A note on the Harnack inequality for elliptic equations in divergence form

TL;DR

This paper highlights that the Harnack inequality, a fundamental result for elliptic equations, was implicitly contained in De Giorgi's 1957 work, clarifying its historical development.

Contribution

It reveals that the Harnack inequality was implicitly present in De Giorgi's original proof, providing historical insight into elliptic equation theory.

Findings

Harnack inequality was hidden in De Giorgi's 1957 work

Clarifies the historical development of elliptic regularity theory

Connects De Giorgi's work with Moser's results

Abstract

In 1957, De Giorgi [3] proved the H\"{o}lder continuity for elliptic equations in divergence form and Moser [7] gave a new proof in 1960. Next year, Moser [8] obtained the Harnack inequality. In this note, we point out that the Harnack inequality was hidden in [3].

Equations

$$(a^{ij}{u}_i{)}_j=0\text{mbox}in{Q}_6,$$

$$\underset{Q_1}{\sup }u\le C\underset{Q_1}{\inf }u,$$

$$\Vert u{\Vert }_{L^{\infty }(Q_1)}\le C\Vert u{\Vert }_{L^2(Q_3)},$$

$$m\{x\in Q_1:u(x)>c\}>c_0\Rightarrow u>1\text{mbox}in{Q}_3,$$

$$\Vert u{\Vert }_{L^{\infty }(Q_1)}\le C\Vert u{\Vert }_{L^{p_0}(Q_3)},$$

$$\Vert u{\Vert }_{L^{\infty }(Q_1)}\le \epsilon \Vert u{\Vert }_{L^{\infty }(Q_3)}+C(\epsilon )\Vert u{\Vert }_{L^{p_0}(Q_3)}$$

$$r^{\frac{n}{p_0}}\Vert u{\Vert }_{L^{\infty }(Q_r(x_0))}\le \epsilon r^{\frac{n}{p_0}}\Vert u{\Vert }_{L^{\infty }(Q_{3r}(x_0))}+C(\epsilon )\Vert u{\Vert }_{L^{p_0}(Q_{3r}(x_0))}$$

$$d_{x_0}^{\frac{n}{p_0}}|u(x_0)|$$

$$\le \epsilon C(n)r^{\frac{n}{p_0}}\Vert u{\Vert }_{L^{\infty }(Q_{3r}(x_0))}+C(\epsilon )\Vert u{\Vert }_{L^{p_0}(Q_{3r}(x_0))}(\text{mbox}By\text{eqref}e2.2)$$

$$\le \epsilon C(n)\underset{x\in Q_3}{\sup }{d}_x^{\frac{n}{p_0}}|u(x)|+C(\epsilon )\Vert u{\Vert }_{L^{p_0}(Q_3)}.$$

$$\underset{x\in Q_3}{\sup }{d}_x^{\frac{n}{p_0}}|u(x)|\le C\Vert u{\Vert }_{L^{p_0}(Q_3)},$$

$$\Vert u{\Vert }_{L^p(Q_1)}\le C\underset{Q_3}{\inf }u,$$

$$m\{x\in Q_1:u(x)>c^k\}\le \frac{1}{2^k},$$

$$A:=\{x\in Q_1:u(x)>c^{k_0}\}\text{mbox}andB:=\{x\in Q_1:u(x)>c^{k_0-1}\}.$$

$$m(A\cap Q_r(x_0))>\frac{1}{2}m(Q_r(x_0))\Rightarrow Q_{3r}(x_0)\cap Q_1\subset B,$$

Full text

This research is supported by NSFC 11171266.

1. The Harnack inequality

Consider the following elliptic equation:

[TABLE]

where $(a^{ij})_{n\times n}$ is uniformly elliptic with ellipticity constants $\lambda$ and $\Lambda$. In this note, $Q_{r}(x_{0})$ denotes the cube with center $x_{0}$ and side-length $r$, and $Q_{r}:=Q_{r}(0)$.

In 1961, Moser [8] obtained the following Harnack inequality:

Theorem 1.1**.**

Let $u\geq 0$ be a weak solution of (1.1). Then

[TABLE]

where $C$ depends only on $n$, $\lambda$ and $\Lambda$.

