a unique $\beta$ such that $Q^k_\alpha \subset Q^\ell _\beta$.

• (iv) ${\text {diam}}Q^k_\alpha \le C_1 \delta ^k$.

• (v) Each $Q^k_\alpha$ contains some “ball” $B(z^k_\alpha , a_0 \delta ^k) \cap \Omega$.

From the above, we can infer the following.

• (a) There is constant $A_1\ge 1$ such that if $Q^{k-1}_\beta$ is the parent of $Q^k_\alpha$ (resp. if $B^k_\alpha$ is the Euclidean ball in $\mathbb {R}^n$ centered at $z^k_\alpha$ with radius $r={\text {diam}}Q^k_\alpha$ ), then we have

  $$\begin{aligned} |Q^{k-1}_\beta | \le A_1 |Q^k_\alpha |\quad (\text {resp. }\;|B^{k}_\alpha | \le A_1 |Q^k_\alpha |\,). \end{aligned}$$

  (4.3)

• (b) The Lebesgue differentiation theorem is available for the chain of cubes shrinking to a point because the maximal function defined as

  $$\begin{aligned} M(f)(x)= {\left\{ \begin{array}{ll} \sup _{x\in Q^k_\alpha }\fint _{Q^k_\alpha }|f|\, dx,\ &{}\quad \text {when}\ x\in \bigcap _k\bigcup _{\alpha \in I_k}Q^k_\alpha ,\\ 0,\ &{}\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

  is of weak type-(1, 1) over $\Omega$.

By (i)–(v) above and (1.5), choose $k_0\in \mathbb {Z}$ with $\theta :=\inf _{\alpha \in I_{k_0}}|Q^{k_0}_\alpha |>0$. To get (4.2) when

$$\begin{aligned} \frac{1}{t}\int _{\Omega } |f|\ dx>\theta , \end{aligned}$$

it suffices to choose $C'\ge \theta ^{-1}|\Omega |$. Otherwise,

$$\begin{aligned} \fint _{Q^{k_0}_\alpha }|f|\ dx\le t,\quad \text {for all}\;\; \alpha \in I_{k_0}. \end{aligned}$$

Let then $\{Q_l\}$ denote the set of cubes chosen as follows. For $k=k_0+1$ and $\alpha \in I_k$, the cube $Q=Q^k_\alpha$ satisfies either $\fint _{Q}\, |f| \le t$ or $\fint _{Q}\, |f| > t$. In the second case, we select $Q=Q^k_\alpha$ as one of the cubes in $\{Q_l\}$. Note that in this case, we have by (4.3)

$$\begin{aligned} t<\fint _{Q} |f|\,dx \le A_1 t. \end{aligned}$$

In the first case, we subdivide $Q=Q^k_\alpha$ further into subcubes $Q'=Q^{k+1}_\beta$, and repeat the process until (if ever) we are forced into the second case. By observation (b), we find that $|f(x)| \le t$ for a.e. $x \in \Omega {\setminus } \bigcup _l Q_l$.

We decompose $f=g+b$, with $b=\sum _l b_l$, such that

$$\begin{aligned} g= m_l(f):=\fint _{Q_l} f \,dx\;\text { on }\;Q_l, \end{aligned}$$

$g=f$ on $\Omega {\setminus } \bigcup _l Q_l$, and set

$$\begin{aligned} b_l= \chi _{Q_l}(f - m_l(f) ). \end{aligned}$$

It is obvious that $\int _{Q_l} b_l \,dx =0$ and we have

$$\begin{aligned} \int _{Q_l} |b_l| \,dx \le \int _{Q_l} |f|\,dx + |Q_l| m_l(f) \le 2 \int _{Q_l} |f| \,dx \le 2A_1 t |Q_l|. \end{aligned}$$

(4.4)

Also, we see that

$$\begin{aligned} |g(x)| \le A_1 t \quad \text {for a.e. }\; x \in \Omega . \end{aligned}$$

(4.5)

Indeed, for a.e. $x \in \Omega {\setminus } \bigcup _l Q_l$, we have $|g(x)| =|f(x)| \le t$ and $|g(x)| \le A_1 t$ on $Q_l$. By Chebyshev’s inequality and the $L^2$ boundedness of T, we have

$$\begin{aligned} |\{x \in \Omega : |Tg(x)| > \tfrac{1}{2} t\}|&\lesssim \frac{1}{t^2} \int _{\Omega }\, |Tg|^2\,dx \lesssim \frac{1}{t^2} \int _{\Omega }\, |g|^2\,dx \nonumber \\&\lesssim \frac{1}{t} \int _{\Omega {\setminus } \bigcup _l Q_l} |f|\,dx + \sum _l \,|Q_l| \lesssim \frac{1}{t} \int _{\Omega } |f|\,dx, \end{aligned}$$

(4.6)

where we used (4.5) and the property that

$$\begin{aligned} \sum _l \,|Q_l| \le \frac{1}{t} \int _{\Omega } |f|\,dx. \end{aligned}$$

(4.7)

We associate each $Q_l=Q^k_\alpha$ with a Euclidean ball $B_l=B(x_l, r_l)$, where $x_l=z^k_\alpha \in \Omega$ and $r_l={\text {diam}}Q^k_\alpha$. Let us denote $B_l^*=B(x_l, cr_l)$. Since $Tb =\sum _l Tb_l$, we have

$$\begin{aligned} \int _{\Omega {\setminus } \bigcup _l B_l^*} |Tb|\,dx \le \sum _l \int _{\Omega {\setminus } B_l^*} |T b_l|\,dx. \end{aligned}$$