The method used in [8] is that first to estimate the upper and lower bound of $u$ in terms of $||u||_{L^{p_{0}}}$ and $||u^{-1}||_{L^{p_{0}}}$ respectively by an iteration for some $p_{0}>0$ and then to join these two estimates together to obtain (1.2) by the John-Nirenberg inequality.

In 1957, De Giorgi [3] proved the Hölder continuity for weak solutions of (1.1) and Moser [7] gave a new proof later. The following are two of the main results in [3] and [7] (see also [1]):

Theorem 1.2**.**

Let $u\geq 0$ be a weak subsolution of (1.1). Then

[TABLE]

where $C$ depends only on $n$, $\lambda$ and $\Lambda$.

Theorem 1.3**.**

Let $u\geq 0$ be a weak supsolution of (1.1). Then for any $c_{0}>0$, there exists a constant $c$ depending only on $n$, $\lambda$, $\Lambda$ and $c_{0}$ such that

[TABLE]

where $m$ denotes the Lebesgue measure.

In this note, we will prove Theorem 1.1 by the above two theorems directly. That is, De Giorgi’s proof implies Harnack inequality. This was first noticed by DiBenedetto [4]. Some other new approaches to Harnack inequality can be founded in [5] and [6], where U. Gianazza and V. Vespri [6] requires only a qualitative boundedness of solutions, which is different from here.

2. Proof of Theorem1.1

In the following, we present the key points for obtaining Theorem 1.1 by Theorem 1.2 and 1.3. First, we show that Theorem 1.2 implies the following local maximum principle:

Lemma 2.1**.**

Let $u\geq 0$ be a weak subsolution of (1.1) and $p_{0}>0$. Then

[TABLE]

where $C$ depends only on $n$, $\lambda$, $\Lambda$ and $p_{0}$.

Proof.

By the interpolation for $L^{p}$ functions and (1.3), for any $\varepsilon>0$, we have

[TABLE]

whose scaling version is

[TABLE]

for any $Q_{6r}(x_{0})\subset Q_{6}$.

Given $x_{0}\in Q_{3}$, denotes by $d_{x_{0}}$ the distance between $x_{0}$ and $\partial Q_{3}$. Then the cube $Q_{6r}(x_{0})\subset Q_{3}$ where $r=d_{x_{0}}/3\sqrt{n}$, and we have

[TABLE]

Take the supremum over $Q_{3}$ and $\varepsilon$ small such that $\varepsilon C(n)<1/2$. Then,

[TABLE]

which implies (2.1). ∎

Next, we show that Theorem 1.3 implies the weak Harnack inequality:

Lemma 2.2**.**

Let $u\geq 0$ be a weak supsolution of (1.1). Then

[TABLE]

where $p>0$ and $C$ depend only on $n$, $\lambda$ and $\Lambda$.

Proof.

Without loss of generality, we assume that $\inf_{Q_{3}}u=1$ and we only need to prove that there exists a constant $c$ depending only on $n$, $\lambda$ and $\Lambda$ such that

[TABLE]

for $k=1,2,...$.

We prove (2.4) by induction. Take $c_{0}=1/2$ in Theorem 1.3. Then there exists a constant $c$ depending only on $n$, $\lambda$ and $\Lambda$ such that (1.4) holds. Hence, (2.4) holds for $k=1$ since we assume that $\inf_{Q_{3}}u=1$. Suppose that (2.4) holds for $k\leq k_{0}-1$. Let

[TABLE]

We need to prove $m(A)\leq m(B)/2$. By the Calderón-Zygmund cube decomposition (see [2, Lemma 4.2]), we only need to prove that for any $Q_{r}(x_{0})\subset Q_{1}$,

[TABLE]

which is exactly the scaling version of (1.4) for $v=u/c^{k_{0}-1}$. ∎

Now, Theorem 1.1 follows clearly by combining (2.1) and (2.3).

Remark 2.3*.*

“$u\geq 0$” can be removed in Lemma 2.1 and a corresponding estimate for $u^{+}$ holds. As for elliptic equations in non-divergence form, we also have the local maximum principle (Lemma 2.1) and the weak Harnack inequality (Lemma 2.2) respectively (see [2, Theorem 4.8]). In fact, this note is inspired by [2].