By the hypothesis (4.1) together with (4.4) and (4.7), we get

$$\begin{aligned} \int _{\Omega {\setminus } \bigcup _{l} B_l^*} |Tb|\,dx \le C \sum _l \int _{Q_l} |b_l|\,dx \lesssim t \sum _l |Q_l| \lesssim \int _{\Omega } |f|\,dx, \end{aligned}$$

which via Chebyshev’s inequality shows that

$$\begin{aligned} |\{x \in \Omega : |Tb(x)| > \tfrac{1}{2} t\} {\setminus } \textstyle \bigcup _l B_l^*\,| \lesssim \frac{1}{t} \int _{\Omega } |f|\,dx. \end{aligned}$$

Also, by (4.3), we have

$$\begin{aligned} |\textstyle \bigcup _l B_l^*| \le c^n \sum _l\, |B_l|\ \lesssim \sum _l \,|Q_l| \lesssim \frac{1}{t} \int _{\Omega } |f|\,dx. \end{aligned}$$

Together then, the last two estimates imply

$$\begin{aligned} |\{x \in \Omega : |Tb(x)| > \tfrac{1}{2} t\}| \lesssim \frac{1}{t} \int _\Omega |f|\,dx, \end{aligned}$$

which combined with (4.6) gives (4.2) since $Tf=Tg+Tb$. $\square$

Finally we prove the following Harnack type inequality for nonnegative adjoint solutions.

Lemma 4.2

Assume the coefficients $\mathbf {A}=(a^{ij})$ are of Dini mean oscillation and satisfy the condition (1.7). Let $w\in L^2(B_4)$ be a nonnegative solution to $D_{ij}(a^{ij} w)=0$ in $B_4=B(0,4)$ and $||w||_{L^1(B_3)}=1$. Then we have

$$\begin{aligned} c\le \inf _{B_1}\,w,\quad \sup _{B_1}\,w \le C, \end{aligned}$$

where c and C are positive constants depending only on n, $\lambda$, $\Lambda$, and $\omega _{\mathbf {A}}$.

Proof

The upper bound follows with the same type of reasoning as in the proof of Lemma 3.1, because from [7, (2.25)], we have

$$\begin{aligned} |w(x)-w(y_0)| \le C \left( \left( \frac{|x-y_0|}{R}\right) ^\beta + \int _0^{|x-y_0|} \frac{\tilde{\omega }_{\mathbf {A}}(t)}{t} \,dt \right) R^{-d} ||w||_{L^1(B(y_0,R))} \end{aligned}$$

(4.8)

for $x \in B(y_0, \frac{1}{2} R)$, $y_0\in B_1$ and $R\in (0,1]$. Here $\beta >0$ is an absolute constant and $\tilde{\omega }_{\mathbf {A}}$ is defined as in (2.20).

We prove the lower bound by contradiction. Suppose the claim is not true. Then we can find a sequence of coefficients $\{\mathbf {A}_k\}$ satisfying

$$\begin{aligned} \sup _k \omega _{\mathbf {A}_k}(t) \le \omega (t) \end{aligned}$$

for some Dini function $\omega$ and a sequence of nonnegative solutions $\{w_k\}$ to

$$\begin{aligned} D_{ij}(a^{ij}_k w_k)=0\quad \text {in}\,\, B_4 \end{aligned}$$

such that

$$\begin{aligned} ||w_k||_{L^1(B_3)}=1\quad \text {and}\quad w_k(x_k) \le 1/k \end{aligned}$$

for some $x_k\in B_1$. After passing to a subsequence, we may assume that $x_k\rightarrow y_0 \in \bar{B}_1$. By [7, Theorem 1.10], $\{w_k\}$ is uniformly bounded and equicontinuous in $\bar{B}_2$. Of course, $\{\mathbf {A}_k\}$ is also uniformly bounded and equicontinuous in $\bar{B}_2$. Therefore, by the Arzelà–Ascoli theorem, they have subsequences, still denoted by $\{w_k\}$ and $\{\mathbf {A}_k\}$, which converge to w and $\mathbf {A}$ uniformly in $\bar{B}_2$, with the same moduli of continuity. It is easily seen that w is a nonnegative solution of

$$\begin{aligned} D_{ij}(a^{ij}w)=0 \quad \text {in}\quad B_2 \end{aligned}$$

and $w(y_0)=0$. By the doubling property of w (see [11]), $||w_k||_{L^1(B_2)}$ is bounded from below and above uniformly. It then follow from the uniform convergence that $||w||_{L^1(B_2)}$ is also bounded from below and above.

Let $\kappa \in (0,1/2)$ be a small constant to be specified later. From (4.8), for any $R\in (0,1]$, we have

$$\begin{aligned} \fint _{B(y_0, \kappa R)}w\le N\left( \kappa ^\beta +\int _0^{\kappa R} \frac{\tilde{\omega }_{\mathbf {A}}(t)}{t} \,dt \right) \fint _{B(y_0,R)}w, \end{aligned}$$

where N is independent of $\kappa$. We then fix $\kappa$ sufficiently small such that $2N\kappa ^\beta \le \kappa ^{\beta /2}$. Then for any small R such that

$$\begin{aligned} \int _0^{\kappa R}\frac{\tilde{\omega }_{\mathbf {A}}(t)}{t} \,dt\le \kappa ^{\beta }, \end{aligned}$$

we obtain

$$\begin{aligned} \fint _{B(y_0, \kappa R)}w\le \kappa ^{\beta /2} \fint _{B(y_0, R)}w. \end{aligned}$$

By iteration, we deduce $\fint _{B(y_0, r)} w\le Nr^{\beta /2}$. This, however, contradicts with [9, Theorem 1.5], which reads that for any $\varepsilon >0$, it holds that $\fint _{B(y_0, r)}w \gtrsim r^{\varepsilon }$ for all $r\in (0,1)$. $\square